So, this question is more like two mini-questions that are subsets of a single regular-sized question. Say I have two planes: $x-z=1$ and $y+2z=3$. I'm trying to find their line of intersection.
a. Would it be okay to just take the lazy way out and add them, and end up with $2x+y=5$, which is the same as $y=-2x+5$? (I think probably not, because then $z$ might be unspecfied... or maybe $z$ is just $0$? Please tell me how wrong this whole train of reasoning is, and why, so that I can get a better understanding).
b. The explanation in my textbook told me to find a point on the line first and then take the cross product of the normal vectors of the planes, which would give me both a starting point and the line's direction. This sounds like a good idea, but the problem is that I'm not really sure how to find said point on the line. The book says to set $z$ equal to $0$ and then solve... but how would I know that the planes intersect at $z=0$ at all? What if the line I'm finding is parallel to the $xy$ axis or something? (And the same question for setting $x=0$, $y=0$, etc.)