How do you solve a linear equation?

Use four steps: 1) Add or subtract the same number on both sides to remove the constant term. 2) Divide both sides by the coefficient of x. 3) Write down the answer. 4) Check by substituting the answer back into the original equation. This method is called inverse operations or the balance method.

What is a linear equation?

A linear equation is an equation where the unknown (usually written x) appears only to the first power, with no x², √x or 1/x. The general form is ax + b = c, for example 3x + 7 = 22.

How do you solve 3x + 7 = 22?

Subtract 7 from both sides to get 3x = 15, then divide both sides by 3. The answer is x = 5. You can check by substituting: 3 · 5 + 7 = 22.

Why do we do the same thing to both sides?

The equals sign means both sides have the same value. If we change one side, we must change the other in the same way, otherwise the two sides are no longer equal.

What is the inverse of addition and multiplication?

Subtraction undoes addition, and division undoes multiplication. To isolate x, we apply the inverse of each operation in reverse order.

Can a linear equation have no solution?

Yes. If after simplifying you get a false statement like 0 = 5, the equation has no solution. If you get a true statement like 0 = 0, every number is a solution.

Tutorial · 6 min read

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How to solve linear equations — step by step (with a worked example)

A linear equation has the form ax + b = c. To find x, undo each operation on both sides in reverse — this is called inverse operations or the balance method. Worked example: 3x + 7 = 22 → x = 5, in four steps. Suitable for Grade 7 / Year 8 algebra.

By Dr. Lena Weber · Maths teacher · 12 years

Updated April 30, 2026Beginner

Quick answer

To solve a linear equation like 3x + 7 = 22, isolate x by doing the same operation on both sides: subtract 7, then divide by 3. The answer is x = 5.

At a glance

Summary of this tutorial
Equation	3x + 7 = 22
Method	Balance method (inverse operations)
Steps	4
Answer	x = 5
Check	3 · 5 + 7 = 22 ✓
Grade level	Grade 7 (ages 12–13)

Worked example: 3x + 7 = 22

EXAMPLE

3x + 7 = 22

3 x + 7 = 22

Think of x as the unknown. We will undo the “+ 7” first, then undo the “× 3”.

The 4 steps to solve a linear equation

These four steps work for any equation of the form ax + b = c.

Step 1 · Start
3x + 7 = 22
Identify the unknown x and the target 22.
Step 2 · Subtract 7
3x = 22 − 7
Undo the “+ 7”. Same operation on both sides keeps the balance.
Step 3 · Simplify
3x = 15
Tidy up the right-hand side.
Step 4 · Divide by 3
x = 15 ÷ 3
Undo the “× 3”. Now x is alone.
Step 5 · Check
x = 5
Substitute back: 3 · 5 + 7 = 22 ✓

Why the balance method works

Every arithmetic operation has an inverse: subtraction undoes addition, division undoes multiplication. Doing the same thing to both sides of an equals sign keeps the two sides equal — the equals sign is a balance.