Question

Does the point (3, 2, -2) lie on the line through C(-1, 4, 5) and D(3, 2, 8)?
Is this the correct approach I'm taking with these equations?
3 = -1t + 3(1 - t)
2 = 4t + 2(1 - t)
-2 = 5t + 8(1 - t)
With these equations we would solve for t and see if the same t is a solution for the other two. If it is, the point lies on the line at that value of t. Otherwise, it doesn't.

Answer 1

You appear to be using a method of finding the line equation that
I am not familiar with.
I use the following:
1. first find the direction vector of the line which is <4, -2, 3>
2. equation in parametric form
x = 3 + 4t
y = 2 - 2t
z = 8 + 3t

now for checking if (3, 2, -2) lies on it ....
3 = 3 + 4t
t = 0
check if t = 0 works for the other two components:
2 = 2 - 2t ---> t = 0 , so far so good!
-2 = 8 + 3t ----> t = -10/3

nope, the point is not on the line