Simultaneous nonlinear equations

[daudaudaudau]

Hi. I have the following two equations

S_{21}=\frac{(1-\Gamma^2)z}{1-z^2\Gamma^2}
S_{11}=\frac{(1-z^2)\Gamma}{1-z^2\Gamma^2}

How would you go about solving these equations? I want to avoid square roots because they make the results ambiguous.

I myself have found that

z=\pm\sqrt{\frac{\Gamma-S_{11}}{\Gamma-S_{11}\Gamma^2}}

but a better solution is

z=\frac{S_{21}}{1-S_{11}\Gamma}

because it avoids the sign ambiguity. Yet another good solution is

z=\frac{(S_{11}+S_{21})-\Gamma}{1-(S_{11}+S_{21})\Gamma}

but I have no clue how to arrive at these results. Any suggestions?

 

[daudaudaudau]

Are there any general methods for solving nonlinear equations analytically? I only know of the substitution method and then applying the quadratic forumla.