5x-2y+z-3=0 and -x+y+2z-1=0
and you want the intersection of these two planes, which is a line, and 0=0 so you should set these equations equal each other
5x-2y+z-3=-x+y+2z-1,
which gives you
6x-3y-z-2=0,
right?
DO NOT DO THIS.
Do not do this because your are trying to find the points <x,y,z> that make both equations true. Yeah, both equations are equal to zero and so 5x-2y+z-3=-x+y+2z-1 when x,y,z satisfy both equations, but 5x-2y+z-3=-x+y+2z-1 for a lot of triples <x,y,z> that don't make either equation true. You need to figure out how to solve both equations at once and this won't help you do this.
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