Part 19 of 36 · EquationsSee the whole course

How to solve a system by elimination

By The Pip team · October 24, 2026 · 7 min read

Why adding two equations is allowed

An equation says two amounts are equal. If you add equal amounts to equal amounts, the results are still equal. So if x + y = 7 and x − y = 1, then (x + y) + (x − y) = 7 + 1 is a true equation too. The point of the move is that the y terms are +y and −y, and they cancel.

On a graph, both equations are lines, and the solution is the one point where they cross. (If that picture is new, start with solving a system by graphing.) Elimination finds that point with algebra and no drawing.

The steps

1. Line up both equations so that the x terms, the y terms and the numbers are in columns, like standard form: Ax + By = C.
2. Make one letter match. If a letter has opposite numbers (+y and −y), or the same number (2x and 2x), go on. If not, multiply one or both equations to make them match.
3. Cancel. Add the equations if the numbers are opposites. Subtract if they are the same.
4. Solve the one equation that is left.
5. Put the answer back into either original equation to find the other letter.
6. Check the pair in both equations.

Try all three kinds below. Press the step buttons to see each stage on the graph.

Example 1: add, x + y = 7 and x − y = 1

Example 2: subtract, 2x + 3y = 12 and 2x − y = 4

This time the x terms are the same (2x and 2x), so subtract the second equation from the first. Subtract everything, including the signs: minus (−y) becomes plus y.

Example 3: multiply first, 2x + 3y = 13 and 3x − 2y = 0

Nothing matches here: 2x and 3x, 3y and −2y. Make the y terms match. The numbers 3 and 2 both go into 6, so multiply the first equation by 2 and the second by 3. Every term gets multiplied, including the number on the right.

What if both letters cancel?

Sometimes adding or subtracting removes x and y at the same time. If what is left is false, like 0 = 5, the lines are parallel and there is no solution. If what is left is always true, like 0 = 0, the two equations are the same line and there are infinitely many solutions. See no solution or infinitely many solutions for the pictures.

Which method should you pick?

• Graphing shows the picture and gives a quick estimate.
• Substitution is the best fit when one letter is already alone, like y = 3x. See solving a system by substitution.
• Elimination is the best fit when both equations are in standard form with the letters lined up.

Watch it explained in under a minute

The five mistakes to avoid

• Multiplying only part of an equation. Multiply every term, including the number on the right.
• Subtracting only the first term of the second equation. Subtract all of it, so signs flip: −(−y) is +y.
• Adding when the numbers are the same (that makes 4x, not 0), or subtracting when they are opposites.
• Finding one letter and stopping. The solution is a pair (x, y).
• Skipping the check. Put the pair into both original equations.

Try it yourself

Solve each system. Tap a question to see the answer.

Where this fits in

You now have three ways to solve a system: graphing, substitution and elimination. Next in this course: what happens when the lines never meet, or are the same line, in no solution or infinitely many solutions.