Two equations, two unknowns. On its own neither equation pins down x or y — but together they have a single solution that satisfies both. Elimination finds it by combining the equations so one variable cancels, leaving an ordinary equation in the other.

The trick is that you can add two equations, or subtract one from the other, and the result is still true. Choose whichever makes a variable disappear. Take:

The y terms are +y and -y — equal and opposite. Add the two equations and the y vanishes:

Then back-substitute x = 3 into either original equation. From x - y = 2 we get 3 - y = 2, so y = 1. The solution is x = 3,\ y = 1 — and you can check it fits the other equation too, just as you would after .

See it built

Step through the cancellation. The two equations stack up; their y terms are equal and opposite, so adding them wipes y out and leaves a single equation in x. Solve that, then feed x back to find y.

When no pair of terms is already equal and opposite, first scale an equation — multiply it through by a number — so that a variable lines up to cancel. That is the same legal move as : whatever you do to one side, do to the other.

See it explained

Sal Khan works through solving a system by adding the equations to eliminate a variable.