How to estimate reaction products of competitive binding? I have two reactions,

$$\ce{A + C \rightleftharpoons AC}$$

$$\ce{B + C \rightleftharpoons BC}$$

I'd like to estimate the relative steady state proportions 

$$\frac{[\ce{BC}]_{SS}}{[\ce{AC}]_{SS}+[\ce{BC}]_{SS}}$$ 

and 

$$\frac{[\ce{AC}]_{SS}}{[\ce{AC}]_{SS}+[\ce{BC}]_{SS}}$$ 

without numerically solving any differential equations, given $[\ce{A}]_{init} = [\ce{B}]_{init}$ and thermodynamic properties: $\Delta G, \Delta H, $ and $\Delta S$.

I'm hoping that there is a way to generalize this for more than two reactions in a linear way. It is for a DNA microarray optimization problem in which computational speed is important. Any thoughts?
Why would you have to solve differential equations for this problem?  What algebra have you tried so far?  

If C is in excess, then there is no competition!  You should be able to model the reactions as just $\ce{A &lt;=&gt; AC}$ and $\ce{B &lt;=&gt; BC}$.  The equilibrium equation would be (for A) $K_a=\frac{ac}{A_{init}-ac}$, and you can apparently already calculate $K_a$ from the thermodynamic parameters, so that is an equation where the only unknown is $ac$, the amount of AC formed.  The equation is linear!

$ac=A_{init}\frac{K_a}{1+K_a}$

A similar result holds for B of course.  

Once you can calculate $ac$ and $bc$, you should be able to calculate the parameters you want to know very easily.

If C is not in excess, the equilibrium equations will not be linear and the balance on C will couple the two equilibria (competition happens!).