System of 3 Equations Solver

About this calculator

A 3×3 system of linear equations A·x = b has three unknowns (x, y, z) and three equations. This calculator solves it with Cramer's rule: each unknown equals the determinant of a modified coefficient matrix — with that variable's column replaced by the constants vector b — divided by the determinant of the original coefficient matrix A. So x = det(Ax)/det(A), y = det(Ay)/det(A), and z = det(Az)/det(A).

Cramer's rule only works when det(A) ≠ 0, meaning the three equations are independent and describe three planes meeting at exactly one point; a zero determinant signals either no solution or infinitely many. While Gaussian elimination scales better for larger systems, Cramer's rule is a clean, closed-form method well suited to 3×3 systems that appear throughout engineering, circuit analysis, and mixture or balance problems with exactly three constraints.