Algebra I

The big picture

If the word algebra has ever made your stomach drop, you are in the right place, and you are going to be okay. Here is a secret almost no one tells you: you already do algebra in your head all the time. Every time you work out the change from a 20 dollar bill, double a recipe, or split a bill three ways, that is algebra. You are not starting from nothing. You are just going to learn to write down something you already know how to do.

We will go slowly, one small step at a time. Nothing skipped, nothing rushed, and no question is too small. In this first lesson we meet the two most basic pieces of that written-down math: letters that stand for numbers, and short recipes built from those letters.

Key idea: Algebra is a way to write down thinking you already do with numbers.

What is a variable?

Picture a small box with a closed lid. Inside is a number, but we cannot see it yet. To talk about that hidden number, we give the box a name, usually a letter like x. That letter has a fancy name, a variable, but do not let the word worry you. A variable is just the name of a box with a number hiding inside. Read that once more if you like: a variable is the name of a box with a number inside.

We use letters like x, y, or n so we can write one rule that works for many numbers at once. A number that is out in the open, like 7 or -3, is called a constant, because its value never changes. A box can hold different numbers on different days; the number 7 is always just 7.

Here is why this is worth the trouble. Instead of solving a hundred look-alike arithmetic problems one by one, we write a single expression with a box in it and handle all hundred at once. That is the whole power of algebra, and it grows out of this one small idea of naming a hidden number.

Key idea: A variable is just a letter standing in for a number we do not know yet.

What is an expression?

An algebraic expression is a little recipe made from numbers, variables, and operations like add, subtract, multiply, and divide. Here are three: 3x + 5, and 2n - 7, and 4(a + b). An expression has no equals sign. It is a recipe, not a question. We will meet equations, which do have an equals sign, in a later lesson.

One thing surprises almost everyone at first, so let us go slowly. In algebra, 3x means "3 times x." We usually leave out the times sign between a number and a letter. Why? Because the usual times sign, ×, looks far too much like the letter x, and that would get confusing fast. So a number sitting right next to a letter, with no sign between them, always means multiply. Read that once more: 3x means 3 times x.

When two letters sit side by side, that means multiply too. So ab means "a times b."

Key idea: A number next to a letter, like 3x, means multiply: 3 times x.

The parts of an expression

Let us learn to read an expression the way you read a sentence, one piece at a time. A term is a single number, a single letter, or numbers and letters multiplied together. The plus and minus signs are the fences between terms.

Look at 3x + 5. There is a plus sign in the middle, so there are two terms: 3x is one term, and 5 is the other.

• The number multiplied by the letter is the coefficient. In 3x, the coefficient is 3.
• A term that is just a number by itself, like 5, is a constant term.

One quiet trap catches many people, so let us name it now. When a letter appears with no number in front, like a lonely x, its coefficient is 1, not 0. That is because x secretly means 1x. In the same way, -x means -1x, so its coefficient is -1. Nice work noticing that; it is the kind of small detail that saves you later.

Key idea: The coefficient is the number in front of the letter, and a bare x has a coefficient of 1.

Turning words into algebra

Word problems hide the operations inside ordinary English. These clue words point you to the right operation:

• Add: sum, plus, more than, increased by, total, combined.
• Subtract: difference, minus, less than, decreased by, fewer, reduced by.
• Multiply: product, times, of, twice (times 2), doubled, tripled.
• Divide: quotient, divided by, per, ratio, split equally.

Here is the spot where people most often get stuck, so let us slow all the way down. The phrase "5 less than a number" is written x - 5, not 5 - x. The order flips. Think about it in plain words: you start with the number, then you take 5 away from it. You begin with the box, then remove 5. The same flip happens with "subtracted from": "3 subtracted from y" means you start with y and remove 3, so it is y - 3. When you meet "less than" or "subtracted from," pause and ask, "What do I start with?" That one question prevents most mistakes here.

Key idea: "5 less than x" means x - 5, because you start with x and take 5 away.

Let us write one together

Phrase: "Seven more than twice a number." We will build it in tiny pieces.

1. "A number" is our box. Call it x.
2. "Twice a number" means 2 times x, which we write 2x.
3. "Seven more than that" means add 7 to it.
4. Put it together: 2x + 7.

See? You just translated English into algebra. Let us try a slightly bigger one.

Phrase: "The product of 4 and the quantity three less than a number."

1. "Three less than a number" starts with the number and takes 3 away: (x - 3).
2. "The product of 4 and that quantity" means 4 times the whole thing.
3. Keep the parentheses so the 4 multiplies the entire difference: 4(x - 3).

Those parentheses matter. They say the 4 multiplies both the x and the -3, not just the x. Hold onto that thought; it comes back in the next lesson.

Key idea: Build a phrase one small piece at a time, and keep parentheses around a group.

Let us evaluate one together, slowly

To evaluate an expression means to find its value by putting a number in place of the letter. We will do 2x + 1 when x = 5.

1. Remember that 2x means "2 times x." So we really have (2 times x) + 1.
2. We were told x is 5. So everywhere we see x, we gently put a 5 in its place: (2 times 5) + 1.
3. Do the multiply first. 2 times 5 is 10. Now we have 10 + 1.
4. Do the add. 10 + 1 is 11.

That is the answer: 11. Take a second to notice what just happened. You took 2x + 1, put in a number, and got a real answer. That is the whole skill. You just did algebra.

Key idea: To evaluate, put the number in for the letter, then do the arithmetic one step at a time.

Evaluating with a negative number

Negatives are where sign mistakes sneak in, so we will go extra slowly and lean on one habit: always wrap a negative number in parentheses when you put it in. Let us evaluate 3x² - 2x + 5 when x = -2.

1. Write the expression with an empty seat for x: 3( )² - 2( ) + 5.
2. Put -2 into each seat, keeping the parentheses: 3(-2)² - 2(-2) + 5.
3. Do the exponent first. (-2)² means (-2) times (-2), which is 4. Now: 3(4) - 2(-2) + 5.
4. Do the first multiply: 3 times 4 is 12. Now: 12 - 2(-2) + 5.
5. Do the next multiply: -2 times -2 is +4, because a negative times a negative is positive. Now: 12 + 4 + 5.
6. Add left to right: 12 + 4 is 16, then 16 + 5 is 21.

The value is 21. Here is where people get stuck: if you write -2² without the parentheses, many people square only the 2 and get -4, which is the wrong sign. The parentheses keep the negative attached to the number. When in doubt, wrap the negative in parentheses. That habit alone fixes a huge share of sign errors.

Key idea: Always put a negative number in parentheses before you square it or multiply it.

A recipe from real life

A phone plan charges 20 dollars a month plus 5 dollars for each gigabyte of data you use. If g is the number of gigabytes, the monthly cost is 20 + 5g. Let us find the cost for a month with 3 gigabytes.

1. Start with the recipe: 20 + 5g.
2. Put 3 in for g: 20 + 5(3).
3. Do the multiply first: 5 times 3 is 15. Now: 20 + 15.
4. Do the add: 20 + 15 is 35.

So the cost is 35 dollars. Notice that this one small recipe handles every possible month. Change g and you get that month's bill. That is exactly why algebra is worth learning.

Two boxes at once

Sometimes a recipe has two different letters. That is fine; you just fill two seats instead of one. Evaluate 2ab - 3b + 4 when a = 3 and b = -2.

1. Put in both numbers, each in parentheses: 2(3)(-2) - 3(-2) + 4.
2. Handle the first term left to right: 2 times 3 is 6.
3. Continue that term: 6 times -2 is -12. Now: -12 - 3(-2) + 4.
4. Next term: -3 times -2 is +6 (negative times negative is positive). Now: -12 + 6 + 4.
5. Add left to right: -12 + 6 is -6, then -6 + 4 is -2.

The value is -2. Nothing new happened here. You filled two seats instead of one and followed the same careful steps. Science formulas work exactly this way, with several inputs feeding one recipe.

Key idea: With two letters, fill both seats, then do the arithmetic one small step at a time.

Where you will see this

Expressions are the quiet workhorses of everyday calculation:

• Temperature: the recipe (9/5)C + 32 turns Celsius into Fahrenheit. At C = 25: (9/5)(25) is 45, and 45 + 32 is 77 degrees Fahrenheit.
• A fundraiser: a club selling shirts for 12 dollars each after spending 150 dollars on supplies earns 12n - 150. Selling 40 shirts gives 12(40) - 150, which is 480 - 150, which is 330 dollars.
• Spreadsheets: a formula like =B2*12-150 is really the expression 12n - 150, with a cell playing the part of the box.

Common misconceptions

• Thinking 2x means 2 + x. A number next to a letter means multiply. When x = 7, 2x is 14, not 9.
• Writing "5 less than x" as 5 - x. It is x - 5. You start with x and take 5 away.
• Squaring a negative without parentheses. For x = -2, the term x² must become (-2)² = 4. Writing -2² gives -4, the wrong sign.
• Thinking a lonely x has coefficient 0. A bare x means 1x, so its coefficient is 1. And -x has coefficient -1.
• Mixing up x² and 2x. x² means x times x; 2x means x plus x. At x = 3 they are 9 and 6, which are not the same.

Try it

Evaluate 5a - 4 when a = 3, and write "the sum of a number and 9" as an expression. Take your time.

Answer: Put 3 in for a: 5(3) - 4. Multiply first: 15 - 4. Subtract: 11. The phrase becomes x + 9. If you got those, you are ready for the next lesson. If not, that is fine too; read the two worked examples once more and try again.

Recap

• A variable is a letter standing for a number we do not know yet; a constant is a fixed number.
• A number next to a letter means multiply, so 3x is 3 times x.
• Terms are separated by + and - signs, and the coefficient is the number in front of the letter.
• Read "less than" and "subtracted from" carefully, because they flip the order.
• To evaluate, put the number in for the letter, wrap negatives in parentheses, and do one step at a time.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 1: Foundations (openstax.org ↗).
• Khan Academy, Algebra 1, "Algebra foundations" unit: introduction to variables and evaluating expressions (khanacademy.org ↗).
• Paul Dawkins, Paul's Online Math Notes, Algebra Preliminaries (tutorial.math.lamar.edu ↗).
• Math is Fun, "Introduction to Algebra" (mathsisfun.com ↗).

The big picture

This lesson is about tidying up. Before you ever solve anything, you learn to clean an expression until it is short and simple, the way you might straighten a messy desk before starting work. Two gentle skills do all the tidying: doing operations in an agreed order, and gathering pieces that belong together.

You already sense the order in real life. You would tally the cost of three of the same item before adding a separate item; you would work out a group in parentheses before folding it into the rest. Here we just make that instinct into a clear, reliable rule.

Key idea: Simplifying is tidying an expression into its shortest, clearest form.

Why we need an agreed order

Look at 2 + 3 × 4. Read that as "two plus three times four." If you add first you get 20. If you multiply first you get 14. Those are different answers, and that is a problem. So long ago, mathematicians agreed on one order everyone would follow, so every person on Earth gets the same result. A common way to remember it is PEMDAS:

1. Parentheses (and other grouping symbols) first.
2. Exponents next.
3. Multiply and Divide, working left to right (they share one level).
4. Add and Subtract, working left to right (they share one level).

So 2 + 3 × 4 becomes 2 + 12, which is 14. Multiplication happens before addition.

Here is where people get stuck, so let us slow down. The M and D share a level, and the A and S share a level. That word "share" matters. Take 12 ÷ 4 × 3. You do not always multiply before dividing. You go left to right instead: 12 ÷ 4 is 3, then 3 × 3 is 9. Read that once more: within the same level, work left to right, like reading a sentence.

Key idea: PEMDAS sets the order, and within a shared level you always work left to right.

Let us do one together

Simplify 4 + 2(5 - 1)² ÷ 8. We will take one tiny step per line and never rush two steps at once.

1. Parentheses first: 5 - 1 is 4. Now we have 4 + 2(4)² ÷ 8.
2. Exponent next: 4² means 4 times 4, which is 16. Now: 4 + 2(16) ÷ 8.
3. Now multiply and divide, left to right. First the multiply: 2 times 16 is 32. Now: 4 + 32 ÷ 8.
4. Still on that level, do the divide: 32 ÷ 8 is 4. Now: 4 + 4.
5. Finally add: 4 + 4 is 8.

The answer is 8. Notice that each line resolved exactly one thing. Writing every line out fully, instead of doing two steps in your head, is the habit that keeps long problems from going wrong. Nicely done; that was the hard kind.

Key idea: Do one operation per line, in PEMDAS order, and the answer takes care of itself.

The distributive property

Now the second tidying tool. The distributive property is the rule a(b + c) = ab + ac. In plain words: a number sitting outside a set of parentheses gets multiplied by every term inside. This is how we remove parentheses in algebra, and it is one of the most used moves in the whole subject.

A picture helps. Imagine a rectangle that is a tall and (b + c) wide. Its area is a(b + c). But you could also slice it into two rectangles, one a-by-b and one a-by-c, with areas ab and ac. Same rectangle, so a(b + c) = ab + ac. The outside number simply visits each inside piece.

Let us do one with a negative, going slowly. Simplify -2(3x - 4).

1. Send the -2 to the first term: -2 times 3x is -6x.
2. Send the -2 to the second term: -2 times -4 is +8, because a negative times a negative is positive.
3. Put the pieces together: -6x + 8.

Here is where people slip: they multiply the outside number by the first term and forget the second. A good habit is to draw a little arrow from the outside number to each inside term, so you never leave one out.

Key idea: The distributive property sends the outside number to every term inside the parentheses.

Like terms

Like terms are terms with exactly the same letter part. Think of them as the same kind of object. 4x and 7x are like terms; they are both "x's." But 4x and 4x² are not like terms, because one has an exponent and one does not; they are different kinds of object. Plain numbers, like 7 and -3, are like terms with each other.

To combine like terms, you add or subtract the coefficients and keep the letter part the same. A friendly way to picture it: 4x + 7x is like 4 apples plus 7 apples, which is 11 apples, so 4x + 7x = 11x. You are really just using the distributive property in reverse: 4x + 7x = (4 + 7)x = 11x.

Key idea: Combine like terms by adding their coefficients and keeping the same letter part.

Let us tidy an expression together

Simplify 4x + 7 - 2x - 3.

1. Find the x-terms: 4x and -2x. Find the number terms: 7 and -3.
2. Combine the x-terms: 4x - 2x is 2x.
3. Combine the numbers: 7 - 3 is 4.
4. Write the tidy result: 2x + 4.

The answer is 2x + 4. Now one that needs distributing first. Simplify 3(x + 2) - x.

1. Distribute the 3: 3 times x is 3x, and 3 times 2 is 6. Now: 3x + 6 - x.
2. Combine the x-terms: 3x - x is 2x.
3. The number term 6 has nothing to combine with, so it stays.
4. Result: 2x + 6.

The answer is 2x + 6. See how distributing cleared the parentheses so the like terms could meet? That is the usual rhythm: clear parentheses, then gather.

Distribute a negative, then combine

This one trips people up, so we go extra slowly. Simplify 5 - 2(3x - 4) + x.

1. The -2 is the outside number for the parentheses. Send it to 3x: -2 times 3x is -6x.
2. Send the -2 to -4: -2 times -4 is +8.
3. Rewrite the whole line with the parentheses gone: 5 - 6x + 8 + x.
4. Gather the x-terms: -6x + x is -5x.
5. Gather the numbers: 5 + 8 is 13.
6. Result: -5x + 13.

The answer is -5x + 13. The whole game was tracking the sign on every piece, one small step at a time. That care is exactly what separates a right answer from a near miss, and you just did it.

Key idea: Distribute a negative to every inside term, watching each sign, before you combine.

Everything at once

Let us do a full PEMDAS example so every level shows up. Simplify 18 ÷ 3 + 2(8 - 5)².

1. Parentheses: 8 - 5 is 3. Now: 18 ÷ 3 + 2(3)².
2. Exponent: 3² is 9. Now: 18 ÷ 3 + 2(9).
3. Multiply and divide, left to right. Divide first: 18 ÷ 3 is 6. Now: 6 + 2(9).
4. Still that level, multiply: 2 times 9 is 18. Now: 6 + 18.
5. Add: 6 + 18 is 24.

The answer is 24. Every step handled exactly one level of the ladder.

A quick way to check your work

Here is a comforting trick. When you simplify an expression with x in it, you can check by putting in an easy number. Simplify 2(3x + 1) + 4(x - 2) first:

1. Distribute the 2: 6x + 2.
2. Distribute the 4: 4x - 8.
3. Put it together: 6x + 2 + 4x - 8.
4. Combine x-terms: 6x + 4x is 10x. Combine numbers: 2 - 8 is -6.
5. Result: 10x - 6.

Now the check. Try x = 1. The original: 2(4) + 4(-1) is 8 - 4, which is 4. The tidy form: 10(1) - 6 is 4. They match. When both forms give the same number, that is strong evidence you tidied correctly.

Key idea: To check a simplification, plug an easy number into both the original and your answer; they should agree.

Where you will see this

• A shopping total: 3 shirts at 12 dollars and 2 hats at 9 dollars costs 3(12) + 2(9), which is 36 + 18, which is 54 dollars. Multiplying before adding is exactly what a cash register does.
• Sales tax: a 40 dollar item with 8 percent tax costs 40(1 + 0.08), which is 40(1.08), which is 43.20 dollars; the parentheses group the rate before multiplying.
• Spreadsheets and code: every programming language reads 2 + 3 * 4 as 14. People add parentheses to force a different order, just like on paper.

Common misconceptions

• Always multiplying before dividing. They share a level and go left to right. 12 ÷ 2 × 3 is 18, not 2.
• Distributing to the first term only. In -2(3x - 4), the -2 must reach both terms: -6x + 8, not -6x - 4.
• Combining unlike terms. 4x + 3 cannot become 7x. An x and a plain number are different kinds of thing.
• Treating x and x² as like terms. 4x + 4x² does not combine, because the letter parts differ.
• Forgetting parentheses around a squared negative. (-3)² is 9, but -3² is -9. The parentheses decide whether the sign gets squared.

Try it

Simplify 5 - 2(x - 3). Go slowly: distribute the -2 first.

Answer: Distribute: -2 times x is -2x, and -2 times -3 is +6, giving 5 - 2x + 6. Combine the numbers: 5 + 6 is 11. Result: -2x + 11 (you may also write it 11 - 2x). If the +6 surprised you, remember: negative times negative is positive.

Recap

• PEMDAS sets the order: grouping, exponents, multiply/divide left to right, add/subtract left to right.
• The distributive property a(b + c) = ab + ac sends the outside number to every inside term.
• Like terms have the same letter part; combine them by adding coefficients.
• Check a simplification by putting an easy number into both forms.
• Tidy each side fully before you try to solve anything.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 1: Foundations, sections on the language of algebra and properties of real numbers (openstax.org ↗).
• Khan Academy, Algebra 1, "Algebra foundations" unit: order of operations and combining like terms (khanacademy.org ↗).
• Paul Dawkins, Paul's Online Math Notes, Algebra Preliminaries (tutorial.math.lamar.edu ↗).
• Math is Fun, "Order of Operations - PEMDAS" and "Like Terms" (mathsisfun.com ↗).

The big picture

Numbers come in a few families, and once you can name them, it is much easier to see how they fit together. In this lesson we meet those families and work with negative numbers. We will use everyday pictures the whole way.

You already use negatives in real life. A temperature below zero, money you owe, a floor below ground level. You are putting clear words to things you already understand.

Key idea: Numbers come in a few nested families, and negatives describe everyday "below" and "owed" ideas.

Families of numbers

Each family is built by adding one new idea to the family before it. Picture a set of measuring cups that nest inside each other:

• Natural numbers: 1, 2, 3, and so on. These are the counting numbers, the ones you learned first.
• Whole numbers: the naturals plus 0.
• Integers: the whole numbers together with their negatives: ... -2, -1, 0, 1, 2, ...
• Rational numbers: any number you can write as a fraction of two integers, like 1/2, or -3, or 0.25. Their decimals either stop or repeat a pattern.
• Irrational numbers: numbers whose decimals go on forever without ever repeating, like the square root of 2, or the number pi.

Put the rationals and the irrationals together and you get the real numbers, which are all the points on the number line. The families nest: every natural number is also a whole number, an integer, a rational, and a real. So a number can belong to several families at once. The handy question to ask is, "What is the smallest family this number lives in?"

Key idea: The families nest inside one another, and the reals are every point on the number line.

