# Linear Combination Calculator

URL: https://speedycalculator.com/calculators/math/linear-combination/
Last updated: 2026-10-30

> The linear combination calculator solves a system of two linear equations by the linear combination (addition) method: scale the equations so one variable cancels, add them and solve.

## Inputs

- a₁
- b₁
- c₁
- a₂
- b₂
- c₂

## Results

- Solution
- x
- y
- Steps

## Formula

```
m₁ = LCM(a₁, a₂)/a₁,  m₂ = LCM(a₁, a₂)/a₂
m₁·(eq 1) − m₂·(eq 2) eliminates x
```

## Explanation

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**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### 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.

## 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.

## FAQ

**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.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)