Linear Equations
A linear equation contains a variable raised to the first power and can be written in the form ax + b = c, where a, b, and c are constants. These equations have exactly one solution and appear as straight lines when graphed. The goal is to isolate the variable by performing inverse operations on both sides of the equation.
Why it matters
Linear equations model countless real-world relationships where one quantity changes at a constant rate relative to another. A mobile phone contract might charge £25 monthly plus £0.10 per text, represented as C = 25 + 0.1t. Engineers use linear equations to calculate stress-strain relationships in materials, whilst economists model supply and demand curves. In GCSE Foundation mathematics, linear equations form the foundation for simultaneous equations and quadratic functions. They appear in Year 11 examinations worth approximately 15-20 marks across multiple papers. Beyond school, linear programming optimises business operations, from minimising costs to maximising profits. Financial advisors use linear models to project savings growth, and scientists employ them to analyse experimental data where variables maintain proportional relationships.
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 − 3 = -59
Answer: x = -8
- Add 3 to both sides → 7x = -56 — Isolate the x term by removing the constant.
- Divide both sides by 7 → x = -8 — -56 ÷ 7 = -8.
- Verify → 7(-8) − 3 = -59 ✓ — Substitution confirms the solution.
4x + 17 = 6x + 5
Answer: x = 6
- Subtract 6x from both sides → -2x + 17 = 5 — Collect all x terms on one side.
- Subtract 17 from both sides → -2x = -12 — Move constants to the other side.
- Divide both sides by -2 → x = 6 — -12 ÷ -2 = 6.
- Verify → LHS = RHS = 41 ✓ — Both sides equal the same value.
Common mistakes
- When solving 3x + 5 = 17, incorrectly writing x = 17 - 5 = 12 instead of first subtracting 5 from both sides to get 3x = 12, then dividing by 3 to get x = 4.
- In equations like 2x - 7 = x + 3, combining terms incorrectly to get x = 10 instead of properly collecting like terms: 2x - x = 3 + 7, giving x = 10.
- When checking solutions, substituting back into only one side of the original equation rather than verifying both sides produce the same value.