The number line and distance from zero

Every real number has a home on the number line. Numbers to the right are bigger; numbers to the left are smaller. Read that once more, because it is the key to negatives: further left means smaller.

The absolute value of a number is simply its distance from zero, and distance is never negative. We write it with two bars, like this: |-4| = 4, said "the absolute value of negative four is four." And |4| = 4 too. Distance does not care which direction you walked, only how far. That is why absolute value throws away the sign.

Key idea: Absolute value is distance from zero, so it is never negative.

The four sign rules, one at a time

Signed numbers feel like a lot of rules, but there are really just four, and each has a friendly picture. Let us meet them slowly.

Rule 1, adding two of the same sign. Add the sizes and keep the sign. Think of it as owing money twice. If you owe 4 and then owe 3 more, you owe 7: -4 + (-3) = -7.

Rule 2, adding two different signs. Subtract the smaller size from the larger, and keep the sign of the bigger one. Picture steps on the number line. -8 + 3 means start at -8 and take 3 steps to the right, landing at -5. So -8 + 3 = -5.

Rule 3, multiplying or dividing. Same signs give a positive; different signs give a negative. So (-6)(-2) = 12, but (-6)(2) = -12.

Rule 4, subtracting. Subtracting is the same as adding the opposite. So 5 - (-3) becomes 5 + 3, which is 8. Taking away a debt is like receiving money.

Why do two negatives multiply to a positive? Here is a gentle way to feel it: multiplying by a negative flips your direction on the number line. Flip once, you face the other way. Flip twice, and you are back facing the original way. Two flips cancel, so two negatives give a positive.

Key idea: Same signs add and keep the sign; different signs subtract; subtracting means adding the opposite; two negatives multiply to a positive.

Let us do a chain together

Compute -3 - (-7) + (-2)(4). We will handle one small piece at a time.

1. Do the multiply first: (-2)(4) is -8. Now: -3 - (-7) + (-8).
2. Change the "subtract a negative" into "add": -3 - (-7) becomes -3 + 7. Now: -3 + 7 + (-8).
3. Add left to right: -3 + 7 is 4. Now: 4 + (-8).
4. Finish: 4 + (-8) is -4.

The answer is -4. Nice work; a chain of signs is just one link at a time.

Fractions with signs

Compute -3/4 + 1/6. Fractions with different bottoms need a common bottom first, so we go slowly.

1. Find a common denominator for 4 and 6. The smallest one is 12.
2. Rename the first fraction: -3/4 is the same as -9/12 (top and bottom times 3).
3. Rename the second fraction: 1/6 is the same as 2/12 (top and bottom times 2).
4. Now the bottoms match, so add the tops: -9 + 2 is -7, over 12.

The answer is -7/12. Here is where people get stuck: you cannot add fractions with different bottoms by adding tops. Rename first, always, so the bottoms match.

Key idea: To add fractions, give them the same bottom first, then add the tops.

Naming the family of a number

Let us name the smallest family for each of these: -7, 0, 3/5, the square root of 9, the square root of 7, and 0.454545... (repeating).

1. -7 is negative, so it cannot be natural or whole. It is an integer (and also rational and real).
2. 0 is a whole number, since the whole numbers are the naturals plus 0.
3. 3/5 is a fraction of integers, so it is rational.
4. The square root of 9 is 3 in disguise, so it is really a natural number. Always simplify before you name the family.
5. The square root of 7 is not a whole number, and its decimal never stops or repeats, so it is irrational.
6. 0.454545... repeats the block 45 forever, so it is rational (it equals 45/99, which reduces to 5/11).

Key idea: Simplify a number first, then find the smallest family it belongs to.

Ordering signed numbers

Order these from least to greatest: -3/2, -2, 0.75, -0.5, 1. Turning fractions into decimals makes comparing easier, so note -3/2 is -1.5.

1. Place them on the number line in your mind. Further left is smaller.
2. The most negative is -2, then -1.5, then -0.5, then 0.75, then 1.
3. Written out: -2 < -1.5 < -0.5 < 0.75 < 1.

Here is where people get stuck: it is tempting to say -2 is bigger than -1.5 because 2 is bigger than 1.5. But with negatives, the bigger the size, the smaller the value, because you are further left. Read that once more, slowly.

Key idea: For negatives, a bigger size means a smaller value, because it sits further left.

Where you will see this

• Temperature: if it climbs from -8 degrees to 5 degrees, the change is 5 - (-8), which is 5 + 8, which is 13 degrees.
• Money: a balance of 40 dollars after a 55 dollar purchase becomes 40 - 55, which is -15 dollars, an overdraft the negative sign makes clear.
• Elevation: Death Valley sits at about -282 feet and a mountain summit at about 20,310 feet; the vertical difference is 20310 - (-282), which is 20592 feet.
• Measurement error: a rod cut 0.2 mm too long or 0.2 mm too short is off by |0.2| either way; absolute value measures the size of the error without caring about direction.

Common misconceptions

• Thinking -x is always negative. The expression -x means "the opposite of x." If x = -3, then -x = 3, a positive number.
• Using the multiply rule on addition. Two negatives multiply to a positive, but -3 + (-4) = -7; adding two negatives goes further negative.
• Thinking every never-ending decimal is irrational. A repeating decimal like 0.333... is rational; only never-ending and never-repeating decimals are irrational.
• Treating 22/7 as pi. 22/7 is a fraction, so it is rational; pi is irrational and equals no fraction exactly.
• Ranking negatives by size. -100 is less than -1, even though 100 is bigger than 1; position on the number line decides order.

Try it

Compute -7 - (-10), and say whether -2/3 is rational or irrational. Take your time.

Answer: -7 - (-10) becomes -7 + 10, which is 3. Since -2/3 is a fraction of integers, it is rational. If the first one tripped you, remember Rule 4: subtracting a negative turns into adding.

Recap

• The families nest: naturals inside wholes inside integers inside rationals, and rationals plus irrationals make the reals.
• Rational numbers have decimals that stop or repeat; irrational numbers never do either.
• Absolute value is distance from zero and is never negative.
• Same signs add and keep the sign; different signs subtract; subtracting means adding the opposite.
• Two numbers with the same sign multiply to a positive; different signs multiply to a negative.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 1: Foundations, sections on integers, fractions, decimals, and the real numbers (openstax.org ↗).
• OpenStax, Prealgebra 2e, Chapter 3: Integers (openstax.org ↗).
• Khan Academy, Algebra 1, "Irrational numbers" unit (khanacademy.org ↗).
• Math is Fun, "The Number Line" and "Absolute Value" (mathsisfun.com ↗).

The big picture

Today we solve equations, and if that phrase makes your shoulders tense up, you are exactly who this lesson was written for. Here is the honest truth: solving an equation is a guessing game with a method. Somebody hid a number, and your job is to find it. You already do this in your head.

If a friend said "I am thinking of a number, and when I add 5 to it I get 12," you would say 7 without breaking a sweat. That is solving an equation. The written version is x + 5 = 12, and by the end of this lesson you will have a reliable routine for cracking open equations like that.

We will go one small step at a time. Nothing skipped, nothing rushed, and you cannot fall behind, because the lesson waits for you.

Key idea: Solving an equation means finding the hidden number that makes both sides equal.

What an equation really says

An equation is a math sentence with an equals sign in the middle, like x + 5 = 12. Read it out loud in plain English: "some mystery number, plus 5, equals 12." The letter x is the mystery number. The equals sign is a promise: whatever sits on the left is worth exactly the same as whatever sits on the right.

To solve the equation means to find the number x must be for the promise to hold. That winning number is called the solution. Let us test some. Try x = 7: does 7 + 5 equal 12? Yes. So 7 is the solution. Try x = 6: does 6 + 5 equal 12? No, that makes 11. So 6 is not the solution. There is nothing mystical happening: a solution is a number that passes the test. Put it in, and both sides match.

Key idea: A solution is a number that makes the equation true when you test it.

The balance scale

Now for the picture that carries you through all of algebra. Imagine an old-fashioned balance scale, the kind with two hanging pans. On the left pan sits a sealed box (that is x) plus 5 marbles. On the right pan sit 12 marbles. The scale is perfectly level, because both sides weigh the same. That level scale IS the equation x + 5 = 12.

Suppose you lift 5 marbles off the left pan only. The scale tips. To keep it level, you must take 5 marbles off the right pan too. Do that, and what is left? The box alone on the left, 7 marbles on the right, still perfectly level. The box weighs 7. You solved it.

That is the golden rule of algebra, and it is worth saying twice: whatever you do to one side, do the exact same thing to the other side. Read it once more if you like. Every equation you will ever solve, in this course and far beyond, obeys that one rule.

Key idea: An equation is a balanced scale. Same action, both sides, always.

Undoing an operation

One more small tool and we are ready. Operations come in undo pairs. Watch with real numbers. Start with 9. Add 5 and you get 14. Now subtract 5 and you are back to 9, as if nothing happened. Subtraction undoes addition. Same with the other pair: start with 4, multiply by 3 to get 12, divide by 3 and you are back to 4. Division undoes multiplication.

• Adding and subtracting undo each other.
• Multiplying and dividing undo each other.

These undo pairs are called inverse operations. The whole strategy of this lesson fits in one sentence: to get the letter alone, undo whatever was done to it, and do that undoing to both sides.

Key idea: Ask "what happened to x?" and then undo it, on both sides.

Worked example 1: x + 5 = 12

Let us solve one together, slowly.

1. Look at the x. What is being done to it? A 5 is being added.
2. The undo for adding 5 is subtracting 5. So subtract 5 from both sides: x + 5 - 5 = 12 - 5.
3. On the left, plus 5 and minus 5 cancel each other out. The x is alone now.
4. On the right, work out 12 - 5. That is 7.
5. So x = 7.

What we just did: we undid the +5 with a -5 on both sides, and the mystery number appeared.

Check it: put 7 back into the original: 7 + 5 is 12. True. That was your first solved equation of the lesson. And it worked.

Worked example 2: x - 4 = 9

1. What is being done to x? A 4 is being subtracted.
2. The undo for subtracting 4 is adding 4. Add 4 to both sides: x - 4 + 4 = 9 + 4.
3. On the left, minus 4 and plus 4 cancel. The x is alone.
4. On the right, 9 + 4 is 13.
5. So x = 13.

Check it: 13 - 4 is 9. True. Two for two.

Try it: solve x + 7 = 15. Take your time; the answer is right below.

Answer: subtract 7 from both sides: x = 15 - 7, so x = 8. Check: 8 + 7 is 15. If you got 8, you have the pattern. If not, no harm done; reread example 1 and try again. Mistakes here are practice, not failure.

When the letter is multiplied: 3x = 21

A quick memory refresher: 3x means "3 times x." So this equation says "3 times a mystery number is 21."

1. What is being done to x? It is being multiplied by 3.
2. The undo for multiplying by 3 is dividing by 3. Divide both sides by 3: 3x ÷ 3 = 21 ÷ 3.
3. On the left, times 3 and divided by 3 cancel. The x is alone.
4. On the right, 21 divided by 3 is 7.
5. So x = 7.

Where people get stuck: seeing 3x = 21 and subtracting 3, getting 18. It is an easy slip, and nearly everyone makes it once. The 3 is not being ADDED to x, it is MULTIPLYING x, so the undo is dividing, not subtracting. Before you touch anything, always ask: "what is actually happening to x?"

Division works the same way in reverse. To solve x ÷ 4 = 6 (also written x/4 = 6), the x is being divided by 4, so multiply both sides by 4: x = 6 × 4 = 24. Check: 24 divided by 4 is 6. True.

Key idea: Undo multiplication with division, and division with multiplication.

Two-step equations: shoes and socks

Now we are ready for equations where two things happened to x, like 2x + 3 = 11. Reading it aloud: "double a mystery number, then add 3, and you get 11."

Here is the trick, and it comes from getting dressed. In the morning you put on socks first, then shoes. At night you take off the shoes first, then the socks. Undoing always happens in reverse order. This equation was built by multiplying x by 2 first, then adding 3. So to unwrap it, we undo the +3 first, and the times-2 last.

1. Undo the +3 by subtracting 3 from both sides: 2x + 3 - 3 = 11 - 3.
2. On the left, +3 and -3 cancel: 2x remains.
3. On the right, 11 - 3 is 8. Now the equation is 2x = 8.
4. This is a one-step equation now, and you already know how to solve those. Divide both sides by 2: 2x ÷ 2 = 8 ÷ 2.
5. On the left, x is finally alone. On the right, 8 divided by 2 is 4.
6. So x = 4.

Check it: 2 times 4 is 8, and 8 + 3 is 11. True. Pause here for a second, because that was the hard part of the whole lesson, and you made it through. Everything else today is this same dance with different numbers.

One more, a little quicker but still every step. Solve 4x - 5 = 19.

1. Undo the -5 by adding 5 to both sides: 4x = 19 + 5.
2. Work out the right side: 19 + 5 is 24. So 4x = 24.
3. Undo the times 4 by dividing both sides by 4: x = 24 ÷ 4.
4. 24 divided by 4 is 6. So x = 6.

Check it: 4 times 6 is 24, minus 5 is 19. True. See? You just solved a two-step equation almost on your own.

Try it: solve 5x + 2 = 17.

Answer: subtract 2 from both sides: 5x = 15. Divide both sides by 5: x = 3. Check: 5 times 3 is 15, plus 2 is 17. Nicely done.

A fraction in front

What about (2/3)x = 6, which says "two thirds of a mystery number is 6"? Do not let the fraction rattle you. Treat it as two tiny moves instead of one big one.

1. The x is divided by 3 (that is the /3 part). Undo it: multiply both sides by 3. That gives 2x = 18.
2. The x is multiplied by 2. Undo it: divide both sides by 2. That gives x = 9.

Check it: two thirds of 9: 9 divided by 3 is 3, times 2 is 6. True. (Once this feels comfortable, there is a shortcut: multiplying both sides by the flipped fraction 3/2 does both moves at once. Use it whenever you are ready, and not a day sooner.)

When the letter sits on the right

Sometimes the mystery number hides on the right side, like 20 = 4 + 8x. Nothing about the method changes; the scale does not care which pan holds the box. Subtract 4 from both sides: 16 = 8x. Divide both sides by 8: 2 = x. And 2 = x means exactly the same thing as x = 2; we usually flip it around at the end so the answer reads naturally.

Always check your answer

Here is a quiet superpower that most struggling students never realize they have: in algebra, you never have to wonder whether you are right. Take your answer, put it back into the original equation, and do the arithmetic. If both sides match, you are right, guaranteed. If they do not, you found your own slip before anyone else did, and you get to fix it. Checking takes ten seconds and turns anxiety into certainty. Make it a habit.

Key idea: Substitute your answer back in. Matching sides mean you are certainly correct.

From story to equation

A gym charges 25 dollars to join, plus 15 dollars per month. So far you have paid 115 dollars in total. How many months have you been a member? Let m stand for the months.

1. Build the sentence in math: joining fee plus monthly cost equals the total: 25 + 15m = 115.
2. Undo the +25 by subtracting 25 from both sides: 15m = 90.
3. Undo the times 15 by dividing both sides by 15: m = 6.

Check it: 25 plus 15 times 6 is 25 + 90, which is 115. True: 6 months. Notice the anatomy, because you will see it everywhere: the one-time fee is the lonely number, the per-month rate is the number stuck to the letter, and the total sits on the other side of the scale.

Where you will see this

• Saving up: you have 80 dollars and save 30 dollars a week. A 320 dollar phone means 80 + 30w = 320. Subtract 80: 30w = 240. Divide by 30: w = 8 weeks.
• Splitting a bill: four friends share pizzas plus a 6 dollar tip, 38 dollars total: 4p + 6 = 38. Subtract 6: 4p = 32. Divide by 4: each share is p = 8 dollars.
• Cooking: a roast needs 20 minutes per pound plus 15 minutes of resting, 95 minutes total: 20p + 15 = 95 gives p = 4 pounds.

Common misconceptions

• Doing something to one side only. Subtracting 5 from the left but not the right tips the scale, and the answer comes out wrong. Same action, both sides, every time.
• Undoing in the wrong order. In 2x + 3 = 11, remove the +3 first, then divide by 2. Shoes off before socks.
• Subtracting a multiplier. In 3x = 21, the 3 multiplies x, so divide by 3. Subtracting 3 undoes an addition that never happened.
• Stopping at -x = 5. That is not finished; x itself is what we want. Multiply (or divide) both sides by -1 to get x = -5.
• Skipping the check. Ten seconds of substitution catches nearly every slip. It is the cheapest insurance in math.

Try it

Two quick ones before the quiz. Solve 3x + 4 = 19, and solve x - 9 = 2.

Answer: First one: subtract 4 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5. Check: 15 + 4 is 19. Second one: add 9 to both sides to get x = 11. Check: 11 - 9 is 2. If you got both, you are more than ready for the quiz. If one slipped, find the exact step where it went sideways; that is not failure, that is how everyone actually learns this.

Recap

• An equation is a balanced scale with an equals sign; a solution is the number that makes both sides match.
• Golden rule: whatever you do to one side, do exactly the same to the other side.
• To free the letter, undo what happened to it: subtraction undoes addition, division undoes multiplication.
• Two-step equations unwrap in reverse order: clear the added or subtracted number first, then the multiplier.
• Check every answer by substituting it back in. Matching sides mean you are right.

Sources

• OpenStax. (2020). Solve equations using the Subtraction and Addition Properties of Equality (Section 2.1). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Solving equations and inequalities [Unit]. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Linear equations. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Solving equations. Math is Fun. mathsisfun.com

The big picture

If you made it through the last lesson, you already own every tool this lesson needs. Truly, there is nothing new to buy here. Today's equations look longer, and longer can feel scarier, but longer is not harder. It is like tidying a messy room: you do not lift the whole mess at once, you pick up one item at a time, and suddenly the room is clean. We are going to pick up these equations one small piece at a time.

By the end you will handle equations with parentheses, equations with a crowd of terms, and even equations where x shows up on BOTH sides of the equals sign. We go slowly, and every step gets a reason.

Key idea: A long equation is a short equation wearing extra layers. Remove one layer at a time.

The game plan

Every equation in this lesson gives way to the same four-move routine. You do not need to memorize it right now; we will practice it together until it feels natural.

1. Clear parentheses (distribute, that is, multiply through).
2. Tidy each side by combining like terms. Left side tidies itself, right side tidies itself; they do not mix yet.
3. Herd the letters to one side and the plain numbers to the other, using the both-sides rule you already know.
4. Finish like a two-step: clear the added number, then the multiplier.

Key idea: Clear parentheses, tidy each side, herd the letters, finish with two steps.

Step one in action: parentheses

What does 3(x + 2) mean? Read it aloud: "3 times the whole package x plus 2." Picture 3 identical gift bags, and inside each bag there is one x and a 2. Altogether you are holding 3 x's and 3 twos: 3x + 6. Multiplying every item inside the parentheses is called distributing, and it is how we unwrap the package.

Let us solve 3(x + 2) = 18 together, slowly.

1. Distribute the 3 to both items inside: 3 times x is 3x, and 3 times 2 is 6. The equation becomes 3x + 6 = 18.
2. Now it is a two-step equation, and you know this dance. Subtract 6 from both sides: 3x = 12.
3. Divide both sides by 3: x = 4.

Check it: 4 + 2 is 6, and 3 times 6 is 18. True. (A side note for the curious: on this one you could also have divided both sides by 3 right away to get x + 2 = 6. Both roads lead to 4. When two legal moves both work, pick whichever feels friendlier.)

Step two in action: tidying a crowded side

Sometimes one side arrives messy, like 2x + 5 + 3x = 25. Reading it aloud: "two x's, plus 5, plus three more x's, is 25." Before undoing anything, tidy up. The 2x and the 3x are like terms, apples with apples, so gather them.

1. Combine 2x + 3x to get 5x. The equation becomes 5x + 5 = 25.
2. Subtract 5 from both sides: 5x = 20.
3. Divide both sides by 5: x = 4.

Check it: 2(4) + 5 + 3(4) is 8 + 5 + 12, which is 25. True. Tidy first, then solve. That is the whole trick.

Try it: solve 4(x + 1) = 24.

Answer: distribute: 4x + 4 = 24. Subtract 4 from both sides: 4x = 20. Divide by 4: x = 5. Check: 5 + 1 is 6, and 4 times 6 is 24. If you got 5, that is the routine working already.

