I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want to choose $x_1$, ..., $x_7$ arbitrarily in such a way that as many of these $64$ expressions as possible become zero, but not all of them.
Which largest number of zeroes can I achieve and how to do it?
Obviously there must be some general method; but then again, maybe for this specific set of expressions I can do better than by some general method, I don't know. And I do not know that general method anyway.
So far I can do 32 zeroes, by putting $x_1=x_2=x_3=x_4=x_5=1$ and $x_6=x_7=0$. Can one do better?