I'm curious as to whether Gauss-Jordan elimination has an equivalent approach for systems of equations involving non-linear terms, for example:
$$ \begin{align} x^3 y^3 z^3 &= 1\\ xy^5 z^3 &= 2\\ xy^3 z^5 &= 3 \end{align} $$
I can solve these equations manually by reducing the terms, but I'm struggling to find a matrix form (does one exist?) to represent it in.