⚖️ Equations

Grade 8 · Cambridge Lower Secondary Stage 8

The Golden Rule

Whatever you do to one side, do to the other side.

An equation is like a balance scale — both sides must stay equal.

What you'll learn

Solve linear equations with unknowns on both sides
Solve equations with brackets
Solve equations with fractions
Form and solve equations from word problems
Solve equations involving powers and roots

📖 Learn

1. Unknowns on Both Sides

Collect all x terms on one side, numbers on the other.

5x − 3 = 3x + 7 → 5x − 3x = 7 + 3 → 2x = 10 → x = 5

4x + 1 = 7 − 2x → 6x = 6 → x = 1

💡 Choose the side where x will be positive — it's easier to work with.

2. Equations with Brackets

Expand brackets first, then solve as normal.

3(2x − 4) = 18 → 6x − 12 = 18 → 6x = 30 → x = 5

2(x + 3) = 5(x − 3) → 2x + 6 = 5x − 15 → 21 = 3x → x = 7

3. Equations with Fractions

Multiply through by the LCD to clear all fractions.

x/3 + 2 = 5 → multiply by 3 → x + 6 = 15 → x = 9

(x+1)/2 = (x+4)/3 → multiply by 6 → 3(x+1) = 2(x+4) → 3x+3 = 2x+8 → x = 5

4. Forming Equations from Word Problems

Assign a letter, write an equation, solve it, state your answer clearly.

Three consecutive integers sum to 48. Let n, n+1, n+2. → 3n+3=48 → n=15 → 15, 16, 17

A rectangle has length 2x+3, width x−1. Perimeter = 40. → 2(2x+3+x−1)=40 → 6x+4=40 → x=6

5. Equations with Powers and Roots

x² = 49 → x = ±7

x² + 5 = 30 → x² = 25 → x = ±5

√x = 6 → x = 36

3x² = 48 → x² = 16 → x = ±4

💡 x² = k has two solutions: x = +√k and x = −√k (unless context requires positive only).

✏️ Worked Examples

Example 1 — Both Sides

Solve 7x − 5 = 4x + 10

Step 1: Collect x: 7x − 4x = 10 + 5 → 3x = 15

Step 2: x = 15/3 = 5

Check: 7(5)−5 = 30 = 4(5)+10 ✅

Example 2 — Brackets

Solve 4(2x − 1) = 3(x + 5)

Step 1: 8x − 4 = 3x + 15

Step 2: 5x = 19 → x = 19/5 = 3.8

Example 3 — Fractions

Solve (3x − 2)/4 = (x + 3)/2

Step 1: Multiply by 4: 3x − 2 = 2(x + 3)

Step 2: 3x − 2 = 2x + 6 → x = 8

Example 4 — Word Problem

A cinema charges £8 for adults and £5 for children. 3 adults and n children pay £49 total. Find n.

Step 1: 24 + 5n = 49

Step 2: 5n = 25 → n = 5 children

🎨 Visualizer

⚖️ Balance Scale Solver

Solve ax + b = cx + d step by step with balance animations.

a: b: = c: d:

📝 Equation Builder

Build and solve an equation from a word problem scenario.

🎯 Equation Speed Round

Solve 10 equations as fast as you can!

Exercise 1 — Simple Linear Equations

Exercise 2 — Unknowns on Both Sides

Exercise 3 — Equations with Brackets

Exercise 4 — Equations with Fractions

Exercise 5 — Powers, Roots & Word Problems