consistent system

A consistent system is one whose demands can all be met — the equations get along well enough that some assignment of the variables pleases every one of them at the same time. In plain terms, a consistent system has at least one solution.

Geometrically, consistency means the graphs actually share at least one point. For two lines this happens when they cross (giving exactly one solution) or when they coincide as the very same line (giving infinitely many). Consistency does not promise a unique answer; it only promises that an answer exists.

The opposite case is an inconsistent system, where the equations contradict one another and no assignment can satisfy them all — like demanding x + y = 1 and x + y = 4 at once. So consistent is the dividing line between solvable and impossible: among consistent systems, the further split is between independent (one solution) and dependent (infinitely many).