We define the coordinates $u,v,w$ in terms of $x,y,z$: \begin{align*} u & = x + xyz \\ v & = y + xy \\ w & = 2x + z + 3z^2. \end{align*} Determine whether this system can be solved for $x,y,z$ in terms of $u,v,w$ near the point $(0,0,0)$.
I don't even know where to start since the system is nonlinear. The first thing I would try to do is rearrange and substitute to solve for $x,y,z$, but I run into issues with nonlinearity (i.e., I can't solve for $z$ without taking the positive and negative root) and division by zero (breaking apart $xyz$). I feel as though this system is not solvable, but I cannot rigorously explain why.