I am trying to solve a problem of breaking an amatuer cryptography.
The problem boils down to solving a combined system of linear and bilinear equations having $50$ unknowns.
For representational purposes, the equations look similar to the following, with $x$, $y$ and $z$ being the unknowns.
\begin{align} \begin{cases} 3x + 10y + 8z + 5xy &= 1470 \\ 2x + 10y + 3z + yz + xz &= 1210 \\ x + 5y + z + 3xy + 16xz &= 5540 \\ x + 3y + 8z + 12xy + 4yz &= 5110 \end{cases} \end{align}
The above system has the solution $x=10 , y=20, z=30$.
I want to know the method for solving these type of equations. Can gaussian elimination be applied on such a system ?