A system of
has no solutions if it has an equation of the form 0x1 +
0x2 + . . .
+ 0xn = b where b is not zero. This would be the same as
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.