The main event: x on both sides

Now the moment that makes most people's stomach drop the first time: 5x - 2 = 3x + 8. There are x's on the left AND on the right. The balance scale has not changed, and neither has the golden rule. We need one new idea, and it is gentle: you are allowed to subtract x's from both sides, exactly the way you subtract numbers.

Think of it as herding sheep. The x's are sheep wandering on both sides of a fence. We herd them all into one pen, count them, and finish as usual.

1. Choose a pen. Let us gather the x's on the left, because the left has more of them (5x beats 3x, and keeping the bigger flock avoids negatives).
2. Subtract 3x from both sides: on the left, 5x - 3x leaves 2x. On the right, 3x - 3x vanishes. The equation is now 2x - 2 = 8.
3. Look at what remains: an ordinary two-step equation. Add 2 to both sides: 2x = 10.
4. Divide both sides by 2: x = 5.

What we just did: one new move, subtracting 3x from both sides, turned a both-sides equation into last lesson's homework.

Check it: left side: 5(5) - 2 is 23. Right side: 3(5) + 8 is 23. Both sides say 23. True. Pause and enjoy that for a second, because equations with x on both sides are the thing this whole lesson builds to, and you have now solved one.

Where people get stuck: the phrase "move the 3x over." Terms do not really move; what happens is we subtract 3x from BOTH sides, and the term disappears from one side while changing the other. If you remember what the move actually is, you will never mangle the sign. When a term seems to "cross" the equals sign and flip its sign, that flip is the both-sides subtraction quietly doing its work.

One more, with parentheses too

Solve 2(x + 4) = x + 11. This one uses the whole game plan.

1. Clear parentheses: distribute the 2 to get 2x + 8 = x + 11.
2. Each side is already tidy, so herd the letters. Subtract x from both sides: 2x - x leaves x, so x + 8 = 11.
3. Subtract 8 from both sides: x = 3.

Check it: left: 3 + 4 is 7, doubled is 14. Right: 3 + 11 is 14. Both sides 14. True. Notice how short the both-sides part was: one subtraction, and the fear was over.

Careful: distributing a minus

Here is the single most common slip in all of Algebra I, so we will meet it in slow motion. When a minus sign sits in front of parentheses, the minus rides along to EVERY item inside, not only the first one.

With numbers: -2(x - 1) means -2 times x, which is -2x, and -2 times -1, which is +2. So -2(x - 1) = -2x + 2. The second sign flips, because a negative times a negative is a positive.

Let us use it. Solve 10 - 2(x - 1) = 4.

1. Distribute the -2 to both items inside: 10 - 2x + 2 = 4. (See the +2? That is the minus times minus.)
2. Tidy the left side: 10 + 2 is 12, so 12 - 2x = 4.
3. Subtract 12 from both sides: -2x = -8.
4. Divide both sides by -2: x = 4. A negative divided by a negative is positive.

Check it: 4 - 1 is 3, times 2 is 6, and 10 - 6 is 4. True. If the minus-through-parentheses step felt shaky, read it once more; it is the one move in this lesson worth double practice.

When the letters vanish

Once in a while, you follow every rule perfectly and something strange happens: all the x's cancel out and disappear. Your first thought will be "I broke the math." You did not. The equation is telling you something, and there are exactly two possibilities.

Case 1: a lie. Solve 2(x + 3) = 2x + 10. Distribute: 2x + 6 = 2x + 10. Subtract 2x from both sides: 6 = 10. The x's are gone, and what remains is false; 6 does not equal 10, not today, not ever. That means NO value of x can make the original equation true. We say the equation has no solution.

Case 2: a truth that never depended on x. Solve 2(x + 3) = 2x + 6. Distribute: 2x + 6 = 2x + 6. Subtract 2x from both sides: 6 = 6. What remains is true no matter what x is. EVERY number works: x = 1, x = 50, x = -3, all of them. We say the equation has infinitely many solutions (it is an identity, the same thing written two ways).

Key idea: If the letters vanish, read the leftover sentence. A false sentence means no solution; a true one means every number works.

Where you will see this

• Comparing phone plans: Plan A costs 20 dollars plus 5 per gigabyte; Plan B costs 10 dollars plus 7 per gigabyte. When do they cost the same? 20 + 5g = 10 + 7g. Subtract 5g: 20 = 10 + 2g. Subtract 10: 10 = 2g, so g = 5. At 5 gigabytes both cost 45 dollars; below that, B wins, above it, A wins.
• Catching up: your friend has 60 dollars saved and adds 10 a week; you have 20 and add 18 a week. 20 + 18w = 60 + 10w gives 8w = 40, so w = 5 weeks to catch up.
• Splitting work: two printers, one has 300 pages done and prints 40 per minute, another has 500 done printing 20 per minute: equal at 300 + 40m = 500 + 20m, so m = 10 minutes.

Common misconceptions

• Distributing to the first term only. 3(x + 2) is 3x + 6, not 3x + 2. Every item in the bag gets multiplied.
• Losing the minus. -2(x - 1) is -2x + 2, not -2x - 2. A minus in front flips every sign inside.
• Combining unlike terms. 2x + 5 is finished; x-terms and plain numbers cannot merge into 7x.
• Moving a term without the both-sides move. Terms cross the equals sign only because you add or subtract them from both sides. Keep that in mind and signs stop misbehaving.
• Stopping when x disappears. A leftover false sentence like 6 = 10 means no solution; a leftover true one like 6 = 6 means every number is a solution. Neither one means you failed.

Try it

Solve 3(x - 2) = x + 8. Use the game plan: distribute, tidy, herd, finish.

Answer: distribute: 3x - 6 = x + 8. Herd the letters: subtract x from both sides: 2x - 6 = 8. Add 6 to both sides: 2x = 14. Divide by 2: x = 7. Check: left is 3 times 5, which is 15; right is 7 + 8, which is 15. Both sides agree. If you got 7, you have mastered the biggest lesson in the equations unit. If not, find the step where yours differs and you will usually spot a sign slip; that is normal and fixable.

Recap

• Game plan: clear parentheses, tidy each side, herd letters to one side, finish like a two-step.
• Distribute to every term inside parentheses, and let a leading minus flip every inside sign.
• Subtracting an x-term from both sides is how letters "move" across the equals sign.
• Herding the smaller flock into the bigger one helps you avoid negative coefficients.
• If all letters vanish: false leftover means no solution, true leftover means infinitely many.

Sources

• OpenStax. (2020). Solve equations with variables and constants on both sides (Section 2.3). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Solving equations and inequalities [Unit]. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Linear equations. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Balance when adding and subtracting. Math is Fun. mathsisfun.com

The big picture

Settle in; this lesson is friendlier than its title. Ratios and proportions are the math of fairness and recipes: doubling the cookies without ruining them, spotting which bag of rice is actually the better deal, reading a map without getting lost. You have been using this math informally your whole life. Today we only put words and a reliable method around what your instincts already do.

Then, at the end, we will learn to flip formulas around, which sounds fancy but is really the same both-sides moves you already trust from the last two lessons. One small step at a time, as always.

Key idea: Ratios compare, proportions scale, and formulas are recipes you can rearrange.

What a ratio is

A ratio compares two amounts. Suppose a lemonade recipe uses 2 cups of lemon juice for every 3 cups of water. We write the ratio 2:3, or as the fraction 2/3, and we read it aloud as "2 to 3." It does not say how big the batch is; it says how the two parts relate.

Here is the important part. If you double both amounts, 4 cups of juice and 6 cups of water, the lemonade tastes exactly the same. The ratios 2:3 and 4:6 and 6:9 are called equivalent ratios: same relationship, different batch size. You make an equivalent ratio by multiplying (or dividing) BOTH parts by the same number, never only one part. Multiply only the water and you have watery lemonade; that is not math punishing you, that is math agreeing with your taste buds.

Key idea: Scaling both parts of a ratio by the same number keeps the relationship the same.

Rates: when the units differ

A rate is a ratio between different kinds of things, like dollars and pounds, or miles and hours. The most useful version is the unit rate: how much for exactly ONE. If 3 pounds of apples cost 6 dollars, then one pound costs 6 ÷ 3 = 2 dollars. The word "per" is a quiet division sign: dollars per pound means dollars divided by pounds. Unit rates are how you win at grocery stores: work out the price per ounce for two package sizes and the better deal reveals itself.

What a proportion is

A proportion is a statement that two ratios are equal, like 2/3 = 6/9. Read it aloud: "2 is to 3 as 6 is to 9." Usually one of the four numbers is missing, and that is where solving comes in:

3/4 = x/20

Read it: "3 is to 4 as x is to 20." Some mystery amount out of 20 keeps the same relationship as 3 out of 4.

Solving a proportion, three ways to the same answer

Way 1, scaling (use this when the numbers are kind).

1. Look at the two bottoms: 4 and 20. Ask: 4 times what gives 20? Times 5.
2. Whatever the bottom did, the top must do too. So x is 3 times 5.
3. 3 times 5 is 15. So x = 15.

Way 2, the both-sides move you already know.

1. x is being divided by 20, so multiply both sides by 20: x = (3/4) × 20.
2. Work it out in two tiny steps: 20 divided by 4 is 5, then 3 times 5 is 15. So x = 15.

Way 3, cross multiplication (the famous shortcut). In any proportion, the two diagonal products are equal: top-left times bottom-right equals bottom-left times top-right.

1. Multiply along one diagonal: 3 times 20 is 60.
2. Multiply along the other: 4 times x is 4x.
3. Set them equal: 60 = 4x.
4. Divide both sides by 4: x = 15.

What we just did: three different roads, one answer, 15. Cross multiplication is not magic; it is what you get when you multiply both sides of the proportion by both bottoms at once. If it ever feels like a trick you cannot trust, fall back on Way 1 or Way 2, which are slower but see-through.

Where people get stuck: cross multiplication is legal ONLY when the equation is exactly one fraction equal to one fraction, nothing else on either side. If there is a plus term hanging around, like x/4 + 1 = 3/2, you may not cross multiply yet; clear the +1 first. Checking "is it fraction equals fraction, full stop?" before crossing will save you many headaches.

Try it: solve 2/5 = x/30.

Answer: the bottoms go 5 to 30, which is times 6, so the top goes 2 times 6, and x = 12. Or cross multiply: 2 times 30 is 60, so 5x = 60 and x = 12. Same answer either road. Well done.

Proportions out in the world

A map. A map says 1 inch stands for 50 miles. Two towns sit 3.5 inches apart on the paper. How far apart are they really?

1. Set up matching ratios, inches over miles on both sides: 1/50 = 3.5/m.
2. Cross multiply: 1 times m is m, and 50 times 3.5 is 175.
3. So m = 175 miles.

A recipe. If 2 cups of flour make 12 cookies, how much flour for 30 cookies? Set up 2/12 = f/30. Cross multiply: 2 times 30 is 60, and 12 times f is 12f. So 12f = 60, and f = 5 cups. Sanity check with your common sense: 30 cookies is two and a half times the batch, and two and a half times 2 cups is 5 cups. When the algebra and your instincts agree, you know you are on solid ground.

Formulas: recipes with name tags

A formula is an equation that relates real quantities, with each letter wearing a name tag. Two you will meet constantly:

• d = rt, read "distance equals rate times time." Drive at 60 miles per hour for 2 hours: d = 60 times 2 = 120 miles.
• P = 2l + 2w, read "perimeter equals twice the length plus twice the width." A 4 by 3 room: P = 8 + 6 = 14 units around.

Using a formula forward is nothing new: put the numbers in the seats, work one step at a time.

Flipping a formula around

Here is the new skill, and it is smaller than it looks. Sometimes the formula answers the wrong question. d = rt hands you distance, but suppose you know the distance and the rate, and you want the TIME. You could plug numbers in and solve every single trip, or you could rearrange the formula once and be done forever.

The secret: treat the other letters exactly like numbers. They are numbers, wearing name tags. Solve d = rt for t:

1. Look at t. What is being done to it? It is multiplied by r.
2. Undo that: divide both sides by r: d/r = t.
3. Flip it to read naturally: t = d/r.

What we just did: the same one-step move as 3x = 21, except the 3 was called r and the 21 was called d. Time equals distance divided by rate: a 300 mile trip at 60 miles per hour takes 300/60 = 5 hours.

Now a two-step one. Solve P = 2l + 2w for w:

1. The w term is 2w. First get it alone: subtract 2l from both sides: P - 2l = 2w.
2. Now w is multiplied by 2. Divide both sides by 2: (P - 2l)/2 = w.
3. Read it forward: w = (P - 2l)/2.

Check it with real numbers, because this is how you learn to trust a rearranged formula. A 4 by 3 rectangle has P = 14. Our new formula says w = (14 - 2 times 4)/2 = (14 - 8)/2 = 6/2 = 3. And the width really is 3. The rearrangement told the truth. That is the whole skill, and you did it with moves you already knew.

Key idea: To solve a formula for a letter, treat every other letter like a number and use the same undo moves as always.

Common misconceptions

• Scaling one part of a ratio. To keep 2:3 equivalent, multiply BOTH parts by the same number. 2:3 scaled by 5 is 10:15, not 10:3.
• Cross multiplying when there is extra baggage. The shortcut needs exactly one fraction on each side and nothing else. Clear added terms first.
• Mismatched setups. Keep the same quantity on top on both sides: inches over miles equals inches over miles. Swap one side and the answer flips with it.
• Dividing only one term. Solving P = 2l + 2w for w, you must subtract 2l BEFORE dividing by 2, or divide every term. (P - 2l)/2 is right; P - l is not.
• Fearing letters in the answer. t = d/r is a finished answer. A rearranged formula is allowed to be made of letters; they are numbers with name tags.

Try it

Two finishers. First, solve 4/6 = 10/x. Second, the cost of a taxi is C = 5d + 10 (5 dollars per mile plus a 10 dollar flat fee); solve it for d.

Answer: First: cross multiply: 4 times x is 4x, and 6 times 10 is 60, so 4x = 60 and x = 15. Check: 4/6 and 10/15 both reduce to 2/3. Second: subtract 10 from both sides: C - 10 = 5d. Divide both sides by 5: d = (C - 10)/5. Test it: a 30 dollar ride gives d = (30 - 10)/5 = 4 miles, and 5 times 4 plus 10 really is 30. If both of those landed, you are in great shape; if one wobbled, reread the matching worked example and it will click.

Recap

• A ratio compares two amounts; multiplying both parts by the same number keeps it equivalent.
• A unit rate is the amount for exactly one: "per" means divide.
• A proportion says two ratios are equal; solve by scaling, by both-sides moves, or by cross multiplying.
• Cross multiplication needs one fraction on each side and nothing else.
• To rearrange a formula, treat other letters as numbers and undo operations one at a time, both sides.

Sources

• OpenStax. (2020). Solve a formula for a specific variable (Section 2.6). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Solving equations and inequalities [Unit]. In Algebra 1. khanacademy.org
• Pierce, R. (n.d.). Proportions. Math is Fun. mathsisfun.com
• Pierce, R. (n.d.). Ratios. Math is Fun. mathsisfun.com

The big picture

Here is some good news to start with: if you can solve equations, you can already do about 95 percent of this lesson. Take that in for a second. Inequalities are not a new mountain; they are the same hill with one extra signpost.

And they are everywhere, because real life rarely demands an exact number. A ride says you must be AT LEAST 42 inches tall. A password needs AT LEAST 8 characters. The speed limit says AT MOST 65. Your budget says spend NO MORE THAN 100 dollars. None of those pin down one exact value; they each describe a whole range of acceptable values. That is what an inequality is: a math sentence about a range.

Key idea: An equation asks for the one number that works. An inequality asks for every number that works.

Meet the four symbols

Read each one aloud until it feels ordinary:

• < reads "is less than." Example: 3 < 7, "3 is less than 7." True.
• > reads "is greater than." Example: 9 > 2, "9 is greater than 2." True.
• ≤ reads "is less than or equal to," or in everyday words, "at most."
• ≥ reads "is greater than or equal to," or "at least."

A gentle way to keep them straight: the symbol has a small pointy end and a wide open end. The small pointy end always points at the smaller number, and the wide open mouth faces the bigger one. In 3 < 7, the point touches the 3. Check any true example and you will see it hold.

The little extra line under ≤ and ≥ means "or exactly equal is fine too." Age ≥ 13 to sign up means 13-year-olds are allowed in; age > 13 would make them wait a year.

Key idea: The pointy end faces the smaller number, and the underline means "equal counts too."

A solution is a crowd, not a single number

The inequality x > 2 says "x is any number greater than 2." Who passes the test? 3 passes. So does 2.5. So does 100, and so does 2.0001. The number 2 itself does NOT pass, because 2 is not greater than 2. The solution is a whole crowd of numbers, and we picture the crowd on a number line: draw an open circle at 2 (open means 2 itself is not included) and shade everything to the right, forever.

For x ≥ 2, the story changes in one small way: 2 now passes the test, so the circle at 2 is filled in (closed), and the shading still sweeps right.

Key idea: Open circle: the endpoint is not included. Filled circle: it is. Shade toward every number that passes.

Solving: the moves you already know

Here is the wonderful part. You solve an inequality with the same both-sides moves as an equation. Watch.

Solve x + 4 < 9.

1. What is being done to x? A 4 is added.
2. Undo it: subtract 4 from both sides: x + 4 - 4 < 9 - 4.
3. Tidy both sides: x < 5.

Check it with a crowd member: pick something less than 5, say x = 0. Does 0 + 4 < 9? Yes, 4 is less than 9. Now pick a number that should FAIL, say x = 10: is 14 < 9? No. The fence sits exactly where it should. That double-check, one insider and one outsider, is the inequality version of substituting back, and it will catch nearly every slip.

One more. Solve 3x ≥ 12.

1. x is multiplied by 3. Undo it: divide both sides by 3.
2. x ≥ 4.

Done. Positive numbers keep everything friendly. That is genuinely all there is, until one special situation.

Try it: solve x - 2 > 1.

Answer: add 2 to both sides: x > 3. Check an insider: x = 5 gives 3 > 1, true. Check an outsider: x = 0 gives -2 > 1, false. The fence is in the right place. Nicely done.

The one new rule: the flip

Now the single fact that makes inequalities different, and we will discover it with real numbers rather than memorize it on faith.

Start with a sentence everyone agrees on: 2 < 5. True. Now multiply BOTH sides by -1, exactly the way we always do things to both sides. The left becomes -2 and the right becomes -5. Is -2 < -5 still true?

Think about temperatures: negative 2 degrees is warmer than negative 5 degrees. Or money: owing 2 dollars leaves you better off than owing 5. Either way, -2 is the BIGGER number. So the true sentence is -2 > -5. The order reversed. Multiplying by a negative flipped every number to the other side of zero, like looking at the number line in a mirror, and left became right.

So here is the one new rule, worth reading twice: when you multiply or divide both sides of an inequality by a NEGATIVE number, the inequality symbol flips direction. Less-than becomes greater-than, and greater-than becomes less-than.

Adding or subtracting anything, even a negative, never flips the symbol. Multiplying or dividing by a positive never flips it either. The flip happens in exactly one situation: multiply or divide by a negative.

Key idea: Multiply or divide both sides by a negative, and the symbol flips. Nothing else flips it.

Worked example: the flip in action

Solve -2x > 8.

1. x is multiplied by -2. Undo it: divide both sides by -2.
2. We divided by a NEGATIVE, so the symbol flips from > to <.
3. Work the numbers: 8 divided by -2 is -4. So x < -4.

Check it: insider x = -5: -2 times -5 is 10, and 10 > 8 is true. Outsider x = 0: -2 times 0 is 0, and 0 > 8 is false. Both tests pass, so x < -4 is right. Notice how the check quietly proves the flip was needed: without it we would have written x > -4, and the outsider x = 0 would have slipped through.

Prefer to avoid the flip altogether? There is a flip-free road. Solve 5 - x < 2 by adding x to both sides first: 5 < 2 + x. Then subtract 2 from both sides: 3 < x, which reads the other way as x > 3. Same answer, no negative division, no flip to remember. Use whichever road feels steadier; they always agree.

A two-step inequality

Solve 2x - 3 ≤ 7.

1. Undo the -3 first: add 3 to both sides: 2x ≤ 10.
2. Undo the times 2: divide both sides by 2. Positive number, so no flip.
3. x ≤ 5.

Check it: insider x = 5 (allowed, because of the underline): 10 - 3 is 7, and 7 ≤ 7 is true. Outsider x = 6: 12 - 3 is 9, and 9 ≤ 7 is false. Solid. Same shoes-and-socks order as equations, plus one glance at the sign of whatever you multiply or divide by.

