Hi. I have the following two equations(adsbygoogle = window.adsbygoogle || []).push({});

[tex]S_{21}=\frac{(1-\Gamma^2)z}{1-z^2\Gamma^2}[/tex]

[tex]S_{11}=\frac{(1-z^2)\Gamma}{1-z^2\Gamma^2}[/tex]

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

[tex]z=\pm\sqrt{\frac{\Gamma-S_{11}}{\Gamma-S_{11}\Gamma^2}}[/tex]

but a better solution is

[tex]z=\frac{S_{21}}{1-S_{11}\Gamma}[/tex]

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

[tex]z=\frac{(S_{11}+S_{21})-\Gamma}{1-(S_{11}+S_{21})\Gamma}[/tex]

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

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# Simultaneous nonlinear equations

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