I have a quadratic system of $n$ equations that looks like: $$ A_{ij}x_{j}y + B_{ij}x_{j}=0 $$
For $i=0...n$, where $A_{i,j}$ and $B_{ij}$ are integer matrices and sums over $j$ are implied. Is there some efficient way to solve this system for y only, without having to implicitly solve all $x_{j}$?
(Bonus question: what if the system looks like $ A_{ijk}x_{j}y_{k} + B_{ij}x_{j}=0 $?)
Your problem is a texbook example of "generalized eigenvalue problem". It is a standard problem implemented in numerical libraries such as lapack.
The routines that compute solutions of the eigenproblem may be used to yield only the eigenspectrum without computing eigenvectors that is $x.$ They are also quicker in that case.
This is a reference: http://www.netlib.org/lapack/lug/node35.html