What if the system has no solution?

Row reduction produces a contradictory row (0 = c). The calculator reports that the system is inconsistent.

What about infinitely many solutions?

The solution is expressed in terms of free variables (parameters).

System of Equations Calculator

Solve simultaneous equations with full row-reduction steps. Linear systems are solved with Gauss–Jordan elimination and checked with Cramer's rule; nonlinear systems use substitution.

How to use the system of equations calculator

Enter one equation per box.
Use the same variable names in each equation (x, y, z).
See the augmented matrix, each row operation and the final solution checked in every equation.

Examples

2x + 3y = 12, x − y = 1

Answer: x = 3, y = 2

x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2

Answer: x = 1, y = 2, z = 3

Formulas

Cramer's Rule

x_{i} = \frac{det ( A _{i} )}{det ( A )}