I am developing a solution for a Chemical Engineering course, and I came across an interesting (tricky) equation system, for which I am curious if it is possible to solve it explicitly (closed form solutions). I am not interested in the solution "per-se" as I could just use MATLAB or any other program to get such solutions.
I have the following system of 2 nonlinear equations in the 2 unknowns $\xi$ and $\eta$: $$ \begin{cases} A(1-\xi)(B+\xi)=(C+\eta)(D+\eta)\\[3pt] E(1-\xi)(B+\xi)=(F+\xi-\eta)(G+\xi-\eta) \end{cases} $$ where $A,B,C,D,E,F,G$ are all strictly positive reals. The conditions of existence of $\xi$ and $\eta$ are: $$ 0<\eta<\xi\le1 $$ Is it possible to rearrange/transform, or more generally exploit the form of the two equations to obtain $\xi$ and $\eta$ in a closed form? I have tried to do so, but to no avail.
The only thing I noticed is that the first equation is in the form $f(\xi)=g(\eta)$, therefore the only condition is that both functions are equal to a constant. However, I have not been able to move past that point.