Writing the answer: number lines and intervals

You will see solutions written three ways, and they all say the same thing. Take x ≤ 5:

• Symbols: x ≤ 5.
• Number line: a filled circle at 5 (the underline includes 5), shading left forever.
• Interval form: (-∞, 5], read aloud "from negative infinity up to and including 5."

Interval form is a compact address for the shaded region: write where it starts, a comma, where it ends. A square bracket means that endpoint is included (like a filled circle); a round parenthesis means it is not (like an open circle). Infinity always gets a round parenthesis, because infinity is not a number you can actually reach and include. So x > 2 becomes (2, ∞): starts just past 2, runs forever right, neither end included.

Key idea: Square bracket = included, round parenthesis = not included, and infinity is always round.

Where you will see this

• Budgets: a gym costs 25 dollars to join plus 15 per month, and you can spend at most 100: 25 + 15m ≤ 100. Subtract 25: 15m ≤ 75. Divide by 15: m ≤ 5 months.
• Grades: you need an average of at least 80 across tests; each requirement is a ≥ inequality.
• Shipping and safety limits: an elevator rated for at most 2000 pounds with n boxes of 150 pounds: 150n ≤ 2000, so n ≤ 13.3, meaning at most 13 boxes.

Common misconceptions

• Flipping for the wrong reasons. Subtracting a number, even a negative one, never flips the symbol. The flip belongs to one move only: multiplying or dividing both sides by a negative.
• Forgetting the flip entirely. After dividing by a negative, pause and flip. The insider-outsider check catches it if you forget.
• Treating the answer as one number. x < 5 names a whole crowd. Any single member, like 4.9 or -300, is only one face in it.
• Mixing up the circles. Strict symbols (<, >) take open circles and round parentheses; the underlined ones (≤, ≥) take filled circles and square brackets.
• Putting a bracket on infinity. Infinity is a direction, not a destination: always a round parenthesis.

Try it

Solve -3x ≥ 15, then write x < 4 in interval form.

Answer: divide both sides by -3, and because -3 is negative, flip the symbol: x ≤ -5. Check an insider, x = -6: -3 times -6 is 18, and 18 ≥ 15 is true. Check an outsider, x = 0: 0 ≥ 15 is false. The flip earned its keep. Interval form of x < 4: (-∞, 4), round on both ends because neither infinity nor 4 is included. If you flipped correctly on the first one, you have conquered the only genuinely new idea in this lesson.

Recap

• An inequality describes a range; its solution is every number that makes it true.
• Solve with the same both-sides moves as equations.
• Multiplying or dividing both sides by a negative flips the symbol; nothing else does.
• Check with one number that should work and one that should not.
• Open circle and parenthesis exclude an endpoint; filled circle and square bracket include it; infinity is always round.

Sources

• OpenStax. (2020). Solve linear inequalities (Section 2.7). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Solving equations and inequalities [Unit]. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Linear inequalities. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Solving inequalities. Math is Fun. mathsisfun.com

The big picture

This is the gentlest lesson in the whole course. There is nothing to solve today. No equations, no unknowns, nothing to undo. We are going to learn how to describe WHERE something is, and you have been doing that your entire life: "row F, seat 12" at the movies, "B4" in a game of battleship, "two blocks over and three blocks up" when giving directions. Each of those is two pieces of information that pin down one exact spot.

The coordinate plane is math's version of that idea, and it is about to become the stage where the rest of algebra performs. Every graph you will ever draw lives here, so a few minutes now pay off for years.

Key idea: Two numbers, in the right order, pin down one exact spot.

Two number lines, crossed

Take the number line you already know and lay it flat, left to right. That is the x-axis. Now stand a second number line straight up through its zero. That is the y-axis. The point where they cross, where both are at zero, is called the origin. Think of the origin as home base: every trip we describe starts from there.

On the x-axis, numbers grow to the right and turn negative to the left, exactly like the number line from earlier lessons. On the y-axis, numbers grow going up and turn negative going down. Up is positive, down is negative, the same way a thermometer works.

Key idea: The x-axis runs sideways, the y-axis runs up and down, and the origin is home base at (0, 0).

Ordered pairs: over, then up

An address on the plane looks like (3, 2) and is called an ordered pair. Read it aloud: "the point where x is 3 and y is 2." The first number is always the x-coordinate, the second is always the y-coordinate. A friendly way to remember: x comes before y in the alphabet, so the x number comes first in the pair. Another way: in a building you walk ACROSS the lobby to the elevator before you ride UP. Over first, then up.

Let us plot (3, 2) together, one tiny move at a time.

1. Put your pencil on the origin, home base.
2. Read the first number: 3. That is the sideways instruction. Walk 3 units to the right (right, because it is positive).
3. Read the second number: 2. That is the vertical instruction. Climb 2 units up (up, because it is positive).
4. Mark the dot. That spot, and no other spot in the world, is (3, 2).

What we just did: first number moved us sideways, second number moved us vertically. That is the entire skill of plotting.

Where people get stuck: swapping the order. (3, 2) and (2, 3) use the same two numbers but land on different spots, the same way "row 3, seat 2" and "row 2, seat 3" are different chairs at the movies. When in doubt, whisper "over, then up" and trust the first number to be the sideways one.

When the numbers are negative

Negative coordinates are not a new idea; they only change the direction of the walk. Negative x means walk LEFT instead of right. Negative y means climb DOWN instead of up.

Plot (-2, 4):

1. Start at the origin.
2. First number is -2: walk 2 units left.
3. Second number is 4: climb 4 units up.
4. Mark the dot. You are up and to the left of home base.

Plot (-3, -1):

1. Start at the origin.
2. Walk 3 units left (the minus says left).
3. Climb 1 unit down (the minus says down).
4. Mark the dot, down-left of home base.

Try it: describe how you would plot (4, -2).

Answer: start at the origin, walk 4 right, climb 2 down, mark the dot. If you said that, you can now plot any point that exists. That is not an exaggeration; every point is some version of this walk.

The four quadrants

The two axes slice the plane into four rooms, called quadrants, numbered with Roman numerals I, II, III, IV. The numbering starts in the upper right and sweeps counterclockwise, like spreading butter from the top-right corner:

• Quadrant I (upper right): x positive, y positive. Signs (+, +).
• Quadrant II (upper left): x negative, y positive. Signs (-, +).
• Quadrant III (lower left): both negative. Signs (-, -).
• Quadrant IV (lower right): x positive, y negative. Signs (+, -).

You do not need to plot a point to know its room; the signs alone tell you. (-3, 5) has a negative x and positive y, so it lives in Quadrant II, upper left, before your pencil ever moves. Read the two signs, name the room.

Key idea: The pair of signs is the point's room number: (+,+) is I, (-,+) is II, (-,-) is III, (+,-) is IV.

Points that live on the axes

What about a point like (0, 3)? The first number says walk 0 sideways, so you never leave the vertical line. You climb 3 and end up ON the y-axis. Likewise (5, 0) walks 5 right and climbs nowhere, landing ON the x-axis. Points on an axis belong to no quadrant; they sit on the walls between rooms.

Here is the sentence that untangles it, worth reading twice: if x is 0 you never moved sideways, so you are on the y-axis; if y is 0 you never moved vertically, so you are on the x-axis. And (0, 0) moves nowhere at all: that is the origin, home base itself.

Reading a point off the graph

Plotting has a reverse skill: seeing a dot and naming its address. Slide your finger from the dot straight down (or up) to the x-axis and read the number there; that is x. Slide from the dot straight across to the y-axis; that is y. A dot sitting 2 left of the y-axis and 5 above the x-axis is (-2, 5). Slow is fine. Nobody speed-reads coordinates at first.

What the coordinates MEAN

Here is the part that turns this from a game into a tool. In real graphs, each axis carries a meaning. Suppose the x-axis is "hours worked" and the y-axis is "dollars earned." Then the point (3, 45) is not an abstract dot; it is a small sentence: "after 3 hours, 45 dollars were earned." Reading points as sentences is the skill every science class, news chart, and business graph will quietly assume you have. You now have it.

Key idea: On a real-world graph, every point is a sentence: (x, y) means "when this much x, that much y."

Where you will see this

• Screens: every pixel on your phone has coordinates; games place characters by ordered pairs.
• Spreadsheets: cell B4 is a coordinate: column B, row 4. Over, then up (well, down, but the idea is the same).
• Maps and GPS: latitude and longitude are an ordered pair for the whole planet.
• Science: time-versus-temperature, dose-versus-effect; every experiment chart is dots on this plane.

Common misconceptions

• Swapping x and y. The first number is ALWAYS the sideways one. (2, 7) and (7, 2) are different points. Over, then up.
• Thinking (0, 4) is on the x-axis. x = 0 means no sideways walk, so the point sits on the y-axis. It feels backwards until you picture the walk; then it never confuses you again.
• Numbering quadrants clockwise. They sweep counterclockwise from the upper right: I, II, III, IV.
• Assigning axis points to a quadrant. Points with a zero coordinate live on a wall, not in a room.
• Ignoring axis labels on real graphs. Always read what x and y stand for first; the labels are what give every dot its meaning.

Try it

Without plotting, name the quadrant of (-5, 7). Then say what the coordinates of the origin are. Then decode this: on a graph where x is minutes and y is meters run, what does the point (10, 1600) say?

Answer: (-5, 7) has signs (-, +), and that sign pattern is the upper-left room, Quadrant II. The origin is (0, 0), home base. And (10, 1600) is the sentence "after 10 minutes, 1600 meters were run." If you got all three, you have everything this lesson hoped to give you, and the next lesson (slope) will feel far friendlier because of it.

Recap

• The coordinate plane is two crossed number lines: x-axis sideways, y-axis vertical, origin at (0, 0).
• An ordered pair (x, y) is a walk from the origin: over first, then up. Order matters.
• Negative x walks left; negative y climbs down.
• The sign pattern names the quadrant: I (+,+), II (-,+), III (-,-), IV (+,-); zero coordinates sit on the axes.
• On real graphs, every point is a sentence built from the axis labels.

Sources

• OpenStax. (2020). Use the rectangular coordinate system (Section 4.1). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Linear equations and graphs [Unit]. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Graphing. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Cartesian coordinates. Math is Fun. mathsisfun.com

The big picture

Picture a staircase, a wheelchair ramp, and a steep hill on a bike ride. Your legs already understand this lesson: some climbs are gentle and some are brutal, and the difference is STEEPNESS. Slope is nothing more than a number that measures steepness, so that instead of saying "pretty steep, I guess," we can say exactly HOW steep. That is the entire lesson. One number, measuring one familiar feeling.

We will read the formula with the little subscript numbers out loud together, piece by piece. Numbers first, formula second, always.

Key idea: Slope is a single number that says exactly how steep a line is.

Rise over run

Start with stairs. Suppose every step of a staircase goes UP 2 units for every 3 units it goes ACROSS. We capture that steepness as a fraction:

slope = rise / run = 2/3

Two words to make friends with: rise is the vertical change, how much you go up or down. Run is the horizontal change, how much you go across. The rise sits on TOP of the fraction; a way to remember is that rising means going up, and up is where the top is. Read the fraction aloud: "this line rises 2 for every 3 it runs."

A slope of 2/3 is a gentle ramp. A slope of 3 (which is 3/1, rise 3 for every 1 across) is a steep climb, like a ladder leaning against a wall. Bigger number, steeper line. It matches your legs' opinion exactly.

Key idea: Slope = rise over run: how much up, per step across.

Slope from two points, with real numbers

A line through the points (1, 2) and (3, 6): how steep is it? Let us walk from the first point to the second, one small step at a time.

1. How much did y change? It went from 2 up to 6. The rise is 6 - 2, which is 4.
2. How much did x change? It went from 1 to 3. The run is 3 - 1, which is 2.
3. Divide: slope = rise/run = 4/2 = 2.

What we just did: subtracted the y's for the rise, subtracted the x's for the run, divided. The line climbs 2 units for every 1 unit across. That is all slope-finding is, forever.

Now the famous formula, gently

Textbooks write what we just did like this:

m = (y2 - y1) / (x2 - x1)

Do not let the small numbers rattle you; let us read every piece aloud. The letter m stands for slope (an old tradition; the letter itself means nothing special). The little 1s and 2s are NOT math to perform; they are name tags. x1 reads "x sub one" and means "the x from the first point."

y2 reads "y sub two" and means "the y from the second point." So the whole formula reads: "slope equals the second y minus the first y, divided by the second x minus the first x." In other words: rise over run, written with name tags. You already did this a moment ago with 4/2.

Let us use it on (2, 5) and (4, 11).

1. Label the points. First point: x₁ = 2, y₁ = 5. Second point: x₂ = 4, y₂ = 11.
2. Top of the fraction (the rise): y₂ - y₁ = 11 - 5 = 6.
3. Bottom (the run): x₂ - x₁ = 4 - 2 = 2.
4. Divide: m = 6/2 = 3.

The line rises 3 for every 1 across. Nice work; that formula was the scariest thing in this lesson, and it is behind you now.

Where people get stuck: mixing the order halfway through. It does not matter WHICH point you call "second," but whoever you pick must go first in BOTH subtractions. Watch: using the same two points the other way gives (5 - 11)/(2 - 4) = -6/-2 = 3. Same slope. But mix the orders, (11 - 5)/(2 - 4), and you get 6/(-2) = -3, wrong sign. Pick an order, whisper "stay consistent," and keep it.

Try it: find the slope through (0, 1) and (2, 7).

Answer: rise = 7 - 1 = 6; run = 2 - 0 = 2; slope = 6/2 = 3. If you got 3, the formula is officially yours.

Downhill lines: negative slope

Find the slope through (1, 5) and (4, 2).

1. Rise: 2 - 5 = -3. The y went DOWN by 3; a negative rise means a drop.
2. Run: 4 - 1 = 3.
3. Slope: -3/3 = -1.

A negative slope means the line goes downhill as you read it the way you read words, left to right. Positive slopes climb left to right; negative slopes fall. Nothing about the method changed; the minus sign came along for the ride and told us the direction.

The four personalities of slope

• Positive slope: uphill left to right, like climbing. Through (0,0) and (2,4): slope 2.
• Negative slope: downhill left to right, like descending. Through (0,4) and (2,0): slope -2.
• Zero slope: perfectly flat. Through (1, 3) and (5, 3): rise = 3 - 3 = 0, run = 4, slope = 0/4 = 0. A flat road has a slope, and that slope is the number zero. y never changes: a horizontal line.
• Undefined slope: straight up and down, a wall. Through (2, 1) and (2, 6): rise = 5, run = 2 - 2 = 0, and slope = 5/0. Dividing by zero has no answer, so a vertical line's slope is undefined: not zero, not huge, but simply not a number at all.

The pair that trips everyone up is the last two, so here is the sentence to keep: a floor has slope zero; a wall has no slope at all. Zero is a perfectly good number describing flatness. Undefined means the question "rise per step across" cannot even be asked, because a wall never steps across.

Key idea: Uphill positive, downhill negative, flat is zero, vertical is undefined.

Slope is a rate: the "per" number

Here is why slope matters far beyond geometry. When the axes carry real meanings, slope becomes a RATE, the "per" in everyday speech.

Suppose a graph shows hours worked (x) against dollars earned (y), and it passes through (2, 30) and (5, 75).

1. Rise: 75 - 30 = 45 dollars.
2. Run: 5 - 2 = 3 hours.
3. Slope: 45/3 = 15.

That 15 is not an abstract steepness; it is 15 DOLLARS PER HOUR, the pay rate. Miles per gallon, points per game, cost per month: every "per" you have ever used was secretly a slope. A straight line means the rate never changes, which is why slope is called a constant rate of change.

Key idea: On a real-world graph, slope is the "per" number: how much y changes for each single unit of x.

Where you will see this

• Accessibility ramps: building codes cap wheelchair ramps near slope 1/12: 1 unit up per 12 across, a gentle, safe climb.
• Roads: a "6 percent grade" sign warns truckers of slope 6/100.
• Money: on a savings graph, slope is deposits per week; on a phone plan, cost per gigabyte.
• Science: speed is the slope of a distance-time graph; this exact idea becomes the star of physics and, later, calculus.

Common misconceptions

• Putting run on top. Slope is rise over run: y-changes on top, x-changes on the bottom. Up per across, never across per up.
• Mixing point order mid-formula. Whichever point is "second" must lead both subtractions. Consistent order, correct sign.
• Confusing zero and undefined. Horizontal means slope 0 (a floor); vertical means undefined (a wall).
• Thinking a bigger denominator means steeper. Slope 1/2 is gentler than slope 2. Compare by the value of the fraction, not by the size of any digit in it.
• Reading downhill as positive. Read lines left to right, like words; falling to the right means negative.

Try it

Find the slope of the line through (3, 4) and (7, 12). Then tell me the slope of the vertical line through (6, 0) and (6, 9).

Answer: rise = 12 - 4 = 8; run = 7 - 3 = 4; slope = 8/4 = 2. The vertical line: run = 6 - 6 = 0, and dividing by zero cannot be done, so the slope is undefined: a wall, not a floor. Got both? You are ready to graph whole lines in the next lesson, where slope becomes your steering wheel.

Recap

• Slope measures steepness: rise (up-down change) over run (across change).
• From two points: subtract the y's, subtract the x's in the SAME order, divide.
• The subscripts in the slope formula are name tags for "first point" and "second point," not operations.
• Positive climbs, negative falls, zero is flat, vertical is undefined.
• With labeled axes, slope is the rate: dollars per hour, miles per gallon, the everyday "per."

Sources

• OpenStax. (2020). Understand slope of a line (Section 4.4). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Linear equations and graphs [Unit]. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Lines. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Slope (gradient) of a straight line. Math is Fun. mathsisfun.com

The big picture

Today two worlds you have already visited, equations and the coordinate plane, meet each other, and the meeting is friendlier than you might fear. Here is the whole idea in one line: an equation like y = 2x + 1 is a RULE, and its graph is a PHOTOGRAPH of every single pair of numbers that obeys the rule. The rule says "whatever x is, double it and add 1 to get y." The photo shows every (x, y) couple that follows it, and for equations like ours the photo always comes out as a perfectly straight line.

You already know how to plot points, and you already know slope. Today we snap the photo. Three different cameras, all easy, and you get to pick your favorite.

Key idea: A graph is the picture of every point that makes the equation true.

Camera 1: a table of values

The most honest method: pick some x's, compute their y's, plot the couples. Let us photograph y = 2x + 1.

1. Pick an easy x, say x = 0. The rule says: double it (0), add 1. So y = 1. Our first couple: (0, 1).
2. Pick x = 1. Double it (2), add 1: y = 3. Couple: (1, 3).
3. Pick x = 2. Double it (4), add 1: y = 5. Couple: (2, 5).
4. Plot the three dots: (0, 1), (1, 3), (2, 5).
5. Look: they line up perfectly. Lay a ruler through them, draw the line, and put small arrows on both ends (the rule keeps going forever in both directions).

What we just did: asked the rule three questions, plotted three answers, connected them. The dots will ALWAYS line up for equations like this; that is exactly what "linear equation" means. If one dot ever refuses to line up, it is not a broken rule, it is a small arithmetic slip in that dot, and the picture itself just told you which one to recheck. The graph proofreads your work for free.

Key idea: Two points make a line; a third point is your built-in error check.

Camera 2: slope-intercept form, the express lane

Most line equations you meet are dressed like this:

y = mx + b

Read it aloud: "y equals m times x, plus b." The two letters are a starting point and a set of directions:

• b is the y-intercept: the spot where the line crosses the y-axis. It is your FRONT DOOR, the place the line starts its walk, always at the point (0, b). Why? Set x = 0 and the rule gives y = b, every time.
• m is the slope: your walking directions from the front door. Rise over run, exactly as last lesson.

So y = 2x + 1 says: front door at (0, 1), then climb 2 for every 1 across. Check it against our table: (0,1), then (1,3), then (2,5): up 2, over 1 each time. The two cameras agree, and they always will.

Practice reading a few out loud:

• y = 4x - 3: slope m = 4, front door b = -3, at the point (0, -3). Watch the sign: minus 3 means the door is below the origin.
• y = -x + 3: slope m = -1 (a lonely minus means -1), door at (0, 3).
• y = (1/2)x: slope 1/2, and no +b written means b = 0: the door is the origin itself.

Graphing with slope-intercept, step by step

Let us graph y = (2/3)x - 1 together, slowly.

