smc

Substituting in the $(x,y)$ pairs gives the following system of equations: $$a-b+c=12$$ $$c=5$$ $$4a+2b+c=-3$$ We know $c=5$, so plugging this in reduces the system to two variables: $$a-b=7$$ $$4a+2b=-8$$ Dividing the second equation by 2 gives $2a+b=-4$, which can be added to the first equation to get $3a=3$, or $a=1$. So the solution set is $(1,-6,5)$, and the sum is $\boxed{0}$.