Solving One-Step Linear Equations: A KS3 Maths Guide

To solve a one-step linear equation, do the inverse (opposite) operation to both sides to get the letter on its own. The opposite of adding is subtracting, and the opposite of multiplying is dividing. For example, x + 5 = 12 solves to x = 7 by subtracting 5 from both sides.

What is a one-step equation?

A one-step equation needs just one operation to solve. The unknown (usually x) has had one thing done to it — added, subtracted, multiplied or divided — and your job is to undo that single step. The golden rule: whatever you do to one side, do to the other, keeping the equation balanced.

What are inverse operations?

Inverse operations are pairs that undo each other. You use them to isolate the letter.

Operation in the equation | Inverse to apply
Add (+) | Subtract (−)
Subtract (−) | Add (+)
Multiply (×) | Divide (÷)
Divide (÷) | Multiply (×)

How do you solve an addition or subtraction equation?

Undo the addition or subtraction. Solve x + 8 = 15:

The 8 is added, so subtract 8 from both sides.

x + 8 − 8 = 15 − 8

x = 7

Solve x − 4 = 9:

The 4 is subtracted, so add 4 to both sides.

x − 4 + 4 = 9 + 4

x = 13

How do you solve a multiplication or division equation?

Undo the multiplication or division. Solve 3x = 21:

The x is multiplied by 3, so divide both sides by 3.

3x ÷ 3 = 21 ÷ 3

x = 7

Solve x ÷ 5 = 4 (often written x/5 = 4):

The x is divided by 5, so multiply both sides by 5.

x = 4 × 5

x = 20

How do you check your answer?

Always substitute your answer back into the original equation. For 3x = 21 with x = 7: check 3 × 7 = 21. ✓ If both sides match, you are correct. This quick check catches most slips and earns confidence in exams.

Why do one-step equations matter?

One-step equations are the foundation of all equation-solving. Two-step and multi-step equations at GCSE simply repeat the same idea — apply inverse operations to both sides — more times. Mastering the single step makes every harder equation feel familiar.

What does "linear" mean in a linear equation?

The word "linear" tells you the unknown appears only to the power of one — there is no x², no x³, and no x inside a root. If you plotted a linear equation such as y = 2x + 1 on a graph, the points would form a perfectly straight line, which is exactly where the name comes from. One-step equations are the simplest linear equations of all, because the unknown has had only a single operation applied to it. Recognising an equation as linear reassures you that the straightforward inverse-operation method will always work, with no special techniques needed. This is the same reasoning you will extend to two-step and multi-step linear equations later in KS3 and at GCSE.

Frequently asked questions

How do you solve a one-step equation?

Apply the inverse operation to both sides so the letter is left on its own. If something is added to the unknown, subtract it; if it is multiplied, divide. Always keep both sides balanced by doing the same to each.

What are inverse operations in maths?

Inverse operations are pairs that undo each other: addition and subtraction, and multiplication and division. To solve an equation you apply the inverse of whatever has been done to the unknown, isolating the letter.

Why do you do the same thing to both sides?

An equation is a balance: both sides are equal. Doing the same operation to each side keeps them equal, so the equation stays true while you simplify it to find the unknown.

How do I check my answer to an equation?

Substitute your answer back into the original equation and check both sides are equal. For example, if you solve 3x = 21 and get x = 7, check that 3 × 7 = 21. If they match, the answer is correct.

For Socratic equation-solving practice, see aitutors.me.