This is one of life's great dilemmas. No matter what answers you put in the back of the book, someone will find a way to misuse and abuse them. The author MUST put something, but it must be done with the greatest of fear.
One does not want to create dependence.
One cannot guarantee that all are exactly correct.
It remains for the student to gain a little fortitude and discern for himself when an answer is incorrect.
What do you know about "consistent" systems?
If the right answer is 2a + b - c = 0, then pick some values and PROVE it. I tried a = b = 1 and c = 3. Guess what? No solution. The answer cannot be correct unless the question is written incorrectly.
Solve the first for a variable: y = a + z - 2x
Substitute into the other two:
2a - 2x + 2z = b
4a - 3x + 3x = c
Solve the first for a variable: x = a + z - (b/2)
Substitute into the other:
2a + 3b - 2c = 0
There is a relationship. I tried a = b = 1 and c = 5/2.
Why worry about it when you can PROVE it!