Solve one-variable linear equations

SECTION 1

Where Did Equations Come From?

People have been solving equations for thousands of years. Long before we used letters like x and y, ancient civilizations needed to figure out unknown amounts. Farmers needed to split land evenly. Merchants needed to calculate fair prices. Builders needed to find missing measurements.

The idea of using a symbol to stand for an unknown number changed math forever. Instead of guessing and checking, people developed rules to find the answer every time. Let's see how this idea grew over time.

1800 BCE

Babylonian Clay Tablets

Ancient Babylonians carved math problems onto clay tablets. They solved problems like "a number plus 5 equals 12" without any symbols at all — just words and clever reasoning.

250 CE

Diophantus of Alexandria

The Greek mathematician Diophantus wrote a book called Arithmetica. He was one of the first people to use a shorthand symbol for an unknown number. Many call him the "father of algebra."

820 CE

Al-Khwarizmi's Algebra

The Persian scholar al-Khwarizmi wrote a book that gave us the word "algebra" (from the Arabic word al-jabr). He described step-by-step methods to solve equations — methods we still use today.

1637

Descartes Uses x, y, z

French mathematician René Descartes started using the letters x, y, and z for unknowns. This is why we write equations like 2x + 3 = 11 today.

Today, solving one-variable linear equations is a key skill you will use in high school math, science, and even everyday life. The big question this lesson answers is: How do you find the value of an unknown number when it's trapped inside an equation?

SECTION 2

Core Principles & Definitions

Before we start solving, let's make sure we understand the key vocabulary. An equation is a math sentence that says two things are equal. It always has an equals sign (=). A variable is a letter (like x) that stands for a number we don't know yet. Our job is to figure out that number.

Equation

A math sentence with an equals sign. Example: 3x + 2 = 14. The left side must equal the right side.

Variable

A letter (like x or n) that represents an unknown number. In 3x + 2 = 14, the variable is x.

Inverse Operations

Operations that undo each other. Addition undoes subtraction. Multiplication undoes division. These are your main tools for solving.

Isolate the Variable

Get the variable alone on one side of the equation. When you see x = 4, you've found your answer!

Balance Rule

Whatever you do to one side of the equation, you must do the same thing to the other side. This keeps the equation true.

✦ KEY TAKEAWAY

Think of an equation like a balanced seesaw. Both sides weigh the same. If you add a weight to one side, the seesaw tips — unless you add the same weight to the other side. That's the balance rule. You can add, subtract, multiply, or divide — as long as you do it to both sides.

SECTION 3

The Balance Model

The diagram below shows how solving an equation is like balancing a scale. On the left side, we have the expression with the variable. On the right side, we have a number. We remove items from both sides equally until the variable is alone.

The balance scale shows the equation x + 3 = 7. By subtracting 3 from both sides, the variable x is isolated, giving x = 4.

Notice how the scale stays balanced in both steps. In Step 1, the left side (x + 3) equals the right side (7). In Step 2, we removed 3 from both sides. The left side became just x. The right side became 7 − 3 = 4. So x = 4.

SECTION 4

The Mathematical Rules

Every linear equation can be solved by using inverse operations to get the variable alone. Here are the four key moves you need to know.

UNDO ADDITION

If x + a = b, then x = b − a

When a number is added to x, subtract it from both sides. Example: x + 5 = 12 → x = 12 − 5 = 7.

UNDO SUBTRACTION

If x − a = b, then x = b + a

When a number is subtracted from x, add it to both sides. Example: x − 4 = 9 → x = 9 + 4 = 13.

UNDO MULTIPLICATION

If a × x = b, then x = b ÷ a

When x is multiplied by a number, divide both sides by that number. Example: 3x = 18 → x = 18 ÷ 3 = 6.

UNDO DIVISION

If x ÷ a = b, then x = b × a

When x is divided by a number, multiply both sides by that number. Example: x ÷ 5 = 3 → x = 3 × 5 = 15.

💡 Remember This Pattern

Addition and subtraction are inverse operations. Multiplication and division are inverse operations. To solve an equation, figure out what operation is being done to the variable, and then do the opposite operation to both sides.

SECTION 5

Types of One-Variable Equations

Not all equations look the same. Some need just one step to solve. Others need two or more steps. The diagram below shows you how to tell the difference and what strategy to use for each type.

This flowchart helps you decide whether you have a one-step or two-step equation. For two-step equations, always undo addition or subtraction first, then undo multiplication or division. Always check your answer at the end.

