If you have an arbitrary system of linear equations and add to it one equation, the dimension of the space of solutions can decrease by at most 1. Do take any $d$ equations of your hyperplanes, order them arbitrarily, and denote by $n_k$ the dimension of the space of solutions of the first $k$ equations. Then $$d=n_0\geq n_1\geq\ldots\geq n_d=0,$$ and $n_j-n_{j+1}\leq 1$. It follows that all $n_{j}-n_{j+1}=1$, for all $j=0,\ldots,d$, and thus $n_k=d-k$.
Remark: only one condition was used: that any $d$ equations have one solution.