Intersections of hyperplanes

I didn't know where to post this, basically I'm trying to figure out ways of finding the intersection between to hyperplanes in a given dimension, It's easy to find the intersection of:

Two lines in the x,y plane

Two planes in the x,y,z realm

Two realms in the x,y,z,w flune

it's little bit hard (I don't know how) to find the interstection of two lines in the x,y,z realm

I thought maybe it would be easy with vector equations (because they have one equals sign)

lines in x,y,z have two equals signs

I've been trying to find the intersection of two planes in x,y,z,w (I know it won't always be possible, but it should be possible to get the equation of a line)

Are there any techniques for this type of thing