System of Equations Calculator
Solve systems of linear equations with two or three unknowns. Enter the coefficients and constants, and this calculator applies Cramer's rule to find the solution, compute the determinant, and identify whether the system has a unique, infinite, or no solution.
A system of linear equations consists of two or more linear equations with the same variables. For a 2×2 system (two equations, two unknowns), the solution is the point where two lines intersect. For a 3×3 system, it is the point where three planes meet.
This calculator uses Cramer's rule, which expresses each variable as a ratio of determinants. For the system Ax = b, the solution is x_i = det(A_i) / det(A), where A_i is the matrix formed by replacing column i of A with the constant vector b. The method works whenever det(A) is nonzero, indicating a unique solution.
When the determinant is zero, the system is either inconsistent (no solution, meaning the lines or planes are parallel) or dependent (infinitely many solutions, meaning the equations describe the same line or plane). The calculator detects both cases and reports the result accordingly.