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Solving Linear Equations

A linear equation is a mathematical statement that two expressions are equal — and your job is to find the value of the unknown variable that makes it true. Master this skill and you can solve a huge range of real-world problems.

In this lesson

  1. What is a linear equation?
  2. The golden rule of equation solving
  3. Solving one-step, two-step, and multi-step equations
  4. Variables on both sides
  5. Common mistakes

1 What Is a Linear Equation?

A linear equation is an equation where the variable (usually x) appears only to the first power — no x², no √x, just plain x. When you graph a linear equation, it produces a straight line, which is where the name comes from.

Examples of linear equations: x + 5 = 12, 3x − 4 = 11, 2x + 3 = x − 7. Examples of non-linear equations (not covered here): x² + 3 = 7, √x = 4.

What 'solving' means

Solving an equation means finding the value of the variable that makes the equation true. If x + 5 = 12, then x = 7 is the solution because 7 + 5 = 12 is a true statement. Any other value of x makes the equation false.

2 The Golden Rule of Equation Solving

There is one rule that governs all equation solving, and it is simple: whatever you do to one side of the equation, you must do exactly the same thing to the other side.

Think of an equation as a perfectly balanced scale. The equals sign is the pivot point. If you add weight to one side, you must add the same weight to the other side to keep it balanced. If you multiply one side by 3, you must multiply the other by 3 too.

The goal of solving is to isolate the variable — to get x alone on one side. You do this by performing inverse (opposite) operations to "undo" whatever is being done to x, while keeping the equation balanced.

Inverse Operations

Addition undoes subtraction (and vice versa). Multiplication undoes division (and vice versa). If x has 5 added to it, subtract 5 from both sides to undo it. If x is multiplied by 3, divide both sides by 3.

3 Solving One-Step, Two-Step, and Multi-Step Equations

One-Step: Addition
Solve: x + 8 = 15
1We want x alone. x has 8 added to it.
2Subtract 8 from both sides: x + 8 − 8 = 15 − 8
3Simplify: x = 7
4Check: 7 + 8 = 15 ✓
Answer: x = 7
One-Step: Multiplication
Solve: 4x = 28
1x is multiplied by 4. Undo by dividing both sides by 4.
24x ÷ 4 = 28 ÷ 4
3x = 7
4Check: 4 × 7 = 28 ✓
Answer: x = 7
Two-Step Equation
Solve: 3x − 5 = 16
1Step 1 — undo the subtraction. Add 5 to both sides: 3x − 5 + 5 = 16 + 5
2Simplify: 3x = 21
3Step 2 — undo the multiplication. Divide both sides by 3: 3x ÷ 3 = 21 ÷ 3
4x = 7
5Check: 3(7) − 5 = 21 − 5 = 16 ✓
Answer: x = 7
Multi-Step with Parentheses
Solve: 2(x + 3) = 14
1Option A — divide both sides by 2 first: x + 3 = 7
2Then subtract 3: x = 4
3Option B — distribute first: 2x + 6 = 14. Subtract 6: 2x = 8. Divide: x = 4
4Check: 2(4 + 3) = 2(7) = 14 ✓
Answer: x = 4 (either method works)

4 Variables on Both Sides

When x appears on both sides of the equation, your first move is to collect all x terms on one side by adding or subtracting.

Variables on Both Sides
Solve: 5x − 3 = 2x + 9
1Collect x terms on the left. Subtract 2x from both sides: 5x − 2x − 3 = 9
2Simplify: 3x − 3 = 9
3Add 3 to both sides: 3x = 12
4Divide by 3: x = 4
5Check: 5(4) − 3 = 17. And 2(4) + 9 = 17. ✓
Answer: x = 4

It doesn't matter which side you collect terms on — you can move x to the right instead. The answer will be the same. Choose whichever side gives you a positive coefficient on x, as this tends to prevent sign errors.

5 Common Mistakes

Forgetting to do the same operation to both sides

The most common error: subtracting from one side but not the other. Every operation must be applied to the entire expression on each side, not just one term.

Sign errors when moving terms

Writing 3x − 5 = 16 and then 'moving the 5 to the other side' and writing 3x = 16 − 5. Watch the sign: adding 5 to both sides gives 3x = 16 + 5 = 21. The 5 becomes positive when moved.

Not checking the answer

Always substitute your solution back into the original equation to verify it works. This catches sign errors and arithmetic mistakes instantly.

Dividing by the wrong number

In 3x = 21, divide by the coefficient of x (which is 3), not by any other number. The coefficient is whatever number is directly in front of x.

Try the Slope Calculator

Linear equations produce straight lines. Use the slope calculator to find the slope and equation of any line from two points.

Open Calculator →

Practice Problems

Solve: x − 9 = 4
Add 9 to both sides: x = 13. Check: 13 − 9 = 4 ✓
Solve: 5x = 35
Divide both sides by 5: x = 7. Check: 5 × 7 = 35 ✓
Solve: 2x + 7 = 19
Subtract 7: 2x = 12. Divide by 2: x = 6. Check: 2(6) + 7 = 19 ✓
Solve: 4x − 3 = 2x + 11
Subtract 2x from both sides: 2x − 3 = 11. Add 3: 2x = 14. Divide by 2: x = 7. Check: 4(7)−3 = 25 and 2(7)+11 = 25 ✓
Solve: 3(x − 2) = 2x + 4
Distribute: 3x − 6 = 2x + 4. Subtract 2x: x − 6 = 4. Add 6: x = 10. Check: 3(10−2) = 24 and 2(10)+4 = 24 ✓