# System of Equations Solver (2×2)

URL: https://speedycalculator.com/calculators/math/system-of-equations/
Last updated: 2026-10-25

> This solver finds x and y for two linear equations, a₁x + b₁y = c₁ and a₂x + b₂y = c₂, using Cramer’s rule. It reports when the system has no unique solution.

## Inputs

- a₁
- b₁
- c₁
- a₂
- b₂
- c₂

## Results

- Determinant
- x
- y

## Formula

```
D = a₁b₂ − a₂b₁
x = (c₁b₂ − c₂b₁) ÷ D,  y = (a₁c₂ − a₂c₁) ÷ D
```

## Explanation

Two lines in a plane usually cross at exactly one point, which is the solution of the system.

### How to use the System of Equations Solver

- **a₁**
- **b₁**
- **c₁**
- **a₂**
- **b₂**
- **c₂**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Cramer’s rule uses determinants. If the determinant D is zero, the lines are parallel or identical.

### Inputs explained

FieldWhat to entera₁A numberb₁A numberc₁A numbera₂A numberb₂A numberc₂A number

### Example values

How the main result changes when **a₁** changes and everything else stays at the example values above:

a₁x12.51.52.2222222.51.818231.666741.4286

## Worked example

Inputs — a₁: 2; b₁: 3; c₁: 13; a₂: 1; b₂: -1; c₂: -1.

Results — Determinant = -5; **x = 2**; y = 3.

## Key terms

- **Determinant**: A number computed from coefficients that shows whether a unique solution exists.

## FAQ

**What does a zero determinant mean?**
No unique solution: the equations are parallel (none) or the same line (infinitely many).

**Can I solve three equations?**
This tool handles two; larger systems need matrix methods.

**What formula does the System of Equations Solver use?**
D = a₁b₂ − a₂b₁.