System of Equations Solver (2×2)
Solve two linear equations in two unknowns. Shows the determinant, x and y by Cramer's rule, and graphs the two lines with their intersection.
About this calculator
Enter the coefficients for a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The solver computes det = a₁b₂ − a₂b₁. If det ≠ 0 there is a unique solution (Cramer's rule). If det = 0 the lines are parallel (no solution) or coincident (infinitely many solutions).
How to solve a 2×2 system
A system a₁x + b₁y = c₁, a₂x + b₂y = c₂ has a unique solution exactly when the determinant det = a₁b₂ − a₂b₁ is non-zero. Cramer's rule then gives x = (c₁b₂ − c₂b₁) / det and y = (a₁c₂ − a₂c₁) / det. Geometrically each equation is a line; the solution is their intersection point. When det = 0 the lines are parallel: if the constants are proportional too, the lines coincide and every point on the line is a solution (infinitely many); otherwise the lines never meet (no solution).