Common types of one-variable linear equations

Type	Example	Steps to Solve
One-step (addition)	x + 8 = 15	Subtract 8 from both sides → x = 7
One-step (multiplication)	4x = 28	Divide both sides by 4 → x = 7
Two-step	2x + 5 = 17	Subtract 5 → 2x = 12. Divide by 2 → x = 6
Variable on both sides	3x + 2 = x + 10	Subtract x → 2x + 2 = 10. Subtract 2 → 2x = 8. Divide by 2 → x = 4

SECTION 6

Worked Example: Step by Step

Let's solve a two-step equation together. We'll go slowly so you can see every move. The equation is: 5x − 3 = 22.

Solve 5x − 3 = 22

Step 1 — Identify the Operations on xLook at the variable x. Two things are happening to it: x is being multiplied by 5, and then 3 is subtracted. This is a two-step equation. We will undo subtraction first, then undo multiplication.

Step 2 — Undo the SubtractionSince 3 is subtracted, we add 3 to both sides: 5x − 3 + 3 = 22 + 3 5x = 25

5x = 25

Step 3 — Undo the MultiplicationSince x is multiplied by 5, we divide both sides by 5: 5x ÷ 5 = 25 ÷ 5 x = 5

x = 5

Step 4 — Check Your AnswerPlug x = 5 back into the original equation: 5(5) − 3 = 25 − 3 = 22 ✓ The left side equals the right side, so our answer is correct!

22 = 22 ✓ Correct!

✅ Pro Tip: Always Check

On a test like the TACHS, checking your answer only takes a few seconds. Just substitute your answer back into the original equation. If both sides match, you got it right. If they don't match, go back and look for a mistake.

SECTION 7

Common Mistakes & How to Avoid Them

Everyone makes mistakes when learning equations. The good news is that most mistakes follow a pattern. Once you know what to watch for, you can avoid them.

Watch out for these common mistakes on the TACHS

Mistake	Example of the Error	How to Fix It
Only changing one side	x + 4 = 10 → x = 10 (forgot to subtract 4 from right side)	Always do the same operation to BOTH sides. x + 4 − 4 = 10 − 4 → x = 6.
Wrong operation order in two-step	2x + 6 = 14 → dividing by 2 first	Undo addition/subtraction FIRST, then undo multiplication/division.
Sign errors with negatives	x − 5 = 3 → x = 3 − 5 = −2 (wrong!)	Subtraction is undone by ADDITION. x − 5 + 5 = 3 + 5 → x = 8.
Not checking the answer	Getting x = 3 but not verifying	Always substitute your answer back in. If both sides are equal, you're right!

✦ KEY TAKEAWAY

Think of inverse operations like an undo button on your phone. If you accidentally deleted a photo (subtracted), you press undo (add it back). If you zoomed in too much (multiplied), you zoom out (divide). The key is doing the opposite action to get back to where you started — and always doing it to both sides.

SECTION 8

From Linear Equations to Bigger Ideas

Solving one-variable linear equations is the starting point for many exciting topics in math. Once you master this skill, you'll be ready to tackle more complex problems. Here's how today's lesson connects to what comes next.

How this lesson builds toward future math topics

What You Learned Today	What Comes Next
Solving equations like 2x + 3 = 11	Solving inequalities like 2x + 3 > 11
One variable (just x)	Two variables (x and y) — systems of equations
Numbers only on one side	Variables on both sides, and equations with fractions
Finding a number	Graphing equations as lines on a coordinate plane

The same balance rule you learned today works in every algebra course you'll ever take. Whether you're solving simple equations or complex ones with fractions and decimals, the idea is always the same: isolate the variable by using inverse operations on both sides. Master this now, and you'll have a head start in high school math!

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. Write out each step, and don't forget to check your answers!

PROBLEM 1 — CONCEPTUAL

In the equation x + 9 = 15, what inverse operation should you use to solve for x? Why?

PROBLEM 2 — BASIC CALCULATION

Solve for x: 7x = 42

PROBLEM 3 — INTERMEDIATE

Solve for x: 4x − 7 = 21

PROBLEM 4 — APPLIED

Maria has $5 and earns $8 for every hour she babysits. She wants to buy a pair of sneakers that costs $53. Write and solve an equation to find how many hours (h) she needs to babysit.

PROBLEM 5 — CRITICAL THINKING

Solve for x: 3x + 4 = x + 12. (Hint: first get all the x terms on one side.)

SUMMARY

Lesson Summary

A one-variable linear equation is a math sentence with an equals sign and one unknown value (a variable). To solve it, you isolate the variable by using inverse operations. Addition undoes subtraction. Multiplication undoes division. The most important rule is the balance rule: whatever you do to one side, you must do to the other side too.

For one-step equations, use a single inverse operation. For two-step equations, undo addition or subtraction first, then undo multiplication or division. Always check your answer by plugging it back into the original equation. If both sides are equal, you solved it correctly!