Is that particular case of the "minimum weight solution to linear equations" still NP-complete?
Here's a sketch of an NP-hardness proof. Exact cover by 3-sets is a natural generalization of the perfect matching problem to hypergraphs $G=(V,E)$ with all edges $e \in E$ containing 3 vertices (instead of 2) and $|V|$ is divisible by 3. The problem is to find a subset of the hyperedges such that each vertex is incident to exactly one of the selected hyperedges. Let $x_e$ be a 0/1 variable for each hyperedge $e$ indicating whether or not it is used. Clearly the system of equations

$\sum_{e \in E : v \in e} x_e = 1$    for all $v \in V$

has only non-negative coefficients and has a solution with at most $|V|/3$ positive coefficients if $G$ has an exact cover. The other direction, namely showing that this system has such a solution only if $G$ has an exact cover, is less trivial but still easy.