I am trying to solve this problem:
Let $\alpha_1x_1 + \alpha_2 x_2 + \alpha_3x_3 + \alpha_4x_4 + \alpha_5 = 0$ and $x_1x_2+x_3x_4=0$ be equations over field of size $2$. Show that we can't choose $\alpha_1,\ldots,\alpha_5$ so that the first equation has solution and second equation holds for each solution of first.
I need hint to prove this.