Definition of consistent linear system

Definition: A linear system is consistent if it has at least one solution.

Equivalently, the system $Ax = b$ is consistent if and only if the vector $b$ lies in the column space of $A$ (i.e., $b$ can be expressed as a linear combination of the columns of $A$).

In echelon form of the augmented matrix, a system is consistent if and only if there is no row of the form $[0 \ 0 \ \cdots \ 0 \ | \ b]$ where $b \neq 0$.

Example:
$$\begin{cases} x + y = 3 \\ 2x + 2y = 6 \end{cases}$$
is consistent (infinitely many solutions: the equations are multiples of each other).