I would like to find out the easiest way (i.e. minimal number of steps required) to solve this non-linear system w.r.t variables $x_1, x_2, y_1, y_2$ ($a,b,c,d$ are constants):
$$ \begin{cases} \frac{x_1x_2}{y_1y_2}=a \\ \frac{x_1}{y_1} + \frac{x_2}{y_2}=b \\ x_1+x_2=c \\ y_1+y_2=d \end{cases} $$
I solved my numerical problem through different approaches, also exploiting non-linear optimization routines but I would like to see if there are different ways to approach this problem that I did not figure out.
Best regards
Simplify the problem using $x_2=c-x_1$ and $y_2=d-y_1$. So, what remains is $$\frac{x_1 (c-x_1)}{y_1 (d-y_1)}=a\tag 1$$ $$\frac{c-x_1}{d-y_1}+\frac{x_1}{y_1}=b\tag 2$$ From $(1)$ $$\frac{ (c-x_1)}{ (d-y_1)}=a\,\frac {y_1}{x_1} \tag 3$$ Plug in $(2)$ $$a\,\frac {y_1}{x_1}+\frac{x_1}{y_1}=b\tag 4$$ Let $t=\frac {y_1}{x_1}$ and solve $$ a t+\frac 1 t=b\implies a t^2-bt+1=0$$