Closed form solution of tricky nonlinear equation system 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.
 A: As indicated in the comments, there is no "clean" closed-form solution, as a quartic equation has to be eventually solved, either with direct methods or particular algorithms.
A proposed solution method exploits the property of the first equation $f(\xi)=g(\eta)$, since two functions of two separate variables are equated, both must be equal to a constant unique value $K$. 
The first equation (RHS) becomes:
$$
(C+\eta)(D+\eta)=K
$$
which yields (since $\eta>0$)
$$
\eta(K)=\frac{-(C+D)+\sqrt{(C+D)^2-4(CD-K)}}{2} \ 
$$
The first equation (LHS) becomes:
$$
(1-\xi)(B+\xi)=K/A
$$
which yields (since $\xi>0$)
$$
\xi(K)=\frac{-(B-1)+\sqrt{(B-1)^2-4(K/A-B)}}{2} \ 
$$
The second equation becomes:
$$
(F+[\xi-\eta])(G+[\xi-\eta])=EK/A
$$
which yields (since $\xi-\eta>0$)
$$
[\xi-\eta](K)=\frac{-(F+G)+\sqrt{(F+G)^2-4(FG-EK/A)}}{2} \
$$
Then, the unique constant $K$ must satisfy
$$
[\xi-\eta](K)=\xi(K)-\eta(K)\,,
$$
thus by substituting the respective functional form of the above functions, the unique constant $K$ is obtained through an irrational equation.
Note
In the special case when $C=D=F=G=0$, the solutions become:
$$
\eta(K)=\sqrt{K}
$$
and
$$
[\xi-\eta](K)=\sqrt{\frac{EK}{A}} \
$$
Thus, by combining the above solutions, $\xi$ is obtained. Since
$$
\xi(K)=\eta(K)\left(1+\sqrt{\frac{E}{A}}\right)=\sqrt{K}\left(1+\sqrt{\frac{E}{A}}\right) \,,
$$
then by using the previous definition of $\xi(K)$, that is
$$
\xi(K)=\sqrt{K}\left(1+\sqrt{\frac{E}{A}}\right)=\frac{-(B-1)+\sqrt{(B-1)^2-4(K/A-B)}}{2} \,,
$$
becomes an irrational equation with closed form solution, being:
$$
\sqrt{K}=\sqrt{A}\frac{-\alpha\beta+\sqrt{(\alpha\beta)^2+4B(\alpha^2+4)}}{\alpha^2+4} 
$$
where $\alpha=2(\sqrt{A}+\sqrt{E})$ and $\beta=B-1$. In this special case a quartic has been reduced to two decoupled quadratics.
