Simultaneous equations
Maths · GCSE · The School
Elimination
To solve 3x + 2y = 16 and x + 2y = 8: the y terms match, so SUBTRACT the equations — 2x = 8, x = 4. Put x back into either equation: 4 + 2y = 8, y = 2. Always substitute into the OTHER equation to check: 3(4) + 2(2) = 16 ✓. If neither variable matches, multiply one equation through first to make a pair match.
Substitution
When one equation already says "y = something", substitution is faster: y = 2x − 1 and 3x + y = 14 gives 3x + (2x − 1) = 14, so 5x = 15, x = 3, then y = 5. Rearranging first only pays when a variable has a coefficient of 1 — otherwise fractions appear and elimination is cleaner.
What the answer means
Each equation is a straight line; solving them simultaneously finds where the lines CROSS. That is why there is normally exactly one solution. Parallel lines (same gradient, different intercept) never cross, so there is no solution; identical lines cross everywhere, so there are infinitely many. The algebra tells you which: you end with a false statement like 0 = 5, or a true one like 0 = 0.
Solve 3x + 2y = 16 and x + 2y = 8. The y terms match, so subtract the second from the first: 2x = 8, x = 4. Substitute into the second: 4 + 2y = 8, so 2y = 4 and y = 2. Now check in BOTH: 3(4) + 2(2) = 12 + 4 = 16 ✓ and 4 + 2(2) = 8 ✓. Checking only one equation is the standard trap — an arithmetic slip often still satisfies the equation you used to find y.
One equation quadratic: substitution is the natural method
Solve y = x + 1 and y = x² − 5 by substituting the linear equation into the quadratic: x + 1 = x² − 5, so x² − x − 6 = 0, which factorises to (x − 3)(x + 2) = 0, giving x = 3 or x = −2. Find y from the LINEAR equation each time, because it is simpler: x = 3 gives y = 4, and x = −2 gives y = −1. Substitution is the natural method here, because y is already the subject in both equations — there is nothing to eliminate by subtracting that substitution does not already do more directly.
Solving simultaneous equations from a graph
Plot both equations as straight lines and read off the coordinates where they cross — that point satisfies both equations at once, which is exactly what "simultaneous" means. It is less precise than algebra when the answer is not a whole number, but it is the fastest way to see WHY there is one solution, no solutions, or infinitely many, because you are looking directly at whether and how the lines meet.