You must have gone wrong in your row reduction. I row reduced the matrix and I satisfied the condition. Check over your working, you might see where you went wrong.
Show that the system of equations x + y + 2z = a, x + z = b, and 2x + y + 3z = c are consistent if and only if c = a + b.
I put the equations into an augmented matrix, and did row operations to produce row-echelon form. I got the following:
1 1 2 | a
0 -1 -1 | b - a
0 0 0 | c - 2b
To be consistent, the third row must not imply no solutions (i.e. LHS must equal RHS). But this gives me c = 2b, not c = a + b. What am I doing wrong? Thanks!