Having two non-parallel hyperplanes of $ \mathbb{R}^n $, $ S_1 $ and $ S_2 $, prove that $ S_1\cap S_2 \neq\emptyset $ and $\dim(S_1\cap S_2)=n-2$ I've got two affine hyperplanes of $ \mathbb{R}^n $, $ S_1 $ and $ S_2 $, which are non-parallels.
I have to prove that $ S_1\cap S_2 \neq\emptyset $ and $\dim(S_1\cap S_2)=n-2$

What I have done is:
As $ S_1 $ and $ S_2 $ are hyperplanes, $\dim(S_1)=n-1$ and $\dim(S_2)=n-1$
But I have no idea what else to do, I'm stuck :|

What does the non-parallel hypothesis mean ? Try to explore this fact to show the non emptyness of the intersection.
What can you say about the intersection of two linear hyperplanes ? Can you adapt this (maybe by translating the origin ?) to prove what is the dimension of the intersection ?
