Definition of inconsistent linear system

Definition: A linear system is inconsistent if it has no solutions.

In echelon form of the augmented matrix, an inconsistent system always contains a row of the form:
$$[0 \ 0 \ \cdots \ 0 \ | \ b] \quad \text{where } b \neq 0$$

This represents the equation $0 = b$ with $b \neq 0$, which is impossible.

Example:
$$\begin{cases} x + y = 2 \\ x + y = 5 \end{cases}$$

Row reducing: subtract the first from the second gives $0 = 3$, which is inconsistent. Geometrically, these are parallel lines that never intersect.