The last column of the augmented matrix is not a pivot column, which means the system is consistent.
Work Step by Step
We know in order for a linear system to be consistent, the last column of its augmented matrix must not be a pivot column. We also know that we can form the augmented matrix by adjoining a column on the right side of the coefficient matrix. Notice by adjoining this column, we do not add any rows to the coefficient matrix, and, therefore, we do not obtain any more pivot positions. So the pivot positions of the augmented matrix will be the same as the pivot positions in the coefficient matrix. Therefore, if the coefficient matrix has a pivot in each row, then so does the augmented matrix, *AND* since all of the pivots appear in the coefficient matrix, none of the pivots are in the last column of the augmented matrix. Hence the last column of the augmented matrix is not a pivot column, which means the system is consistent.