A system of linear equations has no solutions if it has an equation of the form 0x₁ + 0x₂ + . . . + 0x_n = b where b is not zero.  This would be the same as its augmented matrix having a row

        0    0    . . .     0    b

where b is not zero.  The same is true of a system whose augmented matrix can be row reduced to a matrix having such a row.

Thus, to tell if a system of linear equations has solutions, row reduce its augmented matrix to echelon form.   If this form has a row of all zeros, except for a nonzero last entry, the original system has no solutions.  Another way of saying this is that if the last column in the augmented matrix is a pivot column, then the system is inconsistent.

If the last column is not a pivot column, then the system is consistent.