Solving Systems by Elimination
Elimination (Addition) Method
The elimination method — also called the addition method — solves a system of linear equations by adding or subtracting the equations so that one variable cancels out. It is especially efficient when the coefficients are already set up for easy cancellation.
When the coefficients do not line up naturally, you multiply one or both equations by constants to create matching or opposite coefficients before adding. This lesson walks through every variation of the technique.
Both substitution and elimination always give the same answer. Choosing which to use is a matter of convenience — elimination shines when both equations are in standard form.
- Align equations vertically in standard form ($Ax + By = C$).
- Multiply one or both equations so one variable has opposite coefficients.
- Add the equations to eliminate that variable.
- Solve, then back-substitute.
$2x + 3y = 12$ and $4x - 3y = 6$
Add: $6x = 18 \;\Rightarrow\; x = 3$. Sub: $2(3) + 3y = 12 \;\Rightarrow\; y = 2$.
Solution: $(3, 2)$.
$3x + 2y = 16$ and $5x - 4y = -2$
Multiply first by 2: $6x + 4y = 32$. Add to second: $11x = 30 \;\Rightarrow\; x = \dfrac{30}{11}$.
Sub back: $y = \dfrac{31}{11}$. Solution: $\left(\dfrac{30}{11}, \dfrac{31}{11}\right)$.
$5x + 2y = 3$ and $10x + 4y = 6$
Multiply first by $-2$: $-10x - 4y = -6$. Add: $0 = 0$.
Infinitely many solutions (dependent system).
When to Use Which Method
| Method | Best When |
|---|---|
| Graphing | Estimating; visual check |
| Substitution | One variable already isolated |
| Elimination | Both in standard form $Ax + By = C$ |
Practice Problems
Show Answer Key
1. $(4, 3)$
2. $(3, 2.5)$
3. $(-2, 1)$
4. $(3, \dfrac{2}{3})$
5. $(2, 3)$
6. Dependent (infinitely many)
7. Inconsistent (no solution)
8. $(3, 2.5)$
9. $(2, 3)$ (multiply by 10 first)
10. $(6, 3)$