Chemical equilibrium for simultaneous dissociation reactions While calculating the adiabatic flame temperature for the following combustion reaction:

$$\ce{\Phi\, C2H6 + $3.5$\,(O2 + $3.76$\,N2) -&gt; $a$\,CO2 + $b$\,CO + $d$\,H2O + $e$\,H2 + $f$\,O2 + $g$\,N2}$$

Where $\Phi$ is the equivalence ratio.

To determine the product composition ratios $a$, $b$, $d$, $e$, $f$ I need to make mass balance on $\ce{C}$, $\ce{O}$ and $\ce{H}$ and then use two $K_{p}$ (Chemical equilibrium constant) relations. Then solve this system of equation simultaneously.

My question is about the elementary reactions of dissociation I should use to apply $K_{p}$ equations, As I could have the following scenarios happening (regarding dissociation):
\begin{align}
\ce{CO + H2O &amp;&lt;=&gt; CO2 + H2}\\
\ce{CO2      &amp;&lt;=&gt; CO + $0.5$\,O2}\\
\ce{H2O      &amp;&lt;=&gt; H2 + $0.5$\,O2}\\
\end{align}

I only need two equations for $K_{p}$ which two should I use and why?
Short answer (After some trials): It does not matter which dissociation reaction you use, The system of equation will be solved anyway since the reactions are not independent.

However, using the first reaction as a dissociation mechanism would generate a Kp equation that's a lot easier to solve than the others.