It is kind of hard to follow because of the latex but you are not quite correct :

You have for your normal

N = 6x i + 4y j + k

Further N is parallel to M = -3 i + 8 j - k

Since N and M are parallel N = cM for some c

her c = -1 so the z comp match

-6x = -3 -4y = 8

x= 1/2 y= -2

To find z sub the x and y values into

z=8-3x^2-2y^2 so z = 8 -3/4 -8 = -3/4

so the pt is (1/2,-2, -3/4)

Note for your point N = -3 i + 8j + k which is not parallel to M