1. Find the front door: b = -1, so put a dot at (0, -1) on the y-axis.
2. Read the directions: m = 2/3, so rise 2, run 3.
3. Stand on (0, -1). Climb 2 up, walk 3 right. You are at (3, 1). Dot.
4. Do it once more from (3, 1): up 2, right 3 lands at (6, 3). Dot.
5. Ruler through the dots, arrows on the ends. Done.

That is the whole express lane: door, directions, walk. Two small notes that save headaches. First, when the slope is negative, the rise goes DOWN: for y = -2x + 4, start at (0, 4), then go down 2 and right 1 to reach (1, 2). Second, when the slope is a whole number like 3, read it as 3/1: up 3, right 1.

Try it: describe how you would graph y = 3x - 2.

Answer: front door at (0, -2). Slope 3 means 3/1: from the door, up 3 and right 1 to (1, 1), then again to (2, 4). Ruler, arrows, done. If you said that, you have the express lane license.

Camera 3: the two intercepts

Some equations arrive dressed differently, like 2x + 3y = 6 (this outfit is called standard form). You could rearrange it into y = mx + b, but there is a faster trick: find where the line crosses each axis, and those two crossings are all you need.

1. Find the y-intercept: on the y-axis, x is 0. Cover the x-term with your finger: 3y = 6. Divide by 3: y = 2. Crossing: (0, 2).
2. Find the x-intercept: on the x-axis, y is 0. Cover the y-term: 2x = 6. Divide by 2: x = 3. Crossing: (3, 0).
3. Plot (0, 2) and (3, 0), lay the ruler, draw.

What we just did: set one letter to zero, solved a one-step equation, twice. The cover-up trick works because a crossing point always has a zero in it: every point ON the y-axis has x = 0, and every point on the x-axis has y = 0. (If that sentence feels familiar, it is the axis fact from the coordinate plane lesson, now earning its keep.)

Key idea: To use intercepts: set x = 0 and solve, set y = 0 and solve, plot the two crossings, connect.

The two special lines

Two equations look so short that they worry people, so let us take them slowly.

y = 3 has no x anywhere. It says: "whatever x you pick, y is 3. No exceptions." So (0, 3), (2, 3), (-5, 3) are all on it. Every point at height 3: a perfectly flat horizontal line, slope 0, a floor three units up.

x = 2 says: "x is 2, no matter what y is." So (2, 0), (2, 5), (2, -4) all belong. Every point standing on x = 2: a vertical line, a wall, slope undefined.

Where people get stuck: reading x = 2 as the single point "2" or as a dot at (2, 0). It is a whole line, infinitely many points tall. The equation only pins down x; the y is free to be anything, and "anything" is what draws the wall. A quiet memory hook: the equation y = number draws a floor, the equation x = number draws a wall, each one perpendicular to its own axis... read that once more, and picture the floor and the wall.

Choosing your camera

• Equation looks like y = mx + b? Use the express lane: door, directions, walk.
• Equation looks like ax + by = c? Cover-up trick: two intercepts, two dots, done.
• Feeling unsure, or the numbers are strange? A table of values never fails, and the third point proofreads you.

Where you will see this

• Money plans: a gym costing 25 dollars to join plus 15 per month graphs as y = 15x + 25: door at (0, 25) is the joining fee, slope 15 is the monthly rate. The picture makes the plan legible at a glance.
• Science class: converting Celsius to Fahrenheit, F = 1.8C + 32, is a line with door 32 and slope 1.8.
• Budgeting a mix: spending exactly 6 dollars on apples (2 each) and bananas (3 each) is 2x + 3y = 6, the very line we drew by intercepts.

Common misconceptions

• Swapping m and b. In y = 2x + 1, the slope is the number ATTACHED to x (2), and the door is the loose number (1). Attached walks, loose starts.
• Walking the slope sideways first. Rise is vertical, run is horizontal: up (or down) first, then across, from the door.
• Dropping the sign of b. y = 4x - 3 has its door at (0, -3), below the origin, not at (0, 3).
• Treating x = 2 as a point. It is a vertical line through (2, 0), a wall, with undefined slope.
• Mixing intercepts. The x-intercept is where y = 0 (on the x-axis), and the y-intercept is where x = 0. Set the OTHER letter to zero.

Try it

Graph 3x + y = 3 using the intercept camera, and then name the slope and y-intercept of y = -x + 5.

Answer: intercepts: set x = 0, cover the x-term: y = 3, giving (0, 3). Set y = 0: 3x = 3, so x = 1, giving (1, 0). Plot both, connect, arrows. For y = -x + 5: the slope is -1 (the quiet coefficient on x) and the door is (0, 5). From the door you would step down 1, right 1, again and again. If both of those made sense, you can now graph any line this course will ever hand you. That is not a small thing; enjoy it.

Recap

• A graph shows every (x, y) pair that makes the equation true; linear equations photograph as straight lines.
• Table of values: pick x's, compute y's, plot; the third dot is your error check.
• Slope-intercept y = mx + b: door at (0, b), then walk the slope m: rise, then run.
• Standard form: cover-up trick gives both intercepts; two dots make the line.
• y = number is a floor (slope 0); x = number is a wall (undefined slope).

Sources

• OpenStax. (2020). Graph linear equations in two variables (Section 4.2). In Elementary Algebra 2e. openstax.org
• OpenStax. (2020). Use the slope-intercept form of an equation of a line (Section 4.5). In Elementary Algebra 2e. openstax.org
• Dawkins, P. (n.d.). Lines. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Equation of a straight line. Math is Fun. mathsisfun.com

The big picture

Welcome back. Last lesson you turned equations into pictures. Today we walk the same road in the other direction: someone hands you FACTS about a line (its steepness, a point it passes through, maybe two points) and you write its equation. It is like tasting a dish and writing down the recipe. And here is the calming secret before we start: every single problem in this lesson is a fill-in-the-blank. The blank form is always

y = mx + b

and the whole game is finding the two numbers, m and b, that belong to your line. Two blanks. That is all we are ever hunting for. When a problem looks busy, remember: "I need m, and I need b," and the fog lifts.

Key idea: Writing a line's equation means filling exactly two blanks: the slope m and the y-intercept b.

Case 1: they hand you both blanks

Sometimes the problem is this kind: "a line has slope 2 and y-intercept 5." Both blanks are already in your hands.

1. m = 2, so write 2 in the slope seat: y = 2x + b.
2. b = 5, so write 5 in the door seat: y = 2x + 5.

Done. Truly done. One care point: when b is negative, keep its sign. Slope 3 and y-intercept -4 make y = 3x - 4, and the minus belongs to the 4.

Reading the blanks off a GRAPH is the same case in disguise: find where the line crosses the y-axis (that crossing height is b), then count rise over run between two clean grid points (that is m). Cross at (0, 1) and climb 2 for every 1 across? y = 2x + 1. You are reading the line's diary.

Case 2: a slope and one point (the door is hidden)

Now the classic: "a line has slope 3 and passes through (2, 11)." We have m, but the given point is NOT the door, so b is hiding. Take it slowly; the trick is lovely.

Road A: plug in what you know, solve for what you do not.

1. Start with the form: y = mx + b.
2. Put in the slope we know: y = 3x + b.
3. Here is the key idea: the point (2, 11) is ON the line, so it must OBEY the equation. Put x = 2 and y = 11 into their seats: 11 = 3(2) + b.
4. Work the multiply: 3 times 2 is 6. So 11 = 6 + b.
5. Solve the little equation: subtract 6 from both sides: b = 5.
6. Both blanks are filled now. Write the answer: y = 3x + 5.

Check it: does (2, 11) obey? 3 times 2 is 6, plus 5 is 11. Yes. Notice what you used: only equation-solving moves you have owned since Module 2. Nothing new happened, and yet you found a hidden y-intercept. Take the win.

Road B: point-slope form, the ready-made template. Some people prefer a form built for exactly this situation:

y - y1 = m(x - x1)

Read it aloud: "y minus the point's y equals the slope times x minus the point's x." The name tags x₁ and y₁ mean "the known point." Same problem: point (2, 11), slope 3.

1. Fill the template: y - 11 = 3(x - 2). (Honestly, at this moment you are already finished; this IS an equation of the line. But let us polish it into y = mx + b.)
2. Distribute the 3: y - 11 = 3x - 6.
3. Add 11 to both sides: y = 3x + 5.

Same answer as Road A, and it always will be. Use whichever road feels steadier under your feet; they are two staircases to the same floor.

Try it: write the equation of the line with slope 2 through the point (3, 10).

Answer: y = 2x + b. Feed it the point: 10 = 2(3) + b, so 10 = 6 + b, so b = 4. The line is y = 2x + 4. Check: 2 times 3 plus 4 is 10. If you got it, Case 2 is yours, and Case 3 is about to fall like a domino.

Case 3: two points, no slope given

"Write the equation of the line through (1, 4) and (3, 10)." Neither blank is handed to us, but you own a slope formula, so we earn m first and then we are back in Case 2.

1. Find the slope from the two points: rise = 10 - 4 = 6, run = 3 - 1 = 2, so m = 6/2 = 3.
2. Now it is Case 2: slope 3 through (either) point; take (1, 4). Write y = 3x + b.
3. Feed it the point: 4 = 3(1) + b.
4. 3 times 1 is 3, so 4 = 3 + b, and subtracting 3 gives b = 1.
5. Write the line: y = 3x + 1.

Check it with BOTH points, because both must obey: for (1, 4): 3 + 1 = 4, true. For (3, 10): 9 + 1 = 10, true. Both witnesses agree, so the equation is certainly right. (And a soothing fact: had you fed the other point in step 3, b would still come out 1. Either point works; the line does not care which of its residents you interview.)

Key idea: Two points? Slope first, then it is the one-point problem you already know.

Lines from real stories

This skill quietly runs the real world, because so many costs are "flat fee plus rate."

A plumber charges 50 dollars for the visit plus 40 per hour. The per-hour number is the slope (it repeats with every x) and the flat fee is the door (it happens once, at x = 0): y = 40x + 50. Three hours costs 120 + 50 = 170 dollars.

A burning candle is 10 cm tall one hour after lighting and 6 cm tall after three hours. Two data points: (1, 10) and (3, 6). Slope: (6 - 10)/(3 - 1) = -4/2 = -2, meaning it burns 2 cm PER hour (negative because the height falls). Then b: 10 = -2(1) + b gives b = 12, the candle's full height at lighting. The story-line: y = -2x + 12. From two measurements you recovered the whole history and future of that candle, including that it dies at x = 6 hours. That is the power you have been building toward.

A short word on parallel lines

Parallel lines are lines with the SAME slope (same tilt, never meeting, like train rails). To write a line parallel to y = 2x + 1 through the door (0, 7): keep the slope, change the door: y = 2x + 7. Perpendicular lines (meeting at a perfect corner) use the flipped-and-sign-switched slope: perpendicular to slope 2 is slope -1/2. You will practice this more later; for now, "parallel means same m" is the sentence to keep.

Common misconceptions

• Putting the numbers in the wrong seats. The slope m is glued to x; the intercept b stands alone. "Slope 3, intercept 5" is y = 3x + 5, never y = 5x + 3.
• Dropping a sign while solving for b. In 11 = 6 + b, subtracting 6 gives b = 5; in 4 = 6 + b it gives b = -2, minus and all. Go one small line at a time.
• Using an x-intercept as b. b is the crossing of the Y axis. A line through (3, 0) does NOT have b = 3 unless that crossing is on the y-axis.
• Believing the two-point problem is a new kind of problem. It is slope-formula plus Case 2, stapled together.
• Skipping the final check. Feeding your finished equation the given point(s) takes seconds and certifies the answer. Both points must obey.

Try it

Write the equation of the line through (2, 3) and (4, 7), and check it.

Answer: slope: rise = 7 - 3 = 4, run = 4 - 2 = 2, so m = 2. Then feed a point: 3 = 2(2) + b, so 3 = 4 + b, so b = -1 (careful: 3 minus 4 is negative 1). The line: y = 2x - 1. Check both witnesses: (2, 3): 4 - 1 = 3, true; (4, 7): 8 - 1 = 7, true. If your b came out +1, you met the classic sign slip; run the subtraction once more, slowly, and it will show itself. Finding your own slip is not a stumble, it is the skill.

Recap

• Every problem here fills two blanks: m and b in y = mx + b.
• Given both: write them in, mind the signs.
• Given slope and a point: plug the point in and solve for b, or use point-slope y - y₁ = m(x - x₁).
• Given two points: find the slope first, then treat it as the one-point case with either point.
• In stories, the repeating "per" number is the slope and the one-time fee is b. Always check by substituting the given points.

Sources

• OpenStax. (2020). Find the equation of a line (Section 4.6). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Forms of linear equations [Unit]. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Lines. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Equation of a line from 2 points. Math is Fun. mathsisfun.com

The big picture

Settle in, and let us start with a riddle you could solve at a family dinner: "Two numbers add up to 7, and one is 3 bigger than the other. What are they?" Sit with it a moment... 5 and 2. If you got that, or even if you only nodded along, you already understand today's topic in your bones. You held TWO facts in mind at once and found the pair of numbers satisfying both.

A system of equations is exactly that: two equations about the same two mystery numbers, x and y, both of which must be true at the same time. Guessing works for dinner riddles; today we build methods that work even when the answer is something unguessable. We will learn two: drawing (graphing) and a clever note-passing trick called substitution. Small steps, as always, and the first method is one you can literally see.

Key idea: A system is two equations that must BOTH be true; its solution is the pair (x, y) that satisfies both at once.

What counts as a solution

A solution to a system is a PAIR of numbers, one for x and one for y, written as a point (x, y). And there is a simple citizenship test: the pair must make BOTH equations true. Making one true is not enough.

Let us test whether (2, 3) (that is, x = 2, y = 3) solves this system:

y = x + 1 and y = -x + 5

1. First equation: is 3 equal to 2 + 1? Yes, 3 = 3. Passed.
2. Second equation: is 3 equal to -2 + 5? Yes, 3 = 3. Passed.

Both passed, so (2, 3) is THE solution. If even one had failed, the pair would be rejected. This checking skill is humble, but it is also your safety net for everything ahead: any answer you ever find can be certified in ten seconds.

Key idea: Test a pair in BOTH equations. Two passes and it is the solution; one failure and it is not.

Method 1: solve by graphing

Remember what a graph is: the picture of EVERY point that makes an equation true. So each equation in a system draws its own line, its own club of points. A point that satisfies BOTH equations must belong to both clubs, and there is exactly one place where both lines usually stand: their crossing point.

Let us solve the system above by graphing, step by step.

1. Graph y = x + 1: door at (0, 1), slope 1, so up 1 and right 1: through (1, 2), (2, 3), and so on.
2. Graph y = -x + 5 on the same axes: door at (0, 5), slope -1, so down 1 and right 1: through (1, 4), (2, 3)...
3. Look for the crossing. The two lines meet at exactly one point: (2, 3).
4. Certify it: we already tested (2, 3) in both equations above. Citizen of both clubs.

What we just did: drew both rules and let the picture do the intersection for us. When two lines cross, the crossing point is the one pair in the whole plane loyal to both equations.

Graphing is wonderfully visual, and it has one honest weakness: if the answer is something like (7/3, -5/6), no human eye reads that off a sketch. For exact answers, we need algebra. Enter the note-passing trick.

Method 2: substitution, the note-passing trick

Here is the idea in plain words. Suppose one equation flat-out TELLS you what y is: y = x + 1. That is a note that says "wherever you see y, it is worth x + 1." Substitution means carrying that note into the OTHER equation and swapping y out. Suddenly the other equation has only x in it, and a one-letter equation is old, familiar ground.

Solve the system: y = x + 1 and x + y = 5.

1. The first equation hands us the note: y is x + 1.
2. Walk into the second equation and replace y with the note. x + y = 5 becomes x + (x + 1) = 5. The parentheses show exactly where the note was pasted in.
3. Tidy the left side: x plus x is 2x, so 2x + 1 = 5.
4. An old friend: a two-step equation. Subtract 1 from both sides: 2x = 4.
5. Divide both sides by 2: x = 2.
6. Halfway there; a solution is a PAIR. Find y using the easiest equation, the note itself: y = x + 1 = 2 + 1 = 3.
7. The solution is (2, 3).

Check it: first equation: 3 = 2 + 1, true. Second: 2 + 3 = 5, true. And notice: this is the same system we solved by graphing, and the same (2, 3) appeared, this time with no drawing at all. Two roads, one truth. That agreement is your evidence that substitution can be trusted even when you cannot see the lines.

Where people get stuck: pasting the note back into the SAME equation it came from. Substituting y = x + 1 into itself gives x + 1 = x + 1, which melts into 0 = 0 and tells you nothing. The note must travel to the OTHER equation; it has nothing new to say at home.

One more, with the note needing writing first

Solve: x + y = 7 and 2x + 3y = 18.

Neither equation hands us a ready note, so we write one ourselves. Choose the letter that is easiest to free: in x + y = 7, the x has no number stuck to it, so isolating it is one step.

1. Solve the first equation for x: subtract y from both sides: x = 7 - y. There is our note: "x is worth 7 minus y."
2. Carry it into the OTHER equation: 2x + 3y = 18 becomes 2(7 - y) + 3y = 18. Keep the parentheses; the 2 must multiply the WHOLE note.
3. Distribute the 2: 2 times 7 is 14, and 2 times -y is -2y: 14 - 2y + 3y = 18.
4. Tidy: -2y + 3y is 1y, so 14 + y = 18.
5. Subtract 14 from both sides: y = 4.
6. Find the partner with the note: x = 7 - y = 7 - 4 = 3.
7. Solution: (3, 4).

Check it: 3 + 4 = 7, true; 2(3) + 3(4) = 6 + 12 = 18, true. Beautiful. If dropping those parentheses in step 2 worried you, good: that instinct is exactly right. The parentheses are the seat belt of substitution.

Try it: solve the system y = 2x and x + y = 9.

Answer: the note says y is 2x. Paste into the other equation: x + 2x = 9, so 3x = 9, so x = 3. Then y = 2(3) = 6. Solution: (3, 6). Check: 6 = 2 times 3, true; 3 + 6 = 9, true. If you found both coordinates and checked, you are doing real algebra at full stride now.

When the lines refuse to cooperate

Two special endings, and you have met their fingerprints before in Module 2.

Parallel lines: no solution. Solve y = 2x + 1 and y = 2x + 5 by substitution: 2x + 1 = 2x + 5. Subtract 2x from both sides: 1 = 5. A lie, with no x left to fix it. The picture agrees: two lines with the same slope 2 but different doors are parallel rails that never cross. No crossing, no solution, and the honest answer is "no solution."

The same line twice: infinitely many. If the algebra melts into a truth like 4 = 4 or 0 = 0, the two equations were the same line in different outfits, and every point on that line solves the system: infinitely many solutions.

Key idea: A lie like 1 = 5 means parallel lines, no solution. A bare truth like 0 = 0 means one line twice, infinitely many.

Where you will see this

• The dinner riddle, formally: x + y = 7 and x = y + 3. Substitute: (y + 3) + y = 7, so 2y + 3 = 7, y = 2, x = 5. The numbers are 5 and 2, now provable instead of guessed.
• Break-even points: a lemonade stand paying 12 dollars for supplies and earning 2 per cup: cost y = 12, income y = 2x cross at x = 6 cups, the break-even.
• Comparing plans: two phone plans or streaming bundles cross where their cost lines meet; before the crossing one wins, after it the other does.

Common misconceptions

• Giving only x as the answer. A system's solution is a pair. Finding x = 2 is half the job; go collect y.
• Substituting into the same equation. The note teaches nothing at home; it must visit the other equation.
• Dropping parentheses. Substituting 7 - y into 2x means 2(7 - y): the whole note gets multiplied, both terms.
• Reading (x, y) backwards. The pair (3, 4) means x = 3 and y = 4, in that order, always.
• Treating 1 = 5 as a personal failure. It is information: the lines are parallel and the system has no solution. Answer with those words, full credit, head high.

Try it

Solve the system y = x - 1 and x + 2y = 7, and check your pair in both equations.

Answer: paste the note y = x - 1 into the second equation: x + 2(x - 1) = 7. Distribute the 2: x + 2x - 2 = 7. Tidy: 3x - 2 = 7. Add 2 to both sides: 3x = 9. Divide by 3: x = 3. Collect the partner: y = 3 - 1 = 2. Solution: (3, 2). Check: 2 = 3 - 1, true; 3 + 2(2) = 3 + 4 = 7, true. Both clubs accept the pair. If your parentheses held in that second step, you have mastered the one place this method ever wobbles.

