Determine the equation of the line o intersection between the planes

Question

2x-2y+z-4=0 and 2x-3y-z+8=0
State as vector equation

Answer 1

add them:
4x - 5y + 4 = 0
4x = 5y - 4
let y = 0, then x = -1
plug into first equation: 2x-2y+z-4=0
-2 + 0 + z - 4 = 0 , z = 6
so you have the point (-1, 0, 6) on the line of intersection

let y = 4, then x = 4 and
8 - 8 + z - 4 = 0 , z = 4
and (4,4,4) is another point on the line of intersection.

You know have two points and you can find the vector equation of the line