Consistent and inconsistent systems

Definition

Let $F$ be a field. A system of linear equations over $F$ is consistent when it has at least one solution in $F^n$. It is inconsistent when it has no solution.

Remarks

The system $x + y = 1$, $x + y = 2$ over $\R$ is inconsistent. Every homogeneous system is consistent, because the zero tuple is a solution. The Rouché–Capelli theorem gives a test with ranks: $Ax = b$ is consistent if and only if $\rank A = \rank [A \mid b]$.

Explanations

L1 human

"Consistent" means that the equations can all be true at the same time. Two parallel lines in the plane, like $x + y = 1$ and $x + y = 2$, have no common point, so their system is inconsistent. Elimination shows inconsistency when it produces an equation $0 = c$ with $c \ne 0$.