Suppose, for each given $w_1 \in \mathbb{R}$, there exists unique $x_1,y_1,z_1 \in \mathbb{R}$ that satisfy the following system of equations:
$F_1(w_1,x_1,y_1,z_1)=0\\ F_2(x_1,y_1,z_1)=0\\ F_3(x_1,y_1,z_1)=0$
Note that $F_2$ and $F_3$ do not involve $w$ terms.
Now, for our given $w_1$ , I take the $x_1$ computed above and scale it $x_{\gamma}= \gamma x_1$ and then solve for $y_{\gamma},z_{\gamma}$ where
$F_2(x_{\gamma},y_{\gamma},z_{\gamma})=0\\ F_3(x_{\gamma},y_{\gamma},z_{\gamma})=0$
Can I find a third equation G()=0 so I can say that for each given $w_1 \in \mathbb{R}$, there exists unique $x_{\gamma},y_{\gamma},z_{\gamma}$ that satisfy the following system of equations:
$\textbf{G}(w_1,x_{\gamma},y_{\gamma},z_{\gamma})=0\\ F_2(x_{\gamma},y_{\gamma},z_{\gamma})=0\\ F_3(x_{\gamma},y_{\gamma},z_{\gamma})=0$
where $x_{\gamma},y_{\gamma},z_{\gamma}$ are determined by the process detailed above.
Essentially, I don't want to solve a system of 6 equations to determine $(x_1,y_1,z_1,x_{\gamma},y_{\gamma},z_{\gamma})$
namely: $F_1(w_1,x_1,y_1,z_1)=0\\ F_2(x_1,y_1,z_1)=0\\ F_3(x_1,y_1,z_1)=0\\ x_{\gamma}= \gamma x_1\\ F_2(x_{\gamma},y_{\gamma},z_{\gamma})=0\\ F_3(x_{\gamma},y_{\gamma},z_{\gamma})=0$
I am wondering if I can do it in a faster way, get from $w_1$ to $(x_{\gamma},y_{\gamma},z_{\gamma})$ by solving fewer equations. All the equations are painfully non-linear but we can assume that I am looking in the part of the domain where the solutions exist and are unique.
