Algebra
Systems of equations and inequalities
Solving systems by elimination
18 practice questions
2 video lessons
Theory + worked examples
Theory
Elimination cancels a variable by combining the equations:
- Align like terms.
- Multiply one or both equations so a variable's coefficients are opposite (or equal).
- Add (or subtract) to eliminate that variable.
- Solve and back-substitute.
Opposite coefficients add to zero; equal ones subtract to zero.
The elimination steps.
A worked example.
The idea:
\[(a x+by)+(cx-by)=\dots\ \Rightarrow\ y\ \text{cancels}\]
Make a variable's coefficients opposite, then add.
How to solve by elimination
- Line up like terms.
- Multiply to create opposite coefficients.
- Add the equations to eliminate a variable.
- Solve and back-substitute.
Example 1 — Add to eliminate
Solve \(x+y=10\) and \(x-y=4\).
Solution
Add the equations to cancel \(y\).
| \(2x\) | \(=\) | \(14\) |
| \(x\) | \(=\) | \(7\) |
| \(y\) | \(=\) | \(3\) |
Example 2 — Subtract to eliminate
Solve \(3x+2y=16\) and \(x+2y=8\).
Solution
Subtract to cancel \(2y\).
| \(2x\) | \(=\) | \(8\) |
| \(x\) | \(=\) | \(4\) |
| \(y\) | \(=\) | \(2\) |
Example 3 — Multiply first
Solve \(2x+y=7\) and \(x-y=2\).
Solution
Add directly — \(y\) cancels.
| \(3x\) | \(=\) | \(9\) |
| \(x\) | \(=\) | \(3\) |
| \(y\) | \(=\) | \(1\) |
Example 4 — When to use it
When is elimination convenient?
Solution
When a variable's coefficients are equal or opposite (or easily made so).
Common pitfalls
Multiply the whole equation, both sides.
Make coefficients opposite to add to zero.
Back-substitute to find the second variable.
Frequently asked questions
What is elimination?
Combining the equations so one variable cancels.
When do you multiply first?
When no variable's coefficients are already opposite or equal.
Do you add or subtract?
Add for opposite coefficients; subtract for equal ones.
What comes after eliminating one variable?
Solve for the other, then back-substitute.
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