Solving linear equations
A linear equation has no powers of x higher than 1. To solve it, do the same thing to both sides until x is on its own. Each step undoes one operation, working in the reverse order to how they were done to x.
Part 1 of 3: Learn it
In short
- Whatever you do to one side, do the same to the other.
- Expand brackets first; gather the x terms on the side with more x.
- With fractions, multiply every term by the denominator to clear them.
Where this is in your specification
Spec points: DfE A17 (with A21 for forming equations)
| Board | Topic: Linear equations and inequalities |
|---|---|
| DfE content | A17, A21, A22 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 6 Algebra |
| Eduqas C300 | HA17, HA21, HA22 |
| Cambridge IGCSE 0580 | E2.5, E2.6 |
| National 5 C847 75 | Algebraic skills: linear equations and inequations |
One and two steps
Undo the operations in reverse order. For 5x − 7 = 18: add 7 to get 5x = 25, then divide by 5 to get x = 5.
Brackets and unknowns on both sides
- Expand any brackets: 3(2x + 1) = 4x + 15 becomes 6x + 3 = 4x + 15.
- Move the smaller x term so x stays positive: take 4x from both sides, 2x + 3 = 15.
- Finish as normal: 2x = 12, x = 6.
Equations with fractions
Multiply every term on both sides by the denominator. If there are two denominators, multiply by their lowest common multiple. For (x + 4)/3 = (x − 1)/2, multiply by 6 to get 2(x + 4) = 3(x − 1).
Forming equations
Use the facts in the question to write an equation, then solve it. Common sources: angles in a triangle add to 180°, the perimeter of a shape, or two expressions that are equal. Always answer the question that was asked, which may not be x itself.
Solve 4(x − 3) = 26.
Show the answer
4x − 12 = 26, 4x = 38, x = 9.5.
Part 2 of 3: See it worked
Worked examples
Example 1
Solve 7x − 4 = 3x + 22.
- Subtract 3x: 4x − 4 = 22
- Add 4: 4x = 26
- Divide by 4: x = 6.5
Answer: x = 6.5.
Example 2
Solve (x + 4)/3 = (x − 1)/2.
- Clear both fractions with × 6 on each side: 2(x + 4) = 3(x − 1)
- Expand: 2x + 8 = 3x − 3
- Subtract 2x and add 3: 11 = x
Answer: x = 11.
Example 3
The angles of a triangle are x, 2x + 15 and 3x − 21 degrees. Find the largest angle.
- x + 2x + 15 + 3x − 21 = 180
- 6x − 6 = 180, so 6x = 186 and x = 31
- Angles: 31°, 77° and 72°
Answer: 77°.
Common mistakes
- Only multiplying the first term inside a bracket.
- Doing an operation to one side only, or to only one term on a side.
- Losing a negative sign when moving terms: from 3 − 2x = 11, x = −4, not 4.
- Stopping at x when the question asks for an angle or a length.
Solve 9 − 2x = x − 6.
Show the answer
15 = 3x, so x = 5.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Solve 4(x − 3) = 26.
4x − 12 = 26, 4x = 38, x = 9.5.
Solve 9 − 2x = x − 6.
15 = 3x, so x = 5.
Solve (2x + 1)/5 = 3.
2x + 1 = 15, 2x = 14, x = 7.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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