Linear Equations in One Variable: Step-by-Step + Examples

A linear equation in one variable is a mathematical sentence joined by an equals sign (=) that contains exactly one variable raised to the power of one. For example, 2x + 5 = 11. It is taught in Year 7 and it underpins almost all the algebra that follows: simultaneous equations, functions and quadratics.

How to Recognise One

An equation qualifies only if all three conditions hold:

1. It uses an equals sign (=), not an inequality sign such as < or >.
2. It contains one variable only, for example x.
3. The highest power of that variable is one. If x² appears, it is a quadratic, not a linear equation.

General form: ax + b = 0, where a ≠ 0.

Solving It Step by Step

Solving means finding the value of the variable that makes both sides equal. There is a single principle: whatever you do to the left-hand side, do to the right-hand side. The scales must stay balanced.

Example 1: 2x + 5 = 11

• Subtract 5 from both sides → 2x + 5 − 5 = 11 − 5 → 2x = 6
• Divide both sides by 2 → x = 3
• Check: 2(3) + 5 = 11 ✔

Example 2: 5x − 7 = 2x + 8 (variable on both sides)

• Subtract 2x from both sides → 3x − 7 = 8
• Add 7 to both sides → 3x = 15
• Divide by 3 → x = 5
• Check: 5(5) − 7 = 18 and 2(5) + 8 = 18 ✔

Example 3: 3(x + 4) = 27 (with brackets)

• Expand first → 3x + 12 = 27
• Subtract 12 → 3x = 15
• Divide by 3 → x = 5

A Worked Word Problem

Rani is 4 years older than Budi. Together they are 28. How old is Budi?

Let Budi's age be x, so Rani's age is x + 4.

x + (x + 4) = 28 → 2x + 4 = 28 → 2x = 24 → x = 12.

Budi is 12 and Rani is 16. The key to every word problem is the same: decide what x stands for first, then build the equation.

The Mistakes That Actually Cost Marks

From classroom experience, a child who "cannot do linear equations" almost never has a problem with linear equations.

1. Forgetting to change the sign when moving a term. This is the number one cause. The child writes 2x + 5 = 11 → 2x = 11 + 5. The root is not algebra at all — it is shaky integer arithmetic left over from Year 6. If a child hesitates over −7 + 3, moving terms across the equals sign will go wrong every time.

2. Dividing only one term. From 2x = 6 they write x = 6, because the division was not applied to both sides.

3. Not expanding brackets. 3(x + 4) becomes 3x + 4 instead of 3x + 12.

4. Never checking the answer. Linear equations are the one topic where a child can verify their own work in ten seconds: substitute the value back into the original equation. That habit alone raises test scores without learning anything new.

A quick diagnostic: ask −8 + 5 and 6 − (−2) out loud. If either causes hesitation, fix integers first — the equations will fix themselves.