GCSE Maths · Topic guide
Simultaneous equations appear as short ‘find x and y’ problems and as part of word questions (costs, speeds, mixtures). Linear pairs are usually solved by elimination or substitution; examiners want both values and sometimes a check. If coefficients don’t match, multiply one or both equations first. Always label which equation you’re manipulating so you don’t lose a sign under pressure.
Worked examples & mini quiz
For two linear equations, elimination often works fastest: align coefficients, add or subtract to remove one variable, solve, then substitute back. Use substitution when one equation is already y = … or x = ….
Example 1
Elimination
x + 2y = 7 … (1) 2x − y = 4 … (2) Multiply (1) by 2: 2x + 4y = 14. Subtract (2): (2x + 4y) − (2x − y) = 14 − 4 → 5y = 10 → y = 2. In (1): x + 4 = 7 → x = 3.
Example 2
Substitution
y = 2x − 1 and 3x + y = 14. Substitute: 3x + (2x − 1) = 14 → 5x = 15 → x = 3. Then y = 5.
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1. The same solution is obtained by elimination and by substitution when:
Correct: Equations are manipulated consistentlyBoth methods are valid if algebra steps stay balanced.
2. For 2x + 3y = 12 and x − y = 1, a good first step is:
Correct: Multiply the second equation by 3 and addMatching or eliminating y: x − y = 1 → 3x − 3y = 3; add to 2x + 3y = 12 → 5x = 15.
3. A ‘simultaneous’ system has:
Correct: Two or more equations true for the same unknownsSame (x, y) must satisfy all equations at once.
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