Study guides & original practice
Systems of equations by elimination
A solution to a system must satisfy every equation at once. Elimination creates a new equation by adding suitable multiples so one variable cancels.
By ExamCat · Original study lesson · Published September 24, 2026
Understand the method
Keep each complete equation balanced when multiplying. After finding one variable, substitute into an original equation to find the other. The resulting ordered pair must pass both original checks; one successful substitution is not enough.
Worked example, step by step
Solve 2x + y = 11 and 3x − y = 9.
- Add the equations: the y terms cancel, leaving 5x = 20.
- Divide by five: x = 4.
- Substitute into the first equation: 8 + y = 11, so y = 3.
- Check the second: 3(4) − 3 = 9. The solution is (4, 3).
A mistake to watch for
Adding coefficients does not mean multiplying variables. If elimination produces 0 = 0, the equations may describe the same line; if it produces a contradiction, there is no common solution.
Try it yourself: three questions
Write an answer before opening the explanation. For calculations, keep your units and intermediate steps; for concepts, state the reason for your choice.
1. Solve x + y = 7 and x − y = 1.
Show answer and explanation
Add to get 2x = 8, so x = 4 and y = 3.
2. Do x + y = 2 and 2x + 2y = 8 share a solution?
Show answer and explanation
No. Doubling the first requires 2x + 2y = 4, contradicting the second. The lines are parallel.
3. What happens for x + y = 2 and 2x + 2y = 4?
Show answer and explanation
The second is twice the first. There are infinitely many ordered pairs on the same line, not one unique intersection.
Build your next practice session
Mark the first step where your reasoning differed from the explanation. Turn that step into a question, answer it without looking, then change one condition in the example and explain what would change. A correct guess is a reason to revisit the method.
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