System of Equations Solver
Solve 2×2 and 3×3 linear systems with Cramer's rule, showing every determinant step.
Enter the coefficients a, b, c and so on to find the solution (x, y, z) of a linear system with two or three unknowns. Each step of Cramer's rule is laid out visually, including every determinant that goes into the answer.
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How to use
- Choose the tab that matches your system: two equations in two unknowns, or three in three.
- Enter the coefficient of each unknown (a, b, c…) and the constant on the right-hand side of every equation.
- Review whether a solution exists, along with the numerator and denominator determinants from Cramer's rule and the final answer.
FAQ
- What is Cramer's rule?
- It's a formula from linear algebra that uses determinants to solve a system of linear equations. As long as the determinant D of the coefficient matrix isn't zero, the value of each unknown is found by replacing the matching column with the constant terms, taking that determinant, and dividing by D.
- What happens when the determinant D equals 0?
- When the coefficient matrix's determinant D is 0, Cramer's rule can't be applied. That means the equations are either parallel with no solution, or they're actually the same line with infinitely many solutions — and this calculator tells you which of the two cases you've run into whenever D comes out to 0.
- What happens when the determinant D equals 0?
- When the coefficient matrix's determinant D is 0, Cramer's rule can't be applied. That means the equations are either parallel with no solution, or they're actually the same line with infinitely many solutions — and this calculator tells you which of the two cases you've run into whenever D comes out to 0.