GCSE Maths Linear Equations Notes
A linear equation has a single unknown to the power of one. Solving it means rearranging step by step until the letter is on its own.
Key concepts
- Linear equation
- An equation where the highest power of the unknown is 1, e.g. 4x − 3 = 9. Its graph is a straight line.
- Inverse operations
- To undo + you −, to undo × you ÷. You apply inverse operations to both sides to isolate the unknown.
- Unknown on both sides
- Collect the letters on one side and the numbers on the other, e.g. 5x − 2 = 3x + 8 → 2x = 10.
Key formulas
General linear form
ax + b = c
Solve by subtracting b, then dividing by a.
Worked example
Solve 5x − 2 = 3x + 8.
- 1Subtract 3x from both sides to collect letters: 2x − 2 = 8.
- 2Add 2 to both sides: 2x = 10.
- 3Divide both sides by 2: x = 5.
Answer: x = 5
Common mistakes
✗ Moving a term across the equals sign without changing its sign.
✓ Use inverse operations: a +5 becomes −5 when it crosses, a ×3 becomes ÷3.
✗ Dividing only one term by the coefficient, e.g. 2x = 10 → x = 10 − 2.
✓ Divide the whole side: 2x = 10 → x = 10 ÷ 2 = 5.
✗ Getting confused with negatives when the unknown ends up on the right.
✓ Collect the letters on whichever side keeps the x-coefficient positive.
Quick check
Try these, then reveal the answers.
Solve x + 9 = 17.Reveal answer
x = 8
Solve 4x = 28.Reveal answer
x = 7
Solve 6x − 5 = 2x + 11.Reveal answer
x = 4
Exam tips
- Check your answer by substituting it back into the original equation.
- If there are fractions, multiply every term by the denominator first to clear them.
- Keep equals signs lined up underneath each other — it makes method marks easy to award.
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Frequently asked questions
How do I solve an equation with x on both sides?+
Collect the x-terms on one side and the numbers on the other using inverse operations, then divide to find x.
What does 'solve' mean in GCSE Maths?+
It means find the value of the unknown that makes the equation true — always show your working line by line.
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