x+y+z=u,y+z=uv,z=uvw
You asked:
Solve the system of equations:\[\begin{aligned}x + y + z &= u\\y + z &= u v\\z &= u v w\end{aligned}\]
MathBot Answer:
The system of equations has infinitely many solutions.
$$\begin{aligned}u &= x + y + z\\v &= \left\{\frac{y + z}{x + y + z}\right\} \setminus \left\{0\right\}\\w &= \left\{\frac{z}{y + z}\right\} \setminus \left\{0\right\}\end{aligned}$$\(x\) is a free variable.
\(y\) is a free variable.
\(z\) is a free variable.