Recap

• A system is two equations sharing x and y; the solution is the pair making both true.
• By graphing: each equation is a line, and the crossing point is the solution you can see.
• By substitution: solve one equation for a letter, paste that note into the OTHER equation, solve, then find the partner coordinate.
• Parentheses around the pasted note protect every term.
• A leftover lie means parallel lines, no solution; a leftover truth means the same line, infinitely many.

Sources

• OpenStax. (2020). Solve systems of equations by graphing (Section 5.1). In Elementary Algebra 2e. openstax.org
• OpenStax. (2020). Solving systems of equations by substitution (Section 5.2). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Systems of equations [Unit]. In Algebra 1. khanacademy.org
• Pierce, R. (n.d.). Systems of linear equations. Math is Fun. mathsisfun.com

The big picture

One more method for systems, and it may become your favorite, because you have already used it in real life without knowing its name. Picture two coffee-shop receipts. Monday: 2 coffees and 1 muffin cost 7 dollars. Tuesday: 2 coffees and 3 muffins cost 13 dollars. Same coffees both days; the ONLY difference is 2 extra muffins and 6 extra dollars. So 2 muffins cost 6 dollars, and a muffin is 3. You did not graph anything, and you did not substitute anything. You SUBTRACTED one receipt from the other, and a variable (the coffees) canceled itself out of the conversation.

That everyday move is called elimination, and today we turn it into a reliable method. If the last lesson's note-passing felt fiddly, take heart: many people find elimination the most soothing method of all, because it is mostly neat columns and tidy arithmetic.

Key idea: Add or subtract whole equations so one letter cancels out and disappears.

Why adding equations is legal

First, the permission slip. An equation says two things are equal: in x + y = 7, the left pile and the number 7 weigh the same. Now, if you have two balanced scales and you pour everything from both left pans into one pan, and everything from both right pans into another, the big combined scale still balances: equal added to equal stays equal. That is all "adding two equations" means: add the left sides, add the right sides, and the result is a new true equation.

Adding is always LEGAL. It is only USEFUL when the addition makes a letter vanish, and that happens when the letter's coefficients are opposites, like +y and -y. Watching for opposites is the whole art.

Worked example 1: the setup is a gift

Solve: x + y = 7 and x - y = 3. (You may recognize the dinner riddle from last lesson: two numbers that add to 7 and differ by 3.)

1. Stack the equations so x sits under x, y under y, numbers under numbers.
2. Look at the y column: +y on top, -y below. Opposites! Adding will erase them.
3. Add the columns one at a time. x plus x is 2x. Then +y plus -y is 0, gone. Then 7 plus 3 is 10.
4. The new equation is 2x = 10.
5. Divide both sides by 2: x = 5.
6. A system's answer is a pair, so find y. Drop x = 5 into the friendlier original: 5 + y = 7, so y = 2.
7. Solution: (5, 2).

Check it in BOTH originals: 5 + 2 = 7, true. 5 - 2 = 3, true. The riddle that took guessing at dinner took four lines here. That is elimination on a good day: the problem practically solves itself.

Try it: solve x + y = 10 and x - y = 4.

Answer: add the equations: the y's cancel, leaving 2x = 14, so x = 7. Then 7 + y = 10 gives y = 3. Solution (7, 3). Check: 7 + 3 = 10 and 7 - 3 = 4, both true. See how quickly that went? That speed is why this method earns its keep.

Worked example 2: making your own opposites

Life is not always so generous. Solve: 2x + 3y = 12 and x - y = 1.

Stack them and scan the columns: 2x over x, 3y over -y. Nothing cancels as-is. But remember a legal move from Module 2: you may multiply BOTH SIDES of an equation by any number. So we manufacture opposites ourselves.

1. Target the y column: 3y on top, -y below. If the bottom y became -3y, we would have opposites.
2. Multiply the ENTIRE second equation by 3, every term, both sides: x times 3 is 3x, -y times 3 is -3y, and 1 times 3 is 3. The second equation becomes 3x - 3y = 3. (It is still the same line, wearing a bigger coat.)
3. Stack and add: 2x + 3x is 5x. Then 3y + (-3y) is 0, gone. Then 12 + 3 is 15.
4. New equation: 5x = 15. Divide both sides by 5: x = 3.
5. Find the partner: drop x = 3 into x - y = 1: 3 - y = 1. Subtract 3 from both sides: -y = -2, so y = 2.
6. Solution: (3, 2).

Check it: 2(3) + 3(2) = 6 + 6 = 12, true. 3 - 2 = 1, true. That multiply-then-add rhythm handles most systems you will ever meet.

Where people get stuck: multiplying only the letters and forgetting the right side. When the second equation was scaled by 3, the 1 became 3 too. EVERY term rides the multiplication, both sides of the scale, or the equation stops being true. Before adding, glance at your multiplied equation and ask, "did every single term get the memo?"

Worked example 3: when both equations need scaling

Solve: 3x + 2y = 16 and 2x + 5y = 18.

No column cancels, and no single multiplication fixes it, because 3 and 2 do not divide each other. The move: scale BOTH equations toward a common target, and aim for opposite signs so we can ADD (adding is gentler on the nerves than subtracting; fewer sign slips).

1. Target the x column: 3x and 2x. A common meeting size is 6x. Multiply the first equation by 2: 6x + 4y = 32. Multiply the second by -3 (the minus is on purpose, to make opposites): -6x - 15y = -54.
2. Stack and add: 6x + (-6x) is 0, gone. Then 4y + (-15y) is -11y. Then 32 + (-54) is -22.
3. New equation: -11y = -22. Divide both sides by -11: y = 2.
4. Find the partner: 3x + 2(2) = 16, so 3x + 4 = 16, so 3x = 12, so x = 4.
5. Solution: (4, 2).

Check it: 3(4) + 2(2) = 12 + 4 = 16, true. 2(4) + 5(2) = 8 + 10 = 18, true. That was the heaviest lifting elimination ever asks of you, and you just watched it done in five steps. Everything else is lighter than this.

Key idea: Scale one or both equations so one column holds opposites, then add and watch that column vanish.

The familiar special endings

Elimination meets the same two odd endings as every other method, and they mean the same things. If the letters all cancel and a LIE remains, like 0 = 8, the lines are parallel: no solution. If the letters all cancel and a TRUTH remains, like 0 = 0, the two equations were the same line: infinitely many solutions. Neither ending is an error; both are answers, and now you can name them on sight.

Choosing your method like a mechanic chooses a wrench

• Graphing: best when you want to SEE the situation or an estimate is fine. Slow for exact, fraction-flavored answers.
• Substitution: best when a letter is already alone, like y = 2x + 1, or nearly alone. The note is pre-written.
• Elimination: best when both equations are lined up as ax + by = c, especially if a column already holds opposites or one small multiplication makes it so.

Here is the freeing truth: every method works on every system. There is no wrong choice, only slower and faster ones. With practice your eye will pick the cheap wrench in about two seconds, and if you ever pick the slow one, you still arrive at the same answer.

Key idea: Letter already alone: substitute. Equations lined up in columns: eliminate. Want a picture: graph.

Where you will see this

• Two receipts: our muffins-and-coffee opener is a real technique for splitting bundled prices apart.
• Ticket sales: 3 adult and 2 child tickets for 33 dollars, 1 adult and 2 child for 19: subtract and the child tickets cancel, giving 2a = 14, adult 7, child 6.
• Science and business: mixing solutions, balancing supply and demand, splitting a lump total into parts; two unknowns, two facts, one elimination.

Common misconceptions

• Multiplying only part of an equation. Every term, both sides, no exceptions. The right-hand number is the most-forgotten passenger.
• Sign slips while subtracting. Subtracting an equation means subtracting EVERY term of it. If that feels risky, multiply by -1 first and ADD instead; opposites-and-add is the calmest road.
• Stopping at one coordinate. x = 3 is half an answer. Drop it into an original equation and collect y.
• Adding when nothing cancels. Legal, but it produces a third equation no simpler than the first two. Scale FIRST so the addition earns its keep.
• Fearing the vanished letters. 0 = 8 and 0 = 0 are findings, not failures: no solution, and infinitely many, respectively.

Try it

Solve the system 3x + y = 11 and 2x - y = 4.

Answer: scan the columns: +y over -y, ready-made opposites, a gift. Add: 3x + 2x is 5x, the y's cancel, 11 + 4 is 15. So 5x = 15, and x = 3. Partner: 3(3) + y = 11 gives 9 + y = 11, so y = 2. Solution: (3, 2). Check both: 9 + 2 = 11, true; 6 - 2 = 4, true. If that felt almost easy, believe the feeling: you now hold three working methods, and no system of two lines can hide from you.

Recap

• Adding two true equations gives a true equation; do it when a column of opposites will cancel.
• No opposites? Multiply one or both entire equations (every term!) to manufacture them, then add.
• After one letter falls, solve the leftover one-letter equation and substitute back for the partner.
• 0 = lie means no solution; 0 = 0 means infinitely many, same as always.
• Substitute when a letter is alone; eliminate when equations line up; graph to see. All roads meet at the same pair.

Sources

• OpenStax. (2020). Solve systems of equations by elimination (Section 5.3). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Systems of equations [Unit]. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Linear systems with two variables. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Systems of linear equations. Math is Fun. mathsisfun.com

The big picture

This lesson has a reputation for being a wall of rules to memorize, and we are not going to treat it that way. Here is the promise instead: every exponent law is a pattern you can DISCOVER by writing out a few multiplications and counting. Once you have seen where each rule comes from, you will not need to memorize much of anything, because you can rebuild any rule you forget in ten seconds.

First, what an exponent even is: humans got tired of writing 2 × 2 × 2 × 2 × 2, so they invented a shorthand: 2⁵. That is the entire origin story. Exponents are laziness, organized beautifully.

Key idea: An exponent counts how many copies of a number are being multiplied together.

Reading the notation aloud

Look at 23. Read it aloud: "2 to the power 3," or "2 cubed." The big number, 2, is the base: the thing being copied. The small raised number, 3, is the exponent: how many copies get multiplied. So 2³ means 2 × 2 × 2, which is 8.

Where people get stuck, right at the doorway: 2³ is NOT 2 times 3. It is not 6. The exponent does not multiply the base; it counts copies OF the base. 2 × 3 = 6, but 2³ = 8. Say it once out loud: "the little number counts copies." That one sentence prevents the most common exponent error on earth.

A few more to make it familiar: 5² is 5 × 5 = 25 ("5 squared"). 10⁴ is 10 × 10 × 10 × 10 = 10,000. And x² is x times x, whatever x turns out to be.

Rule 1: multiplying same-base powers (discover it)

What is 2² × 2³? Do not reach for a rule; write out the copies.

1. 2² is (2 × 2). Two copies.
2. 2³ is (2 × 2 × 2). Three copies.
3. Multiplied together: (2 × 2) × (2 × 2 × 2). Count all the 2s: five of them.
4. Five copies of 2 is 2⁵, which is 32.

Where did the 5 come from? From 2 + 3: two copies joined three copies. That is the whole rule, and now you own it because you watched it happen:

am × an = am+n

Read it aloud: "same base multiplied: keep the base, ADD the exponents." So x⁴ × x⁵ = x⁹. One warning label: the bases must MATCH. The rule says nothing about 2² × 3⁴; different bases do not pool their copies.

Rule 2: dividing same-base powers (discover it)

What is 2⁵ ÷ 2²? Write the copies as a fraction:

1. Top: 2 × 2 × 2 × 2 × 2. Bottom: 2 × 2.
2. Cancel pairs, one top 2 with one bottom 2, twice, the same way you simplify any fraction.
3. Left over on top: three 2s. So the answer is 2³ = 8.

Five copies minus two copies left three copies. The rule:

am ÷ an = am-n

"Same base divided: keep the base, SUBTRACT the exponents." So y⁷/y³ = y⁴.

Rule 3: a power of a power (discover it)

What is (2³)²? The outside exponent 2 says "two copies of the thing in parentheses."

1. Two copies of 2³: that is 2³ × 2³.
2. By Rule 1, add: 2³⁺³ = 2⁶ = 64.
3. And 3 + 3 is the same as 3 × 2. Copies of copies multiply.

(am)n = am×n

"A power raised to a power: keep the base, MULTIPLY the exponents." So (x²)⁴ = x⁸.

Feeling the pattern? Each rule's arithmetic is one step gentler than what you see: multiplication of powers becomes addition, division becomes subtraction, and power-of-power becomes multiplication. When in doubt, write out three copies and count; the rule will reintroduce itself.

Try it: simplify x⁴ × x⁵, and then (x³)⁴.

Answer: multiplying same base adds: x⁹. Power of a power multiplies: x¹². If you hesitated over which was which, that is normal this week; the add-versus-multiply distinction settles in with a little practice, and the count-the-copies habit is your safety rope.

The zero exponent: ride the elevator down

What could 2⁰ possibly mean? Zero copies of 2? Watch the pattern instead; it is like riding an elevator down one floor at a time. Each time the exponent drops by 1, the value gets divided by 2:

• 2³ = 8
• 2² = 4 (divided by 2)
• 2¹ = 2 (divided by 2)
• 2⁰ = 1 (divided by 2 again: 2 ÷ 2 = 1)

The pattern lands on 1, not 0. So any nonzero number to the power zero is 1. 5⁰ = 1, 100⁰ = 1, x⁰ = 1. A second witness, using Rule 2: 2³ ÷ 2³ should be 2^3-3 = 2⁰, and dividing anything by itself gives 1. Both roads say the same thing, so we can trust it.

Negative exponents: keep riding down

Do not get off the elevator; ride below the ground floor. Keep dividing by 2:

• 2⁰ = 1
• 2^-1 = 1/2 (1 divided by 2)
• 2^-2 = 1/4 (divided by 2 again)
• 2^-3 = 1/8

A negative exponent did not make anything negative. It made things FLIP into fractions. Read the rule aloud: "a negative exponent means take the reciprocal," that is, one over the positive power:

a-n = 1/an

So 3^-2 = 1/3² = 1/9. Positive one-ninth. The minus sign in an exponent means "flip," never "below zero." Read that once more, because it undoes years of gut instinct: negative exponent means flip, not negative number.

One more pattern: a power of a product

What is (3x)²? Two copies of the whole package: (3x) × (3x) = 3 × 3 × x × x = 9x². Everyone inside the parentheses gets squared, the 3 included.

Compare it with 3x², no parentheses: there, the exponent touches only the x, so at x = 2, 3x² is 3 × 4 = 12, while (3x)² is 6² = 36. Different animals. The parentheses decide who gets raised, and this tiny difference is a favorite exam trap; now it is YOUR favorite, because you can spot it.

Key idea: An exponent applies exactly to what it touches: with parentheses, the whole package; without, only the nearest symbol.

Putting several rules together

Simplify (2x3)2 × x4 ÷ x5, one small step at a time.

1. Inside-out. The power of a product squares everything in the package: (2x³)² = 2² × (x³)² = 4x⁶.
2. Multiply by x⁴: same base, add exponents: 4x⁶⁺⁴ = 4x¹⁰.
3. Divide by x⁵: same base, subtract: 4x^10-5 = 4x⁵.

What we just did: three rules in a row, each doing one small job. Long exponent problems are never deep; they are short rules stacked, and you now own every rule in the stack.

Where you will see this

• Computers: memory sizes run on powers of 2: 2¹⁰ = 1024 is a kilobyte's worth of bytes; doubling storage means adding 1 to an exponent.
• Science notation: huge and tiny numbers travel as powers of 10: the sun is about 1.5 × 10⁸ km away; a virus is about 10^-7 meters wide. The negative exponent means a flip to a fraction, exactly as you learned today.
• Growth: doubling populations, compound interest, and viral spread all speak fluent exponents.

Common misconceptions

• 2³ = 6. The exponent counts copies, it does not multiply: 2 × 2 × 2 = 8.
• x² × x³ = x⁶. Multiplying same-base powers ADDS exponents: x⁵. The multiply-the-exponents move belongs to (x²)³ alone.
• Pooling different bases. 2³ × 5² has no shortcut; the laws need matching bases.
• x⁰ = 0. The elevator lands on 1. Any nonzero base to the zero power is 1.
• 3^-2 = -9. Negative exponents flip, they do not negate: 3^-2 = 1/9, a small positive number.

Try it

Simplify each: y⁷ ÷ y³, then 5⁰, then 3^-2.

Answer: dividing same base subtracts: y⁴. Any nonzero number to the zero power: 1. Negative exponent flips: 3^-2 = 1/3² = 1/9. Three for three? You have every law this course will ever ask of you. And if one slipped, remember the safety rope: write out the copies and count. The rules always come back.

Recap

• An exponent counts copies of the base being multiplied: 2³ = 2 × 2 × 2 = 8.
• Same base: multiply means ADD exponents; divide means SUBTRACT them.
• Power of a power: MULTIPLY exponents; a power of a product raises everything inside.
• Anything nonzero to the 0 power is 1; a negative exponent means flip to the reciprocal.
• Forget a rule? Write three copies and count. The pattern rebuilds itself.

Sources

• OpenStax. (2020). Use multiplication properties of exponents (Section 6.2). In Elementary Algebra 2e. openstax.org
• OpenStax. (2020). Integer exponents and scientific notation (Section 6.7). In Elementary Algebra 2e. openstax.org
• Dawkins, P. (n.d.). Integer exponents. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Laws of exponents. Math is Fun. mathsisfun.com

The big picture

The word "polynomial" sounds like advanced machinery, so let us defuse it immediately. It is built from Greek and Latin bits meaning "many terms," and that is all a polynomial is: an expression with one or more terms, like 3x2 + 2x + 1. Look closely at that thing. You already know every part of it: terms, coefficients, exponents.

You have been handling small polynomials this whole course without anyone announcing it. Today we learn to add them (which is coin-sorting), subtract them (coin-sorting with one careful minus), and multiply them (a handshake game). None of it is beyond you, and we will check our answers with real numbers as we go, so you never have to take my word for anything.

Key idea: A polynomial is a string of terms, and you already know how to handle terms.

The family names

A few names, so problems can talk to us:

• A monomial has one term: 5x², or 7, or -3x.
• A binomial has two terms: x + 3, or 2x² - 5.
• A trinomial has three terms: x² + 5x + 6.
• Polynomial is the family surname covering all of them, however many terms.

One more useful word: the degree of a polynomial is the highest exponent that appears on the variable. The degree of 4x³ + 2x - 7 is 3, because the 3 in x³ is the biggest exponent in sight. Degree matters because it tells you a polynomial's personality: degree 1 graphs as our familiar straight lines, and degree 2 makes the curves coming in the final module. We also like standard form: writing terms from highest exponent down to the plain number, like tallest to shortest in a class photo.

Key idea: Count terms for the name; find the highest exponent for the degree.

Adding polynomials: sort the coins

Imagine emptying two pockets of change onto a table. You would not add a dime to a penny and call it anything; you sort: dimes with dimes, pennies with pennies, then count each pile. Adding polynomials is exactly that. The x² terms are one kind of coin, the x terms another, the plain numbers a third. Only like terms combine.

Let us add (3x2 + 2x + 1) + (x2 + 4x + 5), one pile at a time.

1. Sort the x² coins: 3x² and x². Together: 4x². (A lonely x² means 1x².)
2. Sort the x coins: 2x and 4x. Together: 6x.
3. Sort the plain numbers: 1 and 5. Together: 6.
4. Write the piles in standard form: 4x2 + 6x + 6.

Check it with a real number, a habit that will serve you for years. Try x = 1: the first polynomial is 3 + 2 + 1 = 6; the second is 1 + 4 + 5 = 10; their sum should be 16. Our answer at x = 1: 4 + 6 + 6 = 16. It matches. One number does not prove everything, but it catches nearly every slip, and it costs ten seconds.

Subtracting: the minus visits every term

Subtraction is the same coin-sorting with ONE extra care point, and it is the same one from the equations module: a minus in front of parentheses flips the sign of EVERY term inside, not only the first.

Compute (5x2 + 3x) - (2x2 + 7x).

1. Distribute the minus through the second parentheses: -(2x²) is -2x², and -(+7x) is -7x. The problem becomes 5x2 + 3x - 2x2 - 7x.
2. Sort the x² coins: 5x² - 2x² = 3x².
3. Sort the x coins: 3x - 7x = -4x.
4. Answer: 3x2 - 4x.

