System of Three Equations Solver

About this calculator

Cramer's rule solves a system of three linear equations in three unknowns by expressing each variable as a ratio of determinants: x = Dx/D, y = Dy/D, and z = Dz/D, where D is the determinant of the coefficient matrix and Dx, Dy, Dz replace the corresponding column with the constants column. If D = 0, the system has no unique solution — it's either inconsistent or has infinitely many solutions.

This solver takes the nine coefficients and three constants, computes all four determinants, and returns x, y, and z along with the coefficient determinant so you can confirm the system was actually solvable. Engineering, physics, and economics students use it to check circuit-analysis, statics, or input-output model homework, while professionals use the same method to solve small systems (three-force equilibrium, three-component mixture ratios) without setting up a full matrix solver.