The question:

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.

My attempt:

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!