Would it be possible to help me with this problem Thank you

Would it be possible to help me with this problem? Thank you

Parameterize the line that passes through the point (4,0, -3) which is parallel to the intersection of the plane 2x - y + z = 2 and the plane x - 3z = 4 Find an equation of the plane perpendicular to the planes 2x - y + z = 2 and x - 3.z = 4 that passes through (4, -2,3).

Solution

1) intersection of 2x-y+z =2 and x-3z =4 is x-y+4z = -2 plane parallel to this is x-y +4z = c passes through (4,0,-3) 4-0 + 4 *-3 = -8 c= -8 the plane is x-y +4z +8 =0 2) Let the plane be ax+by +cz = d plane perpendicular to 2x-y+z = 0 2a -b + c=0 plane perpendicular to x-3z= 0 be ax+by +cz = d a - 3c =0 a= 3c b= 7c c= c it passes through (4,-2,3) d = 1 equation is 3x + 7y + z -1 =0