The linear combination method is the elimination method under another name: you form a combination of the equations in which one variable disappears. It is the idea behind Gaussian elimination.
How to use the Linear Combination Calculator
- a₁
- b₁
- c₁
- a₂
- b₂
- c₂
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How it works
Both equations are scaled so the x-coefficients match, subtracted to eliminate x, and the remaining equation is solved for y; substitution gives x.
Formula
m₁ = LCM(a₁, a₂)/a₁, m₂ = LCM(a₁, a₂)/a₂
m₁·(eq 1) − m₂·(eq 2) eliminates x
Worked example
Inputs — a₁: 2; b₁: 3; c₁: 13; a₂: 5; b₂: -2; c₂: 4.
Results — Solution = (x, y) = (2, 3); x = 2; y = 3; Steps = 2x + 3y = 13
5x − 2y = 4
1. Multiply equation 1 by 5 and equation 2 by 2 so the x-terms match: 10x + 15y = 65 and 10x − 4y = 8
2. Subtract to eliminate x: 19y = 57 → y = 3
3. Substitute y into equation 1: 2x = 13 − 3·3 → x = 2.
Key terms
- Linear combination
- A sum of scalar multiples.
Frequently asked questions
What is a linear combination of equations?
A sum of multiples of the equations, such as 3·(eq 1) − 2·(eq 2).
Why does it work?
Any combination of true equations is also true, so the solution is unchanged.
What if the variables cancel completely?
0 = 0 means infinitely many solutions; 0 = k ≠ 0 means none.
Is it the same as elimination?
Yes.