I am in a basic linear algebra course, and we are learning to solve linear equations with augmented matrices. We learned that when an augmented matrix is in row echelon form or reduced echelon form, you can tell if the system has one, infinitely many, or no solutions by looking if there is a pivot in every column or a pivot in every row. When you look for pivots, do entries in the augmented column count?
If the augmented matrix has a pivot in its last column, then the system corresponding to that augmented matrix is inconsistent. This is because in the reduced echolon form of the augmented matrix there will be a row of the form $[0\cdots 0\mid c]$ where $c$ is a nonzero number, namely the pivot. This means that $0=c$, implying the inconsistency of the system.
In fact, the same argument sketched above shows that a linear system is consistent if and only if its augmented matrix does not have a pivot in its last column. If no such pivot exists, you can solve the system by back-substitution.