Last reviewed: 25 October 2026
Linear equations are at the heart of junior school algebra. Once a learner can solve them confidently, a long list of other topics, from formulas to word problems and inequalities, becomes much easier. This guide builds the skill from the simplest equations up to the kind of word problems seen in Grade 9 revision papers.
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What an Equation Is
An equation says two expressions are equal, like 3x + 5 = 20. A linear equation has the unknown only to the power of one, so there is no x² or x³. Solving means finding the value of the unknown that makes both sides equal. Think of the equals sign as a balance: whatever you do to one side, you must do to the other.
One-Step and Two-Step Equations
- x + 7 = 15. Subtract 7 from both sides: x = 8.
- 4x = 28. Divide both sides by 4: x = 7.
- 3x + 5 = 20. Subtract 5: 3x = 15. Divide by 3: x = 5.
- x/2 − 3 = 4. Add 3: x/2 = 7. Multiply by 2: x = 14.
The rule is to undo operations in reverse order: deal with adding and subtracting first, then multiplying and dividing.
Unknowns on Both Sides
Collect the unknown terms on one side and numbers on the other. 5x − 4 = 2x + 11. Subtract 2x from both sides: 3x − 4 = 11. Add 4: 3x = 15. So x = 5. It is usually easier to move the smaller x term so the coefficient stays positive.
Equations With Brackets
Expand the brackets first, multiplying every term inside. 3(x + 4) = 2(x + 9) becomes 3x + 12 = 2x + 18. Subtract 2x: x + 12 = 18, so x = 6. Watch negative signs: −2(x − 3) = −2x + 6, not −2x − 6.
Equations With Fractions
Multiply every term by the lowest common multiple of the denominators to clear the fractions. For x/3 + x/4 = 7, the LCM is 12: 4x + 3x = 84, so 7x = 84 and x = 12. For (x + 1)/2 = (x + 4)/3, cross-multiply: 3(x + 1) = 2(x + 4), giving 3x + 3 = 2x + 8, so x = 5.
Forming Equations From Word Problems
Exam questions often describe a situation in words. Let a letter stand for the unknown, write the equation, then solve.
- Ages: Mother is three times as old as Kamau. In 12 years she will be twice as old as him. Let Kamau be x. Then 3x + 12 = 2(x + 12), so 3x + 12 = 2x + 24 and x = 12. Kamau is 12 and his mother is 36.
- Money: A pen costs KSh 15 more than a pencil. Two pens and three pencils cost KSh 180. Let a pencil be p: 2(p + 15) + 3p = 180, so 5p + 30 = 180 and p = 30. A pencil costs KSh 30 and a pen KSh 45.
- Perimeter: A rectangle’s length is 4 cm more than its width and its perimeter is 48 cm. 2(w + w + 4) = 48, so 4w + 8 = 48 and w = 10. The rectangle is 14 cm by 10 cm.
Always Check Your Answer
Substitute your answer back into the original equation. For 5x − 4 = 2x + 11 with x = 5: left side 21, right side 21. It takes ten seconds and catches most sign errors. In word problems, also check that the answer makes sense; a negative age or a pencil costing KSh 2,000 means something went wrong.
Common Mistakes
- Doing an operation to only one side.
- Expanding only the first term in a bracket.
- Losing a negative sign when moving terms.
- Answering x when the question asks for something else, such as Mother’s age.
Next Steps
Once equations are secure, move on to linear inequalities and graphs of straight lines. Our Grade 9 Maths revision topics list shows how it fits with other topics, and the Grade 9 Mathematics papers give exam practice. Use the exam countdown for free daily questions, the What to Revise report to find gaps, and the KJSEA grade calculator to understand scores. Full papers with worked answers are in the Monthly Pass (KSh 500 by M-Pesa).
Ten-minute check: try the free Grade 7 CBC Challenge. It has 10 questions, each marked on the spot with the working, and it shows which topics to practise next.