Satisfiability of at least k linear equations over Z2 is NP-hard
Consider any clause $x \lor y \lor z$, and the following table:
$$
\begin{array}{c|c|c|c}
\text{satisfied literals} &amp; [x]+[y]+[z] &amp; [x\oplus y] + [x\oplus z] + [y\oplus z] &amp; [x\oplus y\oplus z] \\\hline
0 &amp; 0 &amp; 0 &amp; 0 \\\hline
1 &amp; 1 &amp; 2 &amp; 1 \\\hline
2 &amp; 2 &amp; 2 &amp; 0 \\\hline
3 &amp; 3 &amp; 0 &amp; 1
\end{array}
$$
Here $[x]$ is 1 if $x$ is satisfied and 0 otherwise.

The first row sums to 0, and the rest sum to 4 (ignoring the column labeled "satisfied literals"). Can you use this to your advantage?