GRADE 8 Mathematics ALGEBRA – Linear Equations Notes
Mathematics — ALGEBRA
Subtopic: Linear Equations (for age 13, Kenya)
A linear equation is an equation in which the highest power of the unknown (variable) is 1. We usually write a linear equation in one variable as ax + b = c, where a, b and c are numbers and x is the variable we want to find.
Key steps to solve a linear equation
- Simplify both sides (remove brackets and collect like terms).
- Get all terms with the variable on one side and numbers on the other side.
- Use inverse operations (add/subtract, multiply/divide) to isolate the variable.
- Check by substituting the found value into the original equation.
Worked examples
Example 1: Solve 3x + 5 = 20.
Step 1: Subtract 5 from both sides: 3x + 5 − 5 = 20 − 5 → 3x = 15.
Step 2: Divide both sides by 3: x = 15 ÷ 3 → x = 5.
Check: 3×5 + 5 = 15 + 5 = 20 ✓
Step 2: Divide both sides by 3: x = 15 ÷ 3 → x = 5.
Check: 3×5 + 5 = 15 + 5 = 20 ✓
Example 2 (brackets): Solve 2(x − 3) = 10.
Step 1: Expand brackets: 2x − 6 = 10.
Step 2: Add 6 to both sides: 2x = 16.
Step 3: Divide by 2: x = 8.
Check: 2(8 − 3) = 2×5 = 10 ✓
Step 2: Add 6 to both sides: 2x = 16.
Step 3: Divide by 2: x = 8.
Check: 2(8 − 3) = 2×5 = 10 ✓
Example 3 (fraction): Solve x/4 + 3 = 7.
Step 1: Subtract 3: x/4 = 4.
Step 2: Multiply both sides by 4: x = 16.
Check: 16/4 + 3 = 4 + 3 = 7 ✓
Step 2: Multiply both sides by 4: x = 16.
Check: 16/4 + 3 = 4 + 3 = 7 ✓
Example 4 (variables on both sides): Solve 5x − 2 = 2x + 10.
Step 1: Subtract 2x from both sides: 3x − 2 = 10.
Step 2: Add 2 to both sides: 3x = 12.
Step 3: Divide by 3: x = 4.
Check: 5×4 − 2 = 20 − 2 = 18 and 2×4 + 10 = 8 + 10 = 18 ✓
Step 2: Add 2 to both sides: 3x = 12.
Step 3: Divide by 3: x = 4.
Check: 5×4 − 2 = 20 − 2 = 18 and 2×4 + 10 = 8 + 10 = 18 ✓
Simple visual: balance idea
Think of an equation like a balance (both sides must stay equal). Whatever you do to one side, do the same to the other side.
Left: 3x + 5
=
Right: 20
Common types of linear equations
- ax + b = c (simple one-step or two-step equations)
- Equations with brackets: a(bx + c) = d
- Equations with fractions: x/number + ... = ...
- Variables on both sides: ax + b = cx + d
Common mistakes and tips
- Forgetting to do the same operation on both sides.
- Not distributing negative signs correctly in brackets (for example, −(x − 2) = −x + 2).
- When clearing fractions, multiply every term on both sides by the common denominator.
- Always check your answer by substituting back into the original equation.
Word problem (Kenyan context)
Jane saves KSh x each week. After 6 weeks she has saved KSh 240. Form an equation and find x.
Equation: 6x = 240 → x = 240 ÷ 6 = 40. Jane saves KSh 40 each week.
Practice exercises
- Solve: 4x + 7 = 31.
- Solve: 7(x − 2) = 21.
- Solve: (x/3) + 5 = 9.
- Solve: 6x − 4 = 2x + 12.
- Solve: −3x + 9 = 0.
- Word problem: A matatu fare for two people is KSh 140. If one person pays KSh x and the other pays KSh (x + 10), find x.
Answers (work shown briefly)
- 4x + 7 = 31 → 4x = 24 → x = 6.
- 7(x − 2) = 21 → x − 2 = 3 → x = 5.
- x/3 + 5 = 9 → x/3 = 4 → x = 12.
- 6x − 4 = 2x + 12 → 4x = 16 → x = 4.
- −3x + 9 = 0 → −3x = −9 → x = 3.
- Let fares be x and x + 10. Then x + (x + 10) = 140 → 2x + 10 = 140 → 2x = 130 → x = 65. So one person paid KSh 65 and the other KSh 75.
Final tip: practise many different forms — with brackets, fractions and variables on both sides.
Always write each step clearly and check your answer.