Well, this is a long and complex proof, but it gets the job done, I think.
1. Consistency, in your case, means that
implies For a justification of this statement, see this thread - all of it.
2. Computing these determinants means you get that
3. However, is entirely equivalent to which is entirely equivalent to since by assumption. Hence, if the "if" part is true, then either a or b is zero.
4. But if a or b is zero, the implication means that the other one of a or b is also zero, in contradiction with Hence,
5. Thus, the determinant of the usual coefficient matrix associated with the first two equations is nonzero, hence the matrix will have a unique solution. Thus, you should, at this point, solve the system.
6. Now simplify the expression
with your plugged-in values for x and y.
The conclusion follows.
Does that make sense?