A *system of
linear equations*
has no solutions if it has an equation of the form 0x_{1} +
0x_{2} + . . .
+ 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*.