Consistency of a System of Linear Equations

A system of equations is consistent if it has at least one solution, and inconsistent if it has none. Determinants tell us at a glance which case a system falls into.

Two equations in two unknowns. With $D$, $D_x$, $D_y$ as in Cramer's Rule:

• If $D \neq 0$: the system is consistent with a unique solution $x=D_x/D,\ y=D_y/D$. Geometrically the two lines cross at exactly one point.
• If $D = 0$ and $D_x = D_y = 0$: the system is consistent with infinitely many solutions — the two equations describe the same line (they are dependent).
• If $D = 0$ but at least one of $D_x, D_y$ is non-zero: the system is inconsistent — the two lines are parallel and never meet, so there is no solution. …