Determine if the point (4,-2,3) lies in the plane with vector equation (x, y, z) = (2, 0, -1) + s(4, -2, 1) + t(-3, -1, 2).

Answer 1

#1: 4s - 3t +2 = 4 ----> 4s - 3t = 2
#2: -2s - t + 0 = -2 ----> t = 2 - 2s

using substitution:
4s - 3(2-2s) = 2
4s - 6 + 6s = 2
10s = 8
s = 4/5
then t = 2 - 8/5 = 2/5

#3: is s + 2t - 1 = 3 ???
LS=4/5 + 4/5 - 1
= 3/5
RS = 3
NOPE, given point is not on the plane