§ Expressions & Algebra

Introduction to Equations

CCSS.6.EECCSS.7.EE3 min readApr 13, 2026

Equations form the backbone of algebraic thinking in Year 7, yet many pupils struggle with the concept of 'undoing' operations to isolate the unknown. The transition from arithmetic to algebra requires students to think systematically about inverse operations and balance.

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§ 01

Why it matters

Mastering basic equations prepares pupils for GCSE Foundation topics like solving linear inequalities and simultaneous equations. In real contexts, equations help calculate missing values: if Harry saves £5 weekly and needs £47 for trainers, the equation 5x + 12 = 47 determines he needs 7 more weeks (where £12 is his current savings). Shop discounts use equations too—if a £25 jumper costs £18 after reduction, x + 18 = 25 shows the discount was £7. Without solid equation skills, pupils struggle with ratio problems, percentage calculations, and formulae rearrangement throughout secondary maths. The logical thinking required transfers to science formulae and real-world problem-solving scenarios.

§ 02

How to solve introduction to equations

One-Step Equations

An equation has an unknown (x) and an equals sign.
Use the inverse operation to isolate x.
Addition ↔ subtraction; multiplication ↔ division.
Check by substituting your answer back.

Example: x + 7 = 12 → x = 12 − 7 = 5.

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Worked examples

Beginner§ 01

x + 2 = 10. What is x?

Answer: 8

Subtract 2 from both sides → x = 10 − 2 — To isolate x, subtract the number being added.
Calculate → x = 8 — 10 − 2 = 8.

Easy§ 02

x − 8 = 6. What is x?

Answer: 14

Add 8 to both sides → x = 6 + 8 — To undo subtraction, add the same number to both sides.
Calculate → x = 14 — 6 + 8 = 14.

Medium§ 03

3x = 9. What is x?

Answer: 3

Divide both sides by 3 → x = 9 ÷ 3 — To isolate x, divide by the coefficient 3.
Calculate → x = 3 — 9 ÷ 3 = 3.

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Common mistakes

Adding instead of subtracting when solving x + 5 = 12, giving x = 17 instead of x = 7
Forgetting to apply operations to both sides, writing 3x = 15 becomes x = 15 instead of x = 5
Confusing the inverse operation for subtraction, solving x - 4 = 9 as x = 9 - 4 = 5 instead of x = 13
Not checking answers by substitution, missing errors like writing x = 6 for 2x = 8

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§ 05

Frequently asked questions

Why do we use inverse operations to solve equations?

Inverse operations 'undo' what's happening to x. If x is increased by 3, we subtract 3 to get back to x alone. Think of it as unwrapping a present—each operation is a layer we must remove in reverse order.

How do I know which operation to use first?

Follow BIDMAS in reverse when multiple operations appear. For 2x + 5 = 17, subtract 5 first (undoing addition), then divide by 2 (undoing multiplication). Always work backwards from the order operations were applied.

What if I get a negative answer?

Negative solutions are perfectly valid. For x + 8 = 3, the answer x = -5 makes sense. Always check by substituting back: -5 + 8 = 3 ✓. Many real situations involve negative numbers.

Do I always need to check my answer?

Yes, substitution checking catches arithmetic errors and builds confidence. Replace x with your answer in the original equation. If both sides equal the same number, your solution is correct.

What's the difference between an equation and an expression?

Equations contain an equals sign and can be solved (x + 3 = 7). Expressions don't have equals signs and are simplified (2x + 5). Think: equations have solutions, expressions have values.

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