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 al, $k_j=1$,
and thus $n_k=d-k$.

Remark: only one condition was used: that any $d$ equations
have one solution.