HMMT 2009 U10

Let $a, b$, and $c$ be real numbers. Consider the system of simultaneous equations in variables $x$ and $y$ :

$$\begin{aligned} a x+b y & =c-1 \\ (a+5) x+(b+3) y & =c+1 \end{aligned}$$

Determine the value(s) of $c$ in terms of $a$ such that the system always has a solution for any $a$ and $b$.