Simultaneous equations: substitution vs elimination

If a problem has two unknowns, you need two separate equations to pin them down — that's what "simultaneous" means: both are true at the same time. There are two reliable methods, and choosing the right one saves real time.

Elimination — make one variable cancel

Line the equations up and add or subtract them so one variable disappears. It works when the coefficients already match, or can be made to match by multiplying.

The $y$ terms are $+2y$ and $-2y$. Add the equations and the $y$ vanishes:

Put $x = 4$ back into either equation: $4 - 2y = 0 \Rightarrow y = 2$.

Substitution — when one variable is already alone

If one equation already has a variable by itself (or one step away), substitution is cleaner. Rearrange one equation, then slot it into the other.

Replace the $y$ in the second equation:

Which method to use

• Coefficients match or are easy to match (like $2y$ and $2y$)? → elimination.
• One equation already says "$y = \dots$"? → substitution.

The check that catches everything

Put both answers back into the equation you didn't use to find the last variable. If both sides balance, you're right. This one habit catches sign slips before the examiner does — and in simultaneous equations, a sign slip is the usual cause of death.

Last revised 9 April 2025.