Find a vector that lies along the line of intersection of th

Find a vector that lies along the line of intersection of the planes x-y+z= 5 and 2x+y-2z= 7.

Solution

You need to solve the two equations

x-y+z= 5

2x+y-2z= 7

repalce Z by t then

x-y = 5-t

2x+y=7+2t

on solving 3x = 12+t or x = (12+t)/3

and y = (12+t)/3 - 5+t = -1-2t/3

now solving which we get x =   (12+t)/3 and y = -1-2t/3 and Z = t

so we have the vector( (12+t)/3, -1-2t/3 , t )