While analyzing the chemical equilibrium of a combustion reaction products and finding their dissociation ratios, I got the following set of equations (5 unknowns & 5 equations):
$$ \left\{ \begin{align} b &= 2.6 - a\\ d &= 3.9 - e\\ 2f + a + e &= 0.5\\ \frac{3.9-e}{e\left(\frac{0.5-a+e}{2}\right)^{0.5}} \left(\frac{1}{4.15 - 0.5a + 0.5e}\right)^{-0.5} &= -92.208\\ \frac{a}{(2.6-a) \left(\frac{0.5-a+e}{2} \right)^{0.5}} \left ( \frac{1}{2.85-0.5a+0.5e} \right )^{-0.5} &= -103.762 \end{align} \right. $$
What numerical method should I use to solve this set?
You can use Newton-Rapshon method to find the solution of the method.
by setting
$g_1(a,b,e,f)\equiv b+a-2.6=0$
$g_2(a,b,e,f)\equiv d +e-3.9=0$
$g_3(a,b,e,f)\equiv 2f + a + e - 0.5=0$
$g_4(a,b,e,f)\equiv \frac{3.9-e}{e\left(\frac{0.5-a+e}{2}\right)^{0.5}} * \left(\frac{1}{4.15 - 0.5a + 0.5e}\right)^{-0.5}+92.208=0$
$g_5(a,b,e,f)\equiv \frac{a}{(2.6-a) \left(\frac{0.5-a+e}{2} \right)^{0.5}} \left ( \frac{1}{2.85-0.5a+0.5e} \right )^{-0.5} +103.762=0.$
To apply Newton method you can see http://utkstair.org/clausius/docs/che301/pdf/systems.pdf}