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
| Field | What to enter |
|---|---|
| a₁ | A number |
| b₁ | A number |
| c₁ | A number |
| a₂ | A number |
| b₂ | A number |
| c₂ | A number |
Example values
How the main result changes when a₁ changes and everything else stays at the example values above:
| a₁ | x |
|---|---|
| 1 | 2.5 |
| 1.5 | 2.2222 |
| 2 | 2 |
| 2.5 | 1.8182 |
| 3 | 1.6667 |
| 4 | 1.4286 |
Formula
D = a₁b₂ − a₂b₁x = (c₁b₂ − c₂b₁) ÷ D, y = (a₁c₂ − a₂c₁) ÷ DWorked 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.
Frequently asked questions
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.