Elimination is often the quickest way to solve two simultaneous linear equations by hand, especially when coefficients are whole numbers. It is the idea behind Gaussian elimination for larger systems.
How to use the Elimination Method Calculator
- Enter the coefficients of a₁x + b₁y = c₁ in the first row.
- Enter a₂x + b₂y = c₂ in the second row.
- Read x and y and follow the elimination steps.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Each equation is multiplied so the x-coefficients match, then one is subtracted from the other to eliminate x. Solving for y and substituting back gives x. If the determinant a₁b₂ − a₂b₁ is zero the lines are parallel or identical.
Formula
a₁x + b₁y = c₁, a₂x + b₂y = c₂
Multiply by m₁ = LCM(a₁, a₂)/a₁ and m₂ = LCM(a₁, a₂)/a₂, then subtract
Worked example
Inputs — a₁: 2; b₁: 3; c₁: 12; a₂: 3; b₂: -1; c₂: 7.
Results — Solution = (x, y) = (3, 2); x = 3; y = 2; Step-by-step solution = 2x + 3y = 12
3x − 1y = 7
1. Multiply equation 1 by 3 and equation 2 by 2 so the x-terms match: 6x + 9y = 36 and 6x − 2y = 14
2. Subtract to eliminate x: 11y = 22 → y = 2
3. Substitute y into equation 1: 2x = 12 − 3·2 → x = 3.
Key terms
- Simultaneous equations
- Equations that must all be true at once.
Frequently asked questions
When should I use elimination instead of substitution?
When no variable has a coefficient of 1 – elimination avoids fractions until the end.
What if both variables cancel?
If you get 0 = 0 there are infinitely many solutions; if you get 0 = a non-zero number there is none.
Can elimination handle three equations?
Yes – eliminate the same variable from two pairs, then solve the remaining 2 × 2 system.
Why use the LCM?
Scaling to the least common multiple keeps the numbers as small as possible.