Solving Linear Systems

A system of linear equations asks which values satisfy every equation at once. In 2D, each equation is a line and the solution is their intersection; in 3D, each is a plane. This calculator solves 2×2 and 3×3 systems exactly using Cramer's rule.

Determinants Decide Everything

The coefficient determinant tells you the outcome: nonzero → one unique solution; zero → either no solution (parallel, never meeting) or infinitely many (coincident). The chart makes the 2×2 case visually obvious.

Real Applications

Linear systems model supply-and-demand equilibrium, circuit currents (Kirchhoff's laws), chemical reaction balancing, and economic input-output tables. Solving them reliably is foundational to science and engineering.