Solving by Elimination

≈ 30 min

Elimination adds or subtracts whole equations to cancel a variable.

Worked example. Solve:

x + y = 10 \qquad (1)

x - y = 4 \qquad (2)

Add (1) and (2) — the $y$ 's cancel:

2x = 14 \;\Rightarrow\; x = 7

Substitute into (1): $7 + y = 10 \Rightarrow y = 3$ . Solution $(7, 3)$ .

If no variable cancels immediately, multiply an equation first: to solve $2x + y = 11$ and $x - y = 1$ , the $y$ 's already cancel by adding. But for $2x + 3y = 12$ and $x + y = 5$ , multiply the second by 2 or 3 to line up a matching coefficient, then subtract.

Solve by elimination and give the value of $x$ :

x + y = 10 \qquad x - y = 4

Solve by elimination and give the value of $y$ :

2x + y = 11 \qquad x - y = 1

To eliminate $y$ from $2x + 3y = 12$ and $x + y = 5$ , you could multiply the second equation by…

2, then add

3, then subtract

5, then add

12, then subtract