Let $A$ be a $N\times N$ non-singular real symmetric matrix. Consider the system of equations $$ (x_i - a_i)\sum_j A_{ij} x_j = b_i $$ for $i=1,...,N$. The vectors $a$ and $b$ are arbitrary parameters.
What can one say about the $x$ that solve this system of equations? In the solution unique? Is it exists a close form solution?
I am particularly interested in the case $b_i = b$ for all the $i$.
Every individual equation describes a (hyper-)quadric, and there can be up to $2^N$ intersections (without degeneracies). By painful elimination, you can probably reduce to a univariate equation of degree $2^N$, which won't have a closed-form solution in general (but for $N=1,2$).
For the sake of illustration, consider $A$ to be a unit matrix, all $a_i=0$ and all $b_i=1$. Then every equation describes a pair of hyperplanes $x_i=\pm1$ and the solutions are the $2^N$ tuples $(\pm1,\pm1,\cdots\pm1)$.