Check: at x = 1: first is 8, second is 9, and 8 - 9 = -1. Our answer at 1: 3 - 4 = -1. True. The flip-every-sign step is where nearly all subtraction errors live; write the flipped version out fully, every time, and the errors have nowhere to hide.

Try it: add (2x + 3) + (4x - 5).

Answer: x coins: 2x + 4x = 6x. Numbers: 3 - 5 = -2. Answer: 6x - 2. Check at x = 1: 5 + (-1) = 4, and 6 - 2 = 4. Matches. You are sorting like a bank teller now.

Multiplying: start small

Monomial times monomial. Compute 3x² × 4x³: multiply the numbers (3 times 4 is 12), then the letters (x² times x³: same base, ADD exponents: x⁵). Answer: 12x⁵. Notice last lesson's exponent rule showing up for work; everything in algebra gets reused.

Monomial times polynomial. Compute 2x(3x + 4): distribute, exactly like clearing parentheses in equations: 2x times 3x is 6x², and 2x times 4 is 8x. Answer: 6x2 + 8x.

The handshake: binomial times binomial

Now the famous one: (x + 2)(x + 3). Here is the rule in one sentence: EVERY term in the first parentheses must multiply EVERY term in the second, like two small teams where each player shakes hands with each opponent. Two terms times two terms means 2 × 2 = 4 handshakes, no more, no less.

There is even a picture. A rectangle with width (x + 2) and height (x + 3) splits into four small rooms: an x-by-x room (x²), an x-by-3 room (3x), a 2-by-x room (2x), and a 2-by-3 room (6). The total area is the sum of the four rooms. The handshakes ARE the rooms.

Let us do the four handshakes slowly. People remember them with the nickname FOIL: First, Outer, Inner, Last.

1. First terms of each: x × x = x².
2. Outer pair (the two ends): x × 3 = 3x.
3. Inner pair (the two middles): 2 × x = 2x.
4. Last terms of each: 2 × 3 = 6.
5. Collect: x² + 3x + 2x + 6. Sort the coins: 3x + 2x = 5x.
6. Answer: x2 + 5x + 6.

Check with a number: at x = 1, the original is (1 + 2)(1 + 3) = 3 × 4 = 12. Our answer: 1 + 5 + 6 = 12. The handshake count was honest. Take the win; FOIL is a skill people fear, and you just verified your own answer without any answer key.

One more, with a minus lurking: (2x + 1)(x - 4).

1. First: 2x × x = 2x².
2. Outer: 2x × (-4) = -8x. The minus rides along.
3. Inner: 1 × x = x.
4. Last: 1 × (-4) = -4.
5. Collect: 2x² - 8x + x - 4. Sort: -8x + x = -7x.
6. Answer: 2x2 - 7x - 4.

Check: at x = 1: (3)(-3) = -9, and our answer gives 2 - 7 - 4 = -9. True.

The most famous trap in algebra

What is (x + 4)2? The tempting wrong answer is x² + 16, squaring each piece separately. Let us catch the trap with real numbers first: at x = 1, (1 + 4)² = 5² = 25. But 1 + 16 = 17. Not equal, so the tempting answer is wrong, and now we know it for certain.

The honest road: (x + 4)² means (x + 4)(x + 4), four handshakes: x² + 4x + 4x + 16 = x2 + 8x + 16. At x = 1: 1 + 8 + 16 = 25. Now it matches. The middle term, 8x, is the part the shortcut forgets, and it comes from the two cross handshakes. Squaring a sum needs the full handshake ceremony, every time.

Key idea: (a + b)² is (a + b)(a + b), four handshakes with a middle term. It is never a² + b².

Where you will see this

• Area: a garden x meters wide with 2 extra meters one way and 3 the other has area (x + 2)(x + 3): our exact example.
• Business: revenue is price times quantity; when both depend on x, revenue is a product of binomials.
• Next module: factoring, the star of the coming lessons, is literally this lesson run backwards, so every handshake you practice now pays double later.

Common misconceptions

• Combining unlike terms. x² + x cannot merge; dimes and pennies stay separate piles. And x² × x is different: multiplying gives x³; only ADDING needs like terms.
• Losing the minus in subtraction. The minus flips every sign in the second polynomial. Write the flipped line before sorting.
• Shaking only some hands. Two-term times two-term means exactly four products. If you have three, the Outer or Inner pair got skipped.
• (x + 4)² = x² + 16. Missing the middle: the true square is x² + 8x + 16. Test any number and the shortcut collapses.
• Misreading degree. The degree is the highest exponent, not the first one written or the number of terms.

Try it

Multiply (x + 5)(x + 2), then compute (4x2 - x) - (x2 + 3x).

Answer: handshakes: x × x = x²; x × 2 = 2x; 5 × x = 5x; 5 × 2 = 10. Collect: x² + 7x + 10. Check at x = 1: (6)(3) = 18, and 1 + 7 + 10 = 18. For the subtraction: flip every sign in the second: 4x² - x - x² - 3x. Sort: 3x² - 4x. Check at x = 1: (4 - 1) - (1 + 3) = 3 - 4 = -1, and 3 - 4 = -1. Both certified. You are multiplying polynomials and proving yourself right; that is a real mathematician's workflow.

Recap

• Polynomials are strings of terms; count terms for the name, top exponent for the degree.
• Add by sorting like terms: x² with x², x with x, numbers with numbers.
• Subtract by flipping every sign in the second polynomial first, then sorting.
• Multiply binomials with four handshakes (FOIL), then combine the two middle terms.
• (x + a)² always has a middle term; check any answer instantly by plugging in x = 1.

Sources

• OpenStax. (2020). Add and subtract polynomials (Section 6.1). In Elementary Algebra 2e. openstax.org
• OpenStax. (2020). Multiply polynomials (Section 6.3). In Elementary Algebra 2e. openstax.org
• Dawkins, P. (n.d.). Polynomials. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Multiplying polynomials. Math is Fun. mathsisfun.com

The big picture

Factoring has a big reputation, so let us start by shrinking it down to its true size. You factored numbers in elementary school: 12 is 3 × 4. Nobody thought twice about it. Factoring a polynomial is the same act: writing it as a multiplication. Last lesson you multiplied (x + 2)(x + 3) and got x² + 5x + 6. Factoring is that movie played backwards: you are handed x² + 5x + 6 and asked to rediscover (x + 2)(x + 3).

And here is the kindest fact in this whole topic, worth underlining: every factoring answer can be checked by multiplying it back out. You can never be silently wrong. If your factors multiply back to the original, you are right, guaranteed, no answer key needed. Factoring is a puzzle where you always hold the solution-checker in your own hands. Let us play.

Key idea: Factoring means un-multiplying, and multiplying back is your built-in answer check.

Move 1, always first: pull out the GCF

With numbers: the greatest common factor (GCF) of 12 and 18 is 6, the biggest number dividing both. Polynomials work the same, letters included.

Factor 6x2 + 9x, slowly.

1. Look at the numbers 6 and 9. The biggest number dividing both is 3.
2. Look at the letters: the first term has x × x, the second has one x. Both have AT LEAST one x, so one x is common.
3. The GCF is 3x. Pull it out front, like factoring out a common ingredient.
4. What is left of each term? 6x² ÷ 3x = 2x, and 9x ÷ 3x = 3.
5. Write it: 6x2 + 9x = 3x(2x + 3).

Check it: distribute back: 3x times 2x is 6x², and 3x times 3 is 9x. We recovered the original exactly, so the answer is certainly right. One more quick one: 4x³ + 8x²: the GCF is 4x², leaving 4x2(x + 2).

Why is GCF always move 1? Because pulling it out makes whatever remains smaller and friendlier, and because a final answer is not fully factored until the GCF is out. Make it a reflex: before any fancier move, ask "what do all the terms share?"

Try it: factor 10x2 + 5x.

Answer: numbers share 5; both terms have an x; GCF = 5x. Remainders: 2x and 1 (careful: 5x ÷ 5x is 1, not 0; something must hold the seat). So 5x(2x + 1). Distribute to check: 10x² + 5x. Certified.

Move 2: factoring x² + bx + c, the handshake in reverse

Recall the forward direction from last lesson: (x + 2)(x + 3) = x2 + 5x + 6. Look at where the 5 and the 6 came from. The 6 is 2 × 3, the two numbers MULTIPLIED. The 5 is 2 + 3, the same numbers ADDED. That is not a coincidence; the outer and inner handshakes always add up while the last handshake multiplies.

So the reverse puzzle has a clean job description. To factor x2 + 5x + 6: find two numbers that multiply to 6 (the constant) and add to 5 (the middle coefficient).

1. List the factor pairs of 6: 1 and 6, then 2 and 3.
2. Check their sums: 1 + 6 = 7, no. Then 2 + 3 = 5. Yes.
3. The numbers are 2 and 3, so the factors are (x + 2)(x + 3).

Check: FOIL it back: x² + 3x + 2x + 6 = x² + 5x + 6. Home again. That listing-and-checking is the entire method: a short, honest treasure hunt.

Another: factor x2 + 7x + 10. Pairs of 10: 1 and 10 (sum 11), 2 and 5 (sum 7). Winner: (x + 2)(x + 5).

When minus signs join the hunt

The signs of the two mystery numbers follow the clues in the trinomial, and you can always reason it out rather than memorize.

Factor x2 - x - 6. We need two numbers that multiply to -6 and add to -1.

1. The product is NEGATIVE, and only a positive times a negative gives a negative. So the two numbers have opposite signs.
2. The sum is -1, close to zero, so the pair must nearly cancel: try 2 and -3. Product: 2 × (-3) = -6. Sum: 2 + (-3) = -1. Both clues satisfied.
3. Factors: (x + 2)(x - 3).

Check: FOIL: x² - 3x + 2x - 6 = x² - x - 6. True. Now one with a negative middle but positive end: factor x2 - 8x + 12. Product +12 means SAME signs; sum -8 means both negative. Pairs: -2 and -6 (product 12, sum -8). Factors: (x - 2)(x - 6). Multiply back if any doubt lingers; it will confirm.

Where people get stuck: swapping the clues, hunting for numbers that ADD to the constant and MULTIPLY to the middle. Keep the jobs straight with the last lesson's memory: the constant was born from multiplying, the middle from adding. Product goes with the plain number, sum goes with the x term.

Try it: factor x2 + 8x + 15.

Answer: multiply to 15, add to 8: pairs 1 and 15 (sum 16), 3 and 5 (sum 8). Winner: (x + 3)(x + 5). FOIL check: x² + 5x + 3x + 15. Certified. See how quickly the hunt goes once you list the pairs?

Move 3: the difference of two squares

Multiply (x + 3)(x - 3) and watch something lovely: x² - 3x + 3x - 9. The middle handshakes CANCEL, leaving x2 - 9, no middle term at all. Run it backwards and you get a pattern worth knowing on sight:

a2 - b2 = (a + b)(a - b)

Read it aloud: "something squared minus something-else squared factors into (sum)(difference)." Recognizing it takes two glances: are both terms perfect squares? Is the sign between them a MINUS?

• x2 - 25: both squares (x and 5), minus between: (x + 5)(x - 5).
• 4x2 - 49: both squares (2x and 7): (2x + 7)(2x - 7).
• x2 + 9: both squares but a PLUS between. This one does NOT factor with real numbers. A sum of squares keeps its secrets; leave it be, and write "does not factor." That is a correct, full-credit answer.

Factor completely: peel the onion

Sometimes one move is not the whole job. Factor 2x2 - 18 completely.

1. GCF first, always: both terms share 2. Pull it: 2(x2 - 9).
2. Look inside the parentheses: x² - 9 is a difference of squares. Keep peeling: 2(x + 3)(x - 3).
3. Scan once more: nothing left to factor. NOW it is complete.

Check: (x + 3)(x - 3) = x² - 9, times 2 is 2x² - 18. The onion is fully peeled when no layer inside can factor further, and skipping the GCF in step 1 would have made everything harder, not easier.

Key idea: GCF first, then trinomial hunt or difference of squares, then look again. Stop only when nothing inside can be factored.

Where you will see this

• Next lesson, factoring becomes the key that unlocks quadratic EQUATIONS: the reason this skill exists is about to reveal itself.
• Simplifying fractions: factored tops and bottoms cancel cleanly, in algebra and in calculus later.
• Mental arithmetic: 21 × 19 is (20 + 1)(20 - 1) = 400 - 1 = 399, the difference of squares working a party trick.

Common misconceptions

• Skipping the GCF. Always sweep shared factors out first; 2x² - 18 resists the trinomial hunt but opens instantly after the 2 leaves.
• Sign slips in the hunt. Product negative: opposite signs, and the bigger number carries the middle term's sign. Product positive: matching signs, both wearing the middle's sign.
• Stopping early. 2(x² - 9) is not finished; the difference of squares inside still wants factoring.
• Factoring x² + 9. A SUM of squares does not factor over the reals; only the difference does.
• Never checking. Multiplying back takes fifteen seconds and converts "I hope" into "I know." Use the power; it is free.

Try it

Factor completely: 3x2 + 12x, and then x2 - 16.

Answer: First: GCF of 3x² and 12x is 3x, leaving 3x(x + 4). Check: 3x times x is 3x², 3x times 4 is 12x. Second: difference of squares, x and 4: (x + 4)(x - 4). Check: x² - 4x + 4x - 16 = x² - 16. If both checked out under your own multiplication, notice what that means: you no longer need anyone to tell you whether you are right. That independence is the quiet superpower of this chapter.

Recap

• Factoring writes a polynomial as a multiplication; it is FOIL in reverse.
• Move 1 is always the GCF: pull out everything all terms share.
• For x² + bx + c: find two numbers that multiply to c and add to b; signs follow the clues.
• a² - b² = (a + b)(a - b); a sum of squares does not factor.
• Peel until nothing inside factors further, and certify every answer by multiplying back.

Sources

• OpenStax. (2020). Greatest common factor and factor by grouping (Section 7.1). In Elementary Algebra 2e. openstax.org
• OpenStax. (2020). Factor trinomials of the form x² + bx + c (Section 7.2). In Elementary Algebra 2e. openstax.org
• OpenStax. (2020). Factor special products (Section 7.4). In Elementary Algebra 2e. openstax.org
• Dawkins, P. (n.d.). Factoring polynomials. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Factoring in algebra. Math is Fun. mathsisfun.com

The big picture

The word quadratic sounds like serious, advanced math, and if your shoulders tightened a little when you read it, that is a completely normal reaction. Here is the calming truth: you already own every tool this lesson needs. Last lesson you learned to factor. Long before that, you learned that multiplying by zero always gives zero. Today those two small facts snap together into one of the most satisfying moves in all of algebra: solving equations that have x² in them.

Why does anyone want to? Because x² equations answer real questions: when a thrown ball lands, or how wide a rug must be to cover 40 square meters. By the end of this lesson you will solve both of those problems yourself, one small step at a time. Nothing skipped, nothing rushed, and no question is too small.

Key idea: Solving a quadratic equation is factoring plus one friendly fact about zero, and you already know both pieces.

What counts as a quadratic equation

Numbers first, as always. Here is a quadratic equation: x2 - 5x + 6 = 0. Read it aloud: "x squared minus five x plus six equals zero." The star of the show is the x2 term, "x squared." An equation is called quadratic when x² is the highest power of x anywhere in it.

The general pattern is ax2 + bx + c = 0, read aloud as "a x squared, plus b x, plus c, equals zero." Here a, b, and c stand for ordinary numbers, and a is not zero. That arrangement, with everything gathered on one side and 0 alone on the other, is called standard form. In our example, a is 1, b is -5, and c is 6.

One heads-up so nothing surprises you later: a quadratic equation usually has two solutions, not one. A linear equation like 2x = 10 pins down a single answer. A quadratic is a two-answer kind of puzzle, and finding both answers is the whole job. If you expect two from the start, nothing ahead will feel strange.

Key idea: Quadratic means the highest power is x squared. Standard form means zero alone on one side. Expect two solutions.

The zero-product property: zero always tattles

Before any letters, play a quick game with plain numbers. Try to multiply two numbers and get zero WITHOUT using zero. Go ahead: 2 × 3 = 6. 5 × (-4) = -20. 100 × 0.5 = 50. You cannot do it. No matter how you try, the only way a multiplication ends in zero is if zero was one of the numbers going in: 7 × 0 = 0, and 0 × (-12) = 0.

So zero tattles on its factors. The moment someone tells you "these two numbers multiply to zero," you know a secret about them: at least one of them IS zero. Written with letters: if A × B = 0, then A = 0 or B = 0. Read that aloud: "if A times B equals zero, then A is zero or B is zero." This little fact has a big name, the zero-product property, but the idea fits in your pocket.

One caution: this magic belongs to zero alone. If A × B = 12, you know nothing for certain about A or B by themselves. Maybe they are 3 and 4, maybe 2 and 6, maybe 24 and one half. Only a product of ZERO forces a factor to be zero.

Key idea: A product can only be zero if one of its factors is zero. That single fact is the engine of this whole lesson.

Let us solve one together

Solve x2 - 5x + 6 = 0. We will go slowly, and every move will be one you already know.

1. Check the right side. It is already 0, so the equation is ready. No rearranging needed.
2. Factor the left side with last lesson's treasure hunt: we need two numbers that multiply to 6 and add to -5.
3. The hunt finds -2 and -3, because (-2) × (-3) = 6 and (-2) + (-3) = -5.
4. Write the factored form: (x - 2)(x - 3) = 0.
5. Now zero tattles. A product equals zero, so a factor equals zero: x - 2 = 0 or x - 3 = 0.
6. Solve the first little equation. Add 2 to both sides (to undo the minus 2): x = 2.
7. Solve the second little equation. Add 3 to both sides: x = 3.

The solutions are x = 2 or x = 3. What we just did: we turned one quadratic into two tiny one-step equations, the kind you have been solving for weeks.

And here is the best part, your built-in answer checker. Drop each answer back into the original equation. For x = 2: 2 squared is 4, then 4 - 10 + 6 = 0. True. For x = 3: 9 - 15 + 6 = 0. Also true. Both answers certified, by you, with no answer key. Nice work; that was a complete quadratic, start to finish.

When the equation is not ready yet

Solve x2 + 3x = 4.

Where people get stuck: the most common quadratic mistake in the world happens right here. The right side is 4, not 0, and zero's tattling trick works ONLY on zero. If you factor now and set the pieces equal to 4, the answers come out wrong. The fix is one small chore: move everything to one side first, every time.

1. Subtract 4 from both sides (to make the right side 0): x2 + 3x - 4 = 0.
2. Hunt for two numbers that multiply to -4 and add to +3. They are +4 and -1.
3. Factor: (x + 4)(x - 1) = 0.
4. Set each factor to zero: x + 4 = 0 or x - 1 = 0.
5. Solve the first: subtract 4 from both sides, so x = -4.
6. Solve the second: add 1 to both sides, so x = 1.

Check both in the ORIGINAL equation. For x = 1: 1 + 3 = 4. True. For x = -4: 16 - 12 = 4. True. What we just did: one small subtraction made the equation zero-ready, and after that it was the same routine as before.

Try it: solve x2 + 7x + 10 = 0.

Answer: the right side is already 0. Multiply to 10, add to 7: the numbers are 2 and 5. Factor: (x + 2)(x + 5) = 0. Set each to zero: x = -2 or x = -5. Check x = -2: 4 - 14 + 10 = 0. True. See? You solved a quadratic on your own, and the check proved it.

The sneaky one: do not divide away an answer

Solve x2 - 7x = 0. Only two terms, no plain number at the end, and a strong temptation whispers: "divide both sides by x." Please do not. Watch what the safe route finds that dividing would have destroyed.

1. The right side is already 0. Ready.
2. Both terms share an x, so factor it out (the GCF move from last lesson): x(x - 7) = 0.
3. Zero tattles: x = 0 or x - 7 = 0.
4. The first is already solved: x = 0 is an answer, all by itself.
5. Solve the second: add 7 to both sides, so x = 7.

The solutions are x = 0 or x = 7. Check x = 0: 0 - 0 = 0. True. Check x = 7: 49 - 49 = 0. True.

Where people get stuck: dividing both sides by x makes the x = 0 answer vanish silently, and you would report only x = 7, half the truth. Dividing by the variable is like tearing a page out of the answer book. When every term has an x, factor the x out; never divide it away.

