Cramer's rule is the determinant-powered shortcut for 2×2 systems — a formula that reads the answer straight off the coefficients, no elimination steps required. For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, compute the determinant D = a₁b₂ − a₂b₁, then x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D. It's substitution and elimination with the algebra pre-packaged: the same answer, delivered as a pattern. For 2x + 3y = 8 and x − y = 1: D = (2)(−1) − (3)(1) = −5, x = ((8)(−1) − (1)(3))/(−5) = 11/5 = 2.2, and y = 6/5 = 1.2 — a quick back-substitution into x − y = 1 confirms it.

The determinant D is the diagnostic: D ≠ 0 means exactly one solution (the lines cross), D = 0 means the lines are parallel (no solution) or identical (infinitely many). This calculator runs Cramer's rule on your system: enter the six coefficients a₁, b₁, c₁, a₂, b₂, c₂ and read the Determinant (D), x, and y. It's the ideal checker for elimination homework — solve by hand, then confirm all three outputs match. The worked example 2x + 3y = 8 with x − y = 1 gives you a known-good reference to test against.

With the pre-filled defaults — a₁ (coeff of x, eq 1) 2, b₁ (coeff of y, eq 1) 3, c₁ (constant, eq 1) 8, a₂ (coeff of x, eq 2) 1, b₂ (coeff of y, eq 2) -1, c₂ (constant, eq 2) 1 — the System of Equations Calculator returns Determinant (D) -5.0000, x 2.200000, y 1.200000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.