Your system of equations is of the form $Az = b$, where $z_{i(\alpha)} = x^\alpha$ and $i(\alpha)$ is the grlex index of
$\alpha \in$ $ X_{n,k} \equiv$ {$\beta \in \mathbb{Z}^n: \sum_j \beta_j = k$}.
(See Uniquely generate all permutations of three digits that sum to a particular value? for information on this index.)
Solving $Az = b$ is tantamount to solving the equations (if a solution exists), because it is trivial to obtain $x$ from $z$. So really you just need to solve a matrix equation once you sort through the indexing issues.