Key idea: Factor out a shared x. Dividing by x quietly throws away the x = 0 solution.

An old friend returns: the difference of squares

Solve x2 - 9 = 0. You met this exact shape last lesson: x squared minus 3 squared, a difference of squares.

1. Factor the pattern: (x + 3)(x - 3) = 0.
2. Set each factor to zero: x + 3 = 0 or x - 3 = 0.
3. Solve each: x = -3 or x = 3.

The two answers are opposites, and that always happens with this pattern. It makes sense when you test it: 3 × 3 = 9, and (-3) × (-3) = 9 as well, so both numbers deserve to be answers.

Try it: solve x2 - 25 = 0.

Answer: (x + 5)(x - 5) = 0, so x = -5 or x = 5. Quick check: 25 - 25 = 0. Both work. That one probably took you under a minute. Notice how fast this is getting; that speed is real skill, earned.

The whole routine on one card

Every problem in this lesson follows the same four moves. They are worth copying onto an index card:

1. Move everything to one side so the other side is exactly 0.
2. Factor (GCF first, then the two-number hunt or the difference of squares).
3. Set each factor equal to zero.
4. Solve the little equations, then check each answer in the original.

Key idea: Zero on one side, factor, set each factor to zero, solve and check. Four moves, every time, no exceptions.

Real problem 1: when does the ball land?

A ball is tossed upward, and its height in feet after t seconds is h = -16t2 + 48t. Read that aloud: "h equals negative sixteen t squared, plus forty-eight t." The -16t² part is gravity pulling the ball down, and the 48 is how hard it was thrown. The question: when does the ball come back to the ground?

1. Translate the words into math. "On the ground" means the height is zero, so set h to 0: -16t2 + 48t = 0.
2. Both terms share -16t, so factor it out: -16t(t - 3) = 0.
3. Zero tattles: -16t = 0 or t - 3 = 0.
4. Solve the first: divide both sides by -16, so t = 0.
5. Solve the second: add 3 to both sides, so t = 3.

Two answers, and both are telling the truth: the ball is at ground level at t = 0, the instant it leaves the hand, and again at t = 3 seconds, the landing. The question asked about landing, so the answer is 3 seconds. Check: -16 × 9 + 48 × 3 = -144 + 144 = 0. True. What we just did: real physics, solved with factoring. This is the actual method scientists use, not a classroom imitation.

Real problem 2: the rug with area 40

A rectangular rug is 3 meters longer than it is wide, and its area is 40 square meters. What are its dimensions?

1. Name the mystery. Let w be the width (our mystery box). Then the length is w + 3.
2. Area of a rectangle is width times length: w(w + 3) = 40.
3. Multiply out the left side: w2 + 3w = 40.
4. Zero on one side first: subtract 40 from both sides: w2 + 3w - 40 = 0.
5. Hunt: multiply to -40, add to +3. The numbers are +8 and -5.
6. Factor: (w + 8)(w - 5) = 0.
7. Set each factor to zero and solve: w = -8 or w = 5.

Now pause and picture it. Can a rug be -8 meters wide? No. A width must be a positive length. The algebra faithfully found every number that fits the equation, and the real world then chooses which answer makes sense. So w = 5, and the rug is 5 meters by 8 meters. Check: 8 really is 3 more than 5, and 5 × 8 = 40. Throwing out the impossible root is not cheating; it is the final step of good problem solving.

Where you will meet quadratics

• Motion: flight times of balls, fireworks, and divers all come from setting a height expression equal to zero, exactly like our ball.
• Design and space: finding dimensions that produce a target area, like the rug, or a garden bed, a screen, a photo print.
• Money: a small business's profit often rises and then falls as the price changes, and the break-even prices are the roots of a quadratic.

Common misconceptions

• Factoring while the other side is not zero. From (x + 4)(x - 1) = 6 you may NOT write x + 4 = 6. The tattling trick belongs to zero alone. Rearrange to = 0 first, always.
• Dividing both sides by x. That silently deletes the x = 0 answer. Factor the x out instead, and both answers survive.
• Reporting only one solution. Most quadratics have two. Find both, then let the story (a width, a time) decide which to keep.
• Flipping the sign of a root. The factor (x + 4) gives x = -4, not x = 4. Set the factor equal to zero and solve it honestly; the sign takes care of itself.
• Skipping the check. Substituting your answers back takes seconds and turns "I hope this is right" into "I know this is right."

Try it

One last one, all yours: solve x2 + 2x = 15.

Answer: First make the right side zero: subtract 15 from both sides to get x2 + 2x - 15 = 0. Hunt: multiply to -15, add to +2. The numbers are +5 and -3. Factor: (x + 5)(x - 3) = 0. Zero tattles: x + 5 = 0 or x - 3 = 0, so x = -5 or x = 3. Check x = 3: 9 + 6 = 15. True. Check x = -5: 25 - 10 = 15. True. If you got both answers, take the win and notice what you did: rearrange, factor, tattle, solve, check. Every move was yours.

Recap

• A quadratic equation has x² as its highest power, and it usually has two solutions.
• Standard form is ax² + bx + c = 0: everything on one side, zero alone on the other.
• The zero-product property: if a product equals zero, at least one factor equals zero. Zero always tattles.
• The routine: zero on one side, factor, set each factor to zero, solve, check.
• Factor out a lone x rather than dividing by it, and in word problems keep the root that makes physical sense.

Sources

• OpenStax. (2020). Quadratic equations (Section 7.6). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Quadratic functions and equations. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Quadratic equations, part I. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Factoring quadratics. Math is Fun. mathsisfun.com

The big picture

This lesson contains the most famous formula in all of algebra, the quadratic formula, and yes, at first glance it looks like something carved on a wizard's door. Here is the secret that changes everything: the formula is not a puzzle you have to be clever about. It is a recipe. You find three numbers, you put them into the right slots, you follow the arithmetic one small step at a time, and it hands you the answers. Every single time, for every quadratic ever written. No cleverness required.

This lesson also shows you what quadratics LOOK like. Their graphs are parabolas, the same arc a basketball traces on its way to the hoop and the same curve a water fountain draws in the air. The picture and the algebra tell one story: the top of the arc answers "how high did it go," and the landing spots are the very solutions you computed last lesson. We will go slowly, one small piece at a time, and by the end you will read both the picture and the formula.

Key idea: The quadratic formula is a plug-in recipe that solves ANY quadratic, and the parabola is the picture of the same story.

The shape: meet the parabola

A picture first, built from plain numbers. Take the simplest quadratic function, y = x2, read aloud "y equals x squared," and patiently compute a few points:

• x = -3 gives y = 9. x = -2 gives y = 4. x = -1 gives y = 1.
• x = 0 gives y = 0.
• x = 1 gives y = 1. x = 2 gives y = 4. x = 3 gives y = 9.

Plot those points and connect them smoothly, and you get a gentle U shape called a parabola. Two things are hiding in those numbers, and both matter. First, the heights repeat: x = -2 and x = 2 both give y = 4. The left side is a perfect mirror of the right side. Second, there is one special point at the bottom, (0, 0), where the curve stops falling and turns to rise. That turning point is called the vertex.

The mirror line itself, the vertical line through the vertex, is called the axis of symmetry. Picture printing the parabola and folding the page along that line: the two halves would land exactly on top of each other, like folding a paper heart in half.

Which way the parabola opens is decided by one number. In y = ax2 + bx + c, look at a, the number attached to x². If a is positive, the parabola opens upward like a cup, so it has a lowest point. If a is negative, it opens downward like a dome, so it has a highest point. A thrown ball's height formula has a negative a, which is exactly why its path rises and then falls.

Key idea: A quadratic graph is a mirror-symmetric U called a parabola. The vertex is its turning point, and the sign of a says cup (positive) or dome (negative).

What the graph has to do with your equations

Last lesson you solved x2 - 5x + 6 = 0 and found x = 2 and x = 3. Now graph the function y = x2 - 5x + 6 and something lovely appears: the curve crosses the x-axis at exactly x = 2 and x = 3. The crossing points, called x-intercepts, ARE the solutions. Solving the equation and finding where the graph touches the axis are the same activity seen from two directions.

The picture also explains how many answers to expect. A parabola can cross the x-axis in two places, touch it at exactly one point (the vertex resting on the axis), or float entirely above or below it and never touch at all. That is why a quadratic equation can have two, one, or zero real solutions. Hold onto that image; it will make a number later in this lesson feel completely natural.

Key idea: The real solutions of ax² + bx + c = 0 are exactly where y = ax² + bx + c crosses the x-axis: two crossings, one touch, or none.

The famous formula: a recipe, not a riddle

Factoring is wonderful when the numbers are friendly. But some quadratics refuse to factor with whole numbers, and you deserve a tool that never leaves you stranded. Here it is. For any quadratic in standard form ax2 + bx + c = 0:

x = ( -b ± √(b2 - 4ac) ) / (2a)

Read it aloud, slowly: "x equals negative b, plus or minus the square root of b squared minus four a c, all divided by two a." Read it once more if you like.

The symbol ± is read "plus or minus," and it is the formula's way of packing TWO calculations into one line: you run the arithmetic once using plus and once using minus, and that is how both solutions arrive. The letters a, b, and c are the same a, b, c from standard form. Think of the formula as a vending machine: feed in the three numbers, turn the crank of arithmetic, and the answers drop out. You never have to be inspired; you only have to be careful.

Key idea: Put the equation in standard form, read off a, b, c, substitute, and follow the arithmetic. The ± delivers both answers.

Let us run the recipe together

Solve x2 + 3x - 4 = 0. Tiny steps, one small action each.

1. Name the three numbers. The number with x² is a = 1. The number with x is b = 3. The plain number is c = -4, and the minus sign travels with it.
2. Work out b² first: 3 squared is 9.
3. Work out 4ac: 4 × 1 × (-4) = -16.
4. Under the root we need b² - 4ac, which is 9 - (-16). Subtracting a negative becomes adding: 9 + 16 = 25.
5. Take the square root: √25 = 5.
6. Work out -b: the opposite of 3 is -3.
7. Work out 2a: 2 × 1 = 2.
8. Assemble the recipe: x = (-3 ± 5) / 2.
9. Run the plus branch: -3 + 5 = 2, and 2 ÷ 2 = 1. So x = 1.
10. Run the minus branch: -3 - 5 = -8, and -8 ÷ 2 = -4. So x = -4.

The solutions are x = 1 or x = -4. What we just did: ten tiny steps, and not one of them was harder than a single multiplication or addition. Here is a comfort, too: this equation also factors, as (x - 1)(x + 4) = 0, which gives the same 1 and -4. The formula and factoring always agree. The formula is the road that stays open even when factoring's shortcut is closed. That was the famous quadratic formula, and you drove it yourself.

Try it: solve x2 - 5x + 6 = 0 with the formula (a = 1, b = -5, c = 6). You already know the answers from last lesson, which makes this the perfect practice run.

Answer: b² = (-5) × (-5) = 25. 4ac = 4 × 1 × 6 = 24. Under the root: 25 - 24 = 1, and √1 = 1. Then -b = 5, because the opposite of -5 is +5. And 2a = 2. Assemble: x = (5 ± 1)/2. Plus branch: 6/2 = 3. Minus branch: 4/2 = 2. The formula found x = 2 and x = 3, exactly what factoring found last lesson. Two roads, same destination. You can trust this machine.

Where people get stuck: the two sign traps

• Trap 1: -b when b is already negative. If b = -5, then -b = +5, because the opposite of a negative is a positive. Write the substitution with parentheses, -(-5), and resolve it to +5. Rushing this one sign causes more wrong answers than anything else in the formula.
• Trap 2: dividing only part of the top. The whole numerator, the -b AND the ± root together, sits over 2a. Compute the complete top first, THEN divide. On paper, write the fraction with a real horizontal bar; it helps your eyes keep the grouping honest.

Both traps are sign slips, not understanding failures. Nearly everyone falls into each of them once. Falling in is how you learn exactly where they are, so a slip here means you are practicing, not failing.

The discriminant: a sneak preview of your answers

The part under the square root, b2 - 4ac, read aloud "b squared minus four a c," has its own name: the discriminant. It works like a weather forecast: computing it FIRST tells you what kind of answers are on the way, before you do the rest of the work.

• Positive discriminant: two real solutions. The parabola crosses the x-axis twice. Example: x² + 3x - 4 = 0 gave discriminant 25, and sure enough, two answers (1 and -4).
• Zero discriminant: exactly one solution. Plus zero and minus zero are the same number, so the two branches merge into one. The parabola's vertex rests exactly on the x-axis. Example: x² - 6x + 9 = 0 has discriminant 36 - 36 = 0, and its only solution is x = 3.
• Negative discriminant: no real solutions. No real number times itself gives a negative, so the square root cannot be taken. The parabola floats above or below the axis without touching. Example: x² + x + 3 = 0 has discriminant 1 - 12 = -11, so it has no real solutions.

Hear this part gently: a negative discriminant is an ANSWER, not a failure. Writing "no real solutions, because the discriminant is -11" is complete, correct, full-credit work.

Try it: without solving anything, how many real solutions does x2 + 4x + 1 = 0 have?

Answer: discriminant = b² - 4ac = 16 - 4 = 12. Positive, so two real solutions. That is the entire question, answered by the forecast alone. Notice what you did: you learned something true about an equation without solving it.

Key idea: The discriminant b² - 4ac forecasts the count: positive means two real answers, zero means one, negative means none, and the graph crosses, touches, or misses to match.

When the answers are not whole numbers

Solve x2 - 4x + 1 = 0, where a = 1, b = -4, c = 1.

1. b²: (-4) squared is 16.
2. 4ac: 4 × 1 × 1 = 4.
3. Discriminant: 16 - 4 = 12. Positive, so two real answers are coming.
4. -b: the opposite of -4 is +4 (trap 1, handled).
5. 2a: 2 × 1 = 2.
6. Assemble: x = (4 ± √12) / 2.

Now, √12 is not a whole number, and that is perfectly okay. It can be tidied: 12 = 4 × 3, so √12 = 2√3. Then x = (4 ± 2√3)/2, and dividing each piece of the top by 2 gives the exact answer x = 2 ± √3. As decimals, those are about 3.73 and about 0.27.

What we just did: solved an equation that whole-number factoring could never touch. If your answer has a square root living inside it, that does NOT mean you made a mistake. Some true answers are irrational, and the formula reports them honestly. This is exactly when the formula earns its keep.

A leading coefficient bigger than 1

Solve 2x2 + 5x - 3 = 0, where a = 2, b = 5, c = -3. The only new wrinkle is that a = 2, so give the 4ac and 2a slots your full attention.

1. b²: 5 squared is 25.
2. 4ac: 4 × 2 × (-3) = -24.
3. Discriminant: 25 - (-24) = 25 + 24 = 49. Positive: two answers on the way.
4. Square root: √49 = 7.
5. -b = -5, and 2a = 2 × 2 = 4.
6. Assemble: x = (-5 ± 7) / 4.
7. Plus branch: -5 + 7 = 2, and 2/4 = 1/2.
8. Minus branch: -5 - 7 = -12, and -12/4 = -3.

The solutions are x = 1/2 or x = -3. Check by factoring: (2x - 1)(x + 3) multiplies back to 2x² + 5x - 3, and its roots are the same 1/2 and -3. Everything agrees. Notice that the 4 on the bottom divided the ENTIRE top both times: that is trap 2, handled correctly. Nice work; that was the heaviest computation in this lesson, and you watched every step of it happen.

Finding the vertex: the top of the story

The vertex sits on the axis of symmetry, and that line has a small formula of its own: x = -b / (2a), read aloud "x equals negative b over two a." It uses only slots you already know how to fill, and it lands exactly halfway between the two roots, which is what mirror symmetry demands.

Find the vertex of y = x2 - 4x + 1.

1. Here b = -4 and a = 1, so x = -(-4) / (2 × 1).
2. The top: -(-4) = 4. The bottom: 2 × 1 = 2.
3. Divide: x = 4/2 = 2. The axis of symmetry is the line x = 2.
4. For the height there, substitute x = 2 into the function: y = 2² - 4 × 2 + 1.
5. Piece by piece: 4 - 8 + 1 = -3.

The vertex is (2, -3), the lowest point of this upward-opening parabola. A quiet, beautiful detail: this function's two roots, 2 + √3 and 2 - √3 from the previous section, sit perfectly balanced on either side of x = 2. The symmetry is not a slogan; you can see it in the numbers.

Real problem: the rocket's peak

A toy rocket's height in feet after t seconds is h = -16t2 + 64t + 5. How high does it fly?

1. Read the shape first: a = -16 is negative, so this parabola is a dome, and the vertex is the PEAK. "How high" is a vertex question.
2. Peak time: t = -b/(2a) = -64 / (2 × (-16)).
3. The bottom: 2 × (-16) = -32. So t = -64 / (-32) = 2 seconds. A negative divided by a negative is positive.
4. Peak height: substitute t = 2: h = -16 × 4 + 64 × 2 + 5.
5. Piece by piece: -64 + 128 + 5 = 69.

The rocket peaks at 69 feet, 2 seconds after launch. The lone 5 in the formula is the launch height, the height at t = 0. And symmetry hands you a bonus fact for free: the rocket passes 5 feet again at t = 4, exactly twice the vertex time. What we just did: answered a real physics question with one small formula and four lines of arithmetic.

Where you will see parabolas

• Trajectories: balls, fountains, and fireworks trace parabolas. Vertex questions ask "how high," and root questions ask "when does it land."
• Design: satellite dishes and car headlights are parabolic because the shape focuses signals and light onto one point; suspension bridge cables hang in near-parabolas that spread weight evenly.
• Business: revenue as a function of price is often a downward parabola, and the vertex names the price that earns the most.
• Safety: braking distance grows roughly with the square of speed, so doubling your speed roughly quadruples the distance needed to stop.

Common misconceptions

• Using b instead of -b. If b = -4, the formula begins with -(-4) = +4. Write the parentheses and resolve them slowly; this one sign is the most common slip in the whole topic.
• Dividing only the root by 2a. The entire numerator, -b together with the ± root, is divided by 2a. Compute the whole top first, then divide.
• Naming a, b, c before standard form. The equation must equal zero first. For x² + 3x = 4, move the 4 across before reading a = 1, b = 3, c = -4.
• Keeping only the plus branch. The ± is two instructions, not a decoration. A positive discriminant means two answers; report both.
• Thinking "will not factor" means "no solutions." Answers like 2 ± √3 are real and exact. Only a NEGATIVE discriminant means no real solutions.

Try it

One final run of the recipe, all yours: solve x2 + 6x + 8 = 0 with the quadratic formula.

Answer: a = 1, b = 6, c = 8. b² = 36. 4ac = 4 × 1 × 8 = 32. Discriminant: 36 - 32 = 4, positive, so two answers are coming. √4 = 2. -b = -6. 2a = 2. Assemble: x = (-6 ± 2)/2. Plus branch: -6 + 2 = -4, and -4/2 = -2. Minus branch: -6 - 2 = -8, and -8/2 = -4. So x = -2 or x = -4.

Check by factoring: (x + 2)(x + 4) = 0 gives the same pair. If your answers matched, pause and take the win: you now hold the tool that opens every quadratic ever written, and you know how to read the curve it draws. You finished Algebra I's hardest-looking formula, and it turned out to be a recipe after all.

Recap

• Quadratic graphs are parabolas: cup-shaped when a is positive, dome-shaped when a is negative, mirror-symmetric around the vertex.
• The x-intercepts of the graph are the real solutions of the equation: two crossings, one touch, or none.
• The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), solves any quadratic in standard form.
• The discriminant b² - 4ac forecasts the answer count: positive two, zero one, negative none.
• The vertex sits at x = -b/(2a); substitute back for its height. Factoring is fast when it works, and the formula always works.

Sources

• OpenStax. (2020). Solve quadratic equations using the quadratic formula (Section 10.3). In Elementary Algebra 2e. openstax.org
• OpenStax. (2020). Graphing quadratic equations in two variables (Section 10.5). In Elementary Algebra 2e. openstax.org
• Khan Academy. (n.d.). Quadratic functions and equations. In Algebra 1. khanacademy.org
• Dawkins, P. (n.d.). Quadratic equations, part II. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Dawkins, P. (n.d.). Parabolas. In Paul's Online Math Notes: Algebra. Lamar University. tutorial.math.lamar.edu
• Pierce, R. (n.d.). Quadratic equations. Math is Fun. mathsisfun.com