Solve three nonlinear equations in three unknown I have the set below of three nonlinear equations:
\begin{align}
Y_1=&\;\frac{X_1+GX_2X_3}{1+X_2X_3} \tag1\\
Y_2=&\;\frac{X_1+GX_2X_3+GX_2(1-X_3)^2}{1+X_2X_3+X_2(1-X_3)^2} \tag2\\
Y_3=&\;\frac{X_1+GX_2X_3+FGX_2(1-X_3)^2}{1+X_2X_3+FX_2(1-X_3)^2} \tag3
\end{align}
Note that all variables and constants are complex numbers.
I need to find expressions for $X_1$, $X_2$ and $X_3$ each as a function of $Y_1$, $Y_2$, $Y_3$, $F$ and $G$. 
What I did so far is:
I solved $(1)$ for $X_1$ 
$$
X_1=Y_1+X_2(Y_1X_3-GX_3) \tag4
$$
I then substituted $(4)$ in $(3)$ and solved for $X_2$
$$
X_2=\frac{X_1-Y_3}{X_3(Y_3-G)+(1-X_3)^2(Y_3F-FG)} \tag5
$$
let 
$$ 
R=Y_1 -G \\
W=Y_3-G\\
Q=Y_3F-FG\\
H=Y_1-Y_3\\
M=W-R
$$
Hence
$$
X_2=\frac{H}{MX_3+Q(1-X_3)^2} \tag6
$$
then substituted $(6)$ in $(4)$
$$
X_1= Y_1+\frac{HRX_3}{MX_3+Q(1-X_3)^2}\tag7
$$
Now $(6)$ and $(7)$ are $X_2$ and $X_1$ as functions of $X_3$, $Y1$, $Y_3$, $F$ and $G$.
When substituting $(6)$ and $(7)$ back in $(2)$ and solve for $X_3$, and simplify it to a conventional quadratic form $aX_3^2+bX_3+c=0$, I end up with almost zero values for $a$, $b$ and $c$. And hence cannot find $X_3$.
Thanks a lot in advance.
 A: Define
\begin{eqnarray*}
Z_{1} &=&X_{1}+GX_{2}X_{3}, \\
Z_{2} &=&1+X_{2}X_{3}, \\
Z_{3} &=&X_{2}\left( 1-X_{3}\right) ^{2},
\end{eqnarray*}
then the system becomes
\begin{eqnarray*}
Y_{1} &=&\frac{Z_{1}}{Z_{2}}, \\
Y_{2} &=&\frac{Z_{1}+GZ_{3}}{Z_{2}+Z_{3}}, \\
Y_{3} &=&\frac{Z_{1}+FGZ_{3}}{Z_{2}+FZ_{3}}.
\end{eqnarray*}
Solving for $Z_1$ and $Z_2$ from the first two equations yields
\begin{eqnarray*}
Z_{1} &=&Y_{1}Z_{2},\\
Z_{2} &=&\frac{Y_{2}-G}{Y_{1}-Y_{2}}Z_{3}. \\
\end{eqnarray*}
Substituting this into the third equation implies
$$Y_{3} =\frac{Y_{1}\left( G-Y_{2}\right) -FG\left( Y_{1}-Y_{2}\right) }{%
\left( G-Y_{2}\right) -F\left( Y_{1}-Y_{2}\right) }.$$
Hence, if the parameters are such that this equation does not hold, the system does not have a solution. If this condition is satisfied, then there is an infinity of solutions, since you can drop equation $(3)$, and for any $X_3\neq0$,
\begin{eqnarray*}
X_{1} &=&G-\frac{\left( G-Y_{1}\right) \left( G-Y_{2}\right) \left(
1-X_{3}\right) ^{2}}{\left( Y_{1}-Y_{2}\right) X_{3}+\left( G-Y_{2}\right)
\left( 1-X_{3}\right) ^{2}}, \\
X_{2} &=&\frac{Y_{2}-Y_{1}}{\left( Y_{1}-Y_{2}\right) X_{3}+\left(
G-Y_{2}\right) \left( 1-X_{3}\right) ^{2}}.
\end{eqnarray*}
while for $X_3=0$,
\begin{eqnarray*}
X_{1} &=&Y_{1}, \\
X_{2} &=&\frac{Y_{2}-Y_{1}}{G-Y_{2}}.
\end{eqnarray*}
