Linear Equations
Linear equations form the backbone of GCSE algebra, yet many Year 11 students still struggle with the systematic approach needed to isolate variables. Whether solving x + 6 = 7 or tackling 8x - 32 = 4x, the key lies in performing identical operations to both sides of the equation.
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Why it matters
Linear equations appear everywhere in real-world problem solving, from calculating mobile phone contracts to determining break-even points in business studies. A student working out how many £3 cinema tickets they can buy with £45 is solving 3x = 45. Engineers use linear equations to calculate load distributions, whilst economists model supply and demand curves. In GCSE Foundation papers, linear equations typically account for 15-20 marks across multiple questions. Students who master these skills find simultaneous equations, quadratic formula derivations, and A-level calculus significantly more accessible. The systematic thinking required—collecting like terms, maintaining equation balance, checking solutions—develops logical reasoning that transfers to physics, chemistry, and computer science coursework.
How to solve linear equations
Linear equations — how to
- Collect x-terms on one side, constants on the other.
- Do the same operation to both sides (add, subtract, multiply, divide).
- Divide by the coefficient of x to isolate x.
Example: 3x + 7 = 22 → 3x = 15 → x = 5.
Worked examples
x + 6 = 7
Answer: x = 1
- Subtract 6 from both sides → x = 7 − 6 — To isolate x, undo the addition.
- Calculate → x = 1 — 7 − 6 = 1.
- Verify → 1 + 6 = 7 ✓ — Substitution confirms the solution.
7x + 1 = -6
Answer: x = -1
- Subtract 1 from both sides → 7x = -7 — Isolate the x term by removing the constant.
- Divide both sides by 7 → x = -1 — -7 ÷ 7 = -1.
- Verify → 7(-1) + 1 = -6 ✓ — Substitution confirms the solution.
8x − 32 = 4x + 0
Answer: x = 8
- Subtract 4x from both sides → 4x − 32 = 0 — Collect all x terms on one side.
- Add 32 to both sides → 4x = 32 — Move constants to the other side.
- Divide both sides by 4 → x = 8 — 32 ÷ 4 = 8.
- Verify → LHS = RHS = 32 ✓ — Both sides equal the same value.
Common mistakes
- Adding or subtracting incorrectly when moving terms across the equals sign. Students write 3x + 7 = 22 becomes 3x = 22 + 7 = 29 instead of 3x = 22 - 7 = 15, forgetting to change the sign.
- Dividing only one side by the coefficient. For 4x = 12, students write x = 12 ÷ 4 = 3 but forget to show 4x ÷ 4 = 12 ÷ 4, missing the balanced operation principle.
- Incorrectly combining like terms when variables appear on both sides. In 5x - 3 = 2x + 9, students calculate 5x - 2x = 7x instead of 3x, then solve 7x = 12 getting x = 12/7 rather than x = 4.