On the hardness of satisfying K number of linear constraints
As j_random_hacker suggested, we can reduce MAX-2-SAT to this problem.

Given an instance of MAX-2-SAT with $n$ variables $x_1,\ldots,x_n$ and $m$ clauses, we can encode it as the following constraints:


For each variable $x_i$, add $m$ identical constraints: $x_i\le 0$ (if identical constraints are not allowed, we can use $x_i\le 0,x_i\le 0.1,x_i\le 0.01,\ldots$ instead) and $m$ identical constraints: $x_i\ge 1$.
For each clause, say $x_i\vee x_j$, add a constraint $x_i+x_j\ge 1$. If a negative literal is involved, for example, $x_i\vee\neg x_j$, then the constraint becomes $x_i+(1-x_j)\ge 1$.


Now we can conclude that there exist $nm+k$ constraints that can be simultaneously satisfied if and only if there exist $k$ clauses that can be simultaneously satisfied in the MAX-2-SAT instance.