Given a system of $m$ equations in $n$ boolean variables:
$$\begin{align} a_{11}x_1 + a_{12}x_2 + \dots +a_{1n}x_n &= b_1\pmod2\\ a_{21}x_1 + a_{22}x_2 + \dots +a_{2n}x_n &= b_2\pmod2\\ &\;\vdots\\ a_{m1}{x_1} + a_{m2}x_2 + \dots +a_{mn}x_n &= b_m\pmod2 \end{align}$$
Where all $a_{ij}, x_i$ and $b_i$ are $0$ or $1$, I want to prove that the decision problem of finding if there is a vector $x$ that satisfies at least $k$ equation, is NP hard.
I was given a hint to reduce from 3-SAT, so I am trying to build from each clause in the 3-SAT formula several equations, such that a constant number of them is satisfied iff the clause is satisfied. But I am not sure how to impose this restriction of a constant number of equations satisfied (for all different truth assignments that satisfy the given clause).