How it works
The solver uses Cramer’s rule. It computes the determinant of the coefficients (D). Then it swaps one column at a time for the constants to get Dx, Dy and Dz. Each unknown is its determinant divided by D: x = Dx ÷ D.
Example (2 × 2)
2x + 3y = 13 and x − y = −1. The determinant is 2 × (−1) − 3 × 1 = −5, and the answer is x = 2, y = 3.
Example (3 × 3)
x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2. The determinant is 7 and the answer is x = 1, y = 2, z = 3.
When there is no single answer
- D ≠ 0: exactly one solution.
- D = 0: no solution (the equations contradict each other) or infinitely many (one equation is a combination of the others).
For a single equation with x², use the quadratic equation solver; for determinants and inverses, the matrix calculator.
Frequently asked questions
What if an equation has no y or z?
Type 0 as its coefficient. For example, 2x + z = 5 has b = 0.
Can I use decimals and negative numbers?
Yes. Use a minus sign for negative coefficients and a point or comma for decimals.
Is Cramer’s rule the same as elimination?
They give the same answer. Elimination (adding and subtracting equations) is faster by hand; Cramer’s rule is easier to follow with determinants.
See also
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