Simultaneous equations
Linear/linear simultaneous equations
Two linear equations in two unknowns. Methods: elimination and substitution.
Elimination method
Make the coefficient of one variable the same in both equations, then add or subtract.
Example: 3x + 2y = 11 ... (1) 5x − 2y = 21 ... (2) Add: 8x = 32 → x = 4. Substitute: 12 + 2y = 11 → y = −½.
If coefficients don't match, scale one or both equations first.
Substitution method
Rearrange one equation for one variable, substitute into the other.
Example: y = 2x − 3 and 3x + y = 12. 3x + (2x − 3) = 12 → 5x = 15 → x = 3; y = 3.
Linear/quadratic simultaneous equations (Higher)
Edexcel tests this at Higher: one linear and one quadratic equation.
Method: use substitution (the linear equation is rearranged and substituted into the quadratic).
Example: y = x + 3 ... (1) x² + y² = 29 ... (2) Substitute (1) into (2): x² + (x + 3)² = 29. x² + x² + 6x + 9 = 29. 2x² + 6x − 20 = 0 → x² + 3x − 10 = 0 → (x + 5)(x − 2) = 0. x = −5 or x = 2. Corresponding y values: y = −2 or y = 5. Solutions: (−5, −2) and (2, 5).
Graphical interpretation
Linear/linear: one intersection point (if not parallel). Linear/quadratic: 0, 1, or 2 intersection points (discriminant of the resulting quadratic).
Edexcel exam style
Papers 2 and 3 often present simultaneous equations in a context (e.g. "two shops charge different prices; find when they cost the same"). The answer must be interpreted in context (e.g. "after 4 months").
⚠Common mistakes
- Not checking the solution by substituting back into BOTH original equations.
- Forgetting to find the y value (or second variable) — the question asks for both.
- Linear/quadratic: not substituting the linear into the quadratic — trying to eliminate instead.
- Sign errors when subtracting equations: be careful when the signs require subtraction.
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