The Elimination Method
Adding equations to make a variable vanish.
The explanation
Elimination removes a variable by adding the two equations together.
If the coefficients are already opposites, just add:
3x + 2y = 12
5x − 2y = 4
Adding gives 8x = 16, so x = 2. Then back-substitute for y = 3.
If they are not opposites, multiply one or both equations first. To eliminate x from 2x + 3y = 7 and 3x − y = 5, multiply the first by 3 and the second by −2, giving 6x and −6x.
Multiply *every* term in the equation, both sides. That is where errors live.
Elimination is the linear-combination method: adding a multiple of one equation to another produces an equivalent system, because any solution of both originals satisfies any linear combination of them.
The systematic version scales the two equations by the ratios needed to make one variable's coefficients additive inverses — often LCM of the coefficients — then adds. Either variable may be targeted, and choosing the one with smaller or already-opposite coefficients reduces arithmetic.
For systems in standard form, elimination is generally faster than substitution and avoids fractions, which is why it is the method that scales: Gaussian elimination on larger systems is exactly this procedure applied systematically, and it is what matrix row reduction automates.
The degenerate cases appear as before. Eliminating a variable and obtaining 0 = k for nonzero k means inconsistent; obtaining 0 = 0 means dependent.
Worked example
Solve 3x + 4y = 18 and 5x − 2y = 4 by elimination.
- The y coefficients are 4 and −2. Multiply the second equation by 2: 10x − 4y = 8.
- Add the equations: 13x = 26, since 4y and −4y cancel.
- x = 2.
- Back-substitute into the first: 3(2) + 4y = 18, so 4y = 12 and y = 3.
Answer: (2, 3) — check: 3(2)+4(3) = 18 ✓ and 5(2)−2(3) = 4 ✓
Common mistakes
- Multiplying only one side of an equation.
- Adding when the coefficients match instead of subtracting (or scaling by a negative).