How to Solve a System of Equations by Elimination

Problem

2x + 3y = 12,\quad x - y = 1

Answer

x = 3,\ y = 2

Elimination removes one variable by adding two equations. Make the coefficients of one variable opposites, add, then solve the equation that is left.

Step-by-step solution

Pick a variable and make its coefficients opposites.
$3(x - y) = 3(1) \Rightarrow 3x - 3y = 3$
Why: Multiplying the second equation by 3 turns -y into -3y, which is the opposite of +3y in the first equation.
Add the two equations.
$(2x + 3y) + (3x - 3y) = 12 + 3$
Why: The y terms cancel, leaving one equation in x.
Solve for x.
$5x = 15 \Rightarrow x = 3$
Why: Divide both sides by 5.
Substitute x = 3 into an original equation.
$3 - y = 1 \Rightarrow y = 2$
Why: The solution needs both values, and the simpler equation gives y quickly.

Multiplying only the left side of x - y = 1 by 3. The right side must be multiplied too.
Adding when the coefficients are the same sign. Add only when the coefficients are opposites.
Stopping after finding x. Write the pair (3, 2).

Use the same method. Work each problem on paper, then open the answer to check.