A worked example: 2x + 3y = 13, x − y = 1
Solving that system gives x = 3.2 and y = 2.2 — checkable directly: 2(3.2) + 3(2.2) = 6.4 + 6.6 = 13, and 3.2 − 2.2 = 1, both matching the original equations exactly.
The formula: determinants instead of substitution
For a1x + b1y = c1 and a2x + b2y = c2, Cramer's rule computes a coefficient determinant (a1b2 − a2b1) and two more determinants (swapping in the constants for each variable's column), then divides. It's a direct, mechanical formula — no need to manually isolate one variable and substitute it into the other equation.
What "no unique solution" means here
When the coefficient determinant is zero, the two equations are parallel (no solution at all) or identical (infinitely many solutions) — Cramer's rule can't produce a single answer either way, since dividing by zero is undefined.
Why this is scoped to exactly two equations
Cramer's rule generalizes to larger systems, but the determinant calculations grow quickly more complex — a 3×3 or larger system is better handled with the Matrix Calculator's row-reduction tools instead of hand-expanded determinants.
An older, equally valid alternative
Before determinant-based methods existed, systems like this were solved by substitution — solving one equation for one variable and plugging that expression into the other. Both approaches are solving the identical system, and necessarily agree on the answer.