I am given three equations with two unknowns. I am asked to find an equation which must be satisfied by $a$, $b$, & $c$ in order for this system to have a solution.
$$ x - y = a \\ -2x + 3y = b \\ -x - 2y = c $$
I have used Reduced Row Echelon Form to solve for the pivot variables $x$ and $y$.
$$ x = 3a + b \\ y = 2a + b $$
At this point I am able to arbitrarily pick $a$ and $b$ with success for the first two equations but am confused with the third equation and $c$. At this point I have reasoned $c = -x - 2y = -7a - 3b$ after substituting the values found with rref. Solving in terms of $a$, $b$, and $c$, I finally arrive at $$-7a - 3b - c = 0$$
Logically this would be an equation that if satisfied would give the original system a solution. Is this the correct solution as asked? Is the logic used to derive it correct as well.