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How to Solve Systems of Equations by Elimination

Make one variable disappear. Then you only have one equation to solve.

The method

Line up the variables.
Multiply an equation if needed so one variable has matching coefficients.
Add or subtract the equations to remove that variable.
Solve, then substitute back to find the other variable.
Check in both original equations.

Worked examples

Example: $x + y = 10$ and $x - y = 4$

Add the equations: $2 x = 14$ , so $x = 7$ .

Substitute: $7 + y = 10$ , so $y = 3$ .

Example: $2 x + 3 y = 12$ and $x - y = 1$

Multiply the second equation by 2: $2 x - 2 y = 2$ .

Subtract: $5 y = 10$ , so $y = 2$ .

Substitute: $x - 2 = 1$ , so $x = 3$ .

Common mistake

$(3 x + 2 y) - (3 x - 2 y) = 0 y$

Subtracting $- 2 y$ is the same as adding $2 y$ . You get $4 y$ . So $3 x + 2 y = 8$ and $3 x - 2 y = 4$ give $4 y = 4$ , $y = 1$ and $x = 2$ .

Try it yourself

$x + y = 7$ and $x - y = 1$
Show answer
$x = 4, y = 3$
$2 x + y = 9$ and $x - y = 3$
Show answer
$x = 4, y = 1$
$3 x + 2 y = 12$ and $x + 2 y = 8$
Show answer
$x = 2, y = 3$

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