# Cramer’s Rule Calculator

URL: https://speedycalculator.com/calculators/math/cramers-rule/
Last updated: 2026-10-30

> The Cramer's rule calculator solves a 2 × 2 or 3 × 3 system of linear equations with determinants: each unknown equals |Wₓ| / |W|.

## Inputs

- Number of equations
- a₁ (x)
- b₁ (y)
- c₁ (z)
- d₁ (right side)
- a₂ (x)
- b₂ (y)
- c₂ (z)
- d₂ (right side)
- a₃ (x)
- b₃ (y)
- c₃ (z)
- d₃ (right side)

## Results

- Solution
- |W|
- |Wₓ|
- |W_y|
- |W_z|
- x
- y
- z

## Formula

```
x = |Wₓ| / |W|,  y = |W_y| / |W|,  z = |W_z| / |W|
Wₓ: W with the x-column replaced by the right-hand sides
```

## Explanation

Cramer's rule gives an explicit formula for the solution of a square linear system. It is elegant for small systems and for theory, though elimination is faster for large ones.

### How to use the Cramer's Rule Calculator

- **Number of equations**
- **a₁ (x)**
- **b₁ (y)**
- **c₁ (z)**
- **d₁ (right side)**
- **a₂ (x)**
- **b₂ (y)**
- **c₂ (z)**
- **d₂ (right side)**
- **a₃ (x)**
- **b₃ (y)**
- **c₃ (z)**
- **d₃ (right side)**

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

### How it works

The determinant of the coefficient matrix W is computed, then the determinants with each column replaced by the constants. If |W| = 0 the system has no unique solution.

## Worked example

Inputs — Number of equations: Two equations (x, y); a₁ (x): 2; b₁ (y): 3; d₁ (right side): 12; a₂ (x): 3; b₂ (y): -1; d₂ (right side): 7.

Results — **Solution = x = |Wx| / |W| = −33 / −11 = 3
y = |Wy| / |W| = −22 / −11 = 2**; |W| = −11; |Wₓ| = −33; |W_y| = −22; x = 3; y = 2.

## Key terms

- **Determinant**: A scalar computed from a square matrix.

## FAQ

**When can I use Cramer's rule?**
When the coefficient determinant is non-zero.

**What does**
W| = 0 mean? | No solution or infinitely many – Cramer's rule cannot decide which on its own.

**Is Cramer's rule efficient?**
For 2 × 2 and 3 × 3 systems yes; for larger systems use elimination.

**Who was Cramer?**
Gabriel Cramer, a Swiss mathematician who published the rule in 1750.


## Sources

- [Wolfram MathWorld – Cramer's rule](https://mathworld.wolfram.com/CramersRule.html)
- [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/)