Method guideAlgebra IIntermediate10 min read

Solving systems by elimination

Align like terms, create opposite coefficients, add the equations, and recover the variable that was deliberately removed.

\begin{aligned}ax+by&=c\\-ax+dy&=e\end{aligned}\Rightarrow(b+d)y=c+e

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Updated July 13, 2026

MethodStart here

Multiplying an entire equation by a nonzero number preserves its solution set. Use that freedom to create opposite coefficients.

Standard form keeps columns honest

Put x-terms, y-terms, and constants in aligned columns. Missing terms can be written with coefficient zero.

Before adding, verify that the chosen variable has equal magnitude and opposite signs. Equal signs call for subtraction instead.

\begin{aligned}3x+2y&=11\\5x-2y&=13\end{aligned}

If a coefficient needs scaling, multiply both sides and every term in the equation. Scaling only the convenient term changes the equation.

Choose small multipliers, often using the least common multiple of coefficient magnitudes.

2(x+3y=7)\Rightarrow2x+6y=14

Once one variable is found, substitute into whichever original equation has simpler coefficients. Then check the pair in both originals.

A zero row with a nonzero constant signals no solution; a zero row with zero signals dependent equations.

0=9\Rightarrow\text{no solution}

Worked exampleCancel one variable

8x=24

Add the aligned equations; y cancels.

x=3

Divide by eight.

3(3)+2y=11\Rightarrow2y=2\Rightarrow y=1

Back-substitute.

5(3)-2(1)=13

Check the second equation.

Result

\boxed{(3,1)}

Watch for

Keep