I find it hard to derive an elegant result of the following equation with $x$ and $y$.
$$ \begin{cases} {a_1} = \frac{1}{{x + {k_1}}} + \frac{1}{{y + {k_2}}}\\{a_2} = \frac{1}{{x + {k_3}}} + \frac{1}{{y + {k_4}}} \end{cases} $$
I want to know whether the solution of this system can be expressed as a determinant or some other elegant form.
Solving the first equation for $y$ we get $$y=-{\frac {\alpha_1\,{\it k_1}\,{\it k_2}+\alpha_1\,{\it k_2}\,x-{\it k_1}-{ \it k_2}-x}{\alpha_1\,{\it k_1}+\alpha_1\,x-1}}$$ plugging this in the second one $$\alpha_{{2}}= \left( x+k_{{3}} \right) ^{-1}+ \left( -{\frac {\alpha_1 \,{\it k_1}\,{\it k_2}+\alpha_1\,{\it k_2}\,x-{\it k_1}-{\it k_2}-x}{\alpha_1 \,{\it k_1}+\alpha_1\,x-1}}+k_{{4}} \right) ^{-1} $$ Then you have to solve this equation for $x$,good luck!