System of 2 Equations Solver

About this calculator

A system of two linear equations in two unknowns, a₁x + b₁y = c₁ and a₂x + b₂y = c₂, can be solved directly with Cramer's rule: x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁), where the denominator is the determinant of the coefficient matrix. If that determinant is zero, the lines are parallel and the system has no unique solution (no solution or infinitely many).

The method is a staple of algebra courses because it avoids the substitution/elimination bookkeeping, and it shows up throughout engineering and science wherever two linear relationships must be solved simultaneously — mixing problems, circuit mesh-current analysis, and basic supply-and-demand equilibrium in economics.