All you have to do is take any two points on the line and show those points are on the plane.
Q.Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically.
What I tried doing:
I started by getting the parametric equation of (x, y, z) = (5, -4, 6) + u(1,4,-1)
x=5+u
y=-4+4u
z=6-u
I then subbed in u=0 to get a set of points
x=5
y=-4
z=6
I then got the parametric equation for (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)
x=3+s+2t
y=0+s-t
z=2-s+t
I decided to use the points (5,-4,6) and sub them into
x=3+s+2t
y=0+s-t
z=2-s+t
5=3+s+2t
5-3=s+2t
2=s+2t
-4=0+s-t
-4+0=s-t
-4=s-t
6=2-s+t
6-2=-s+t
4=-s+t
I don't know if I'm doing this correctly. I really don't know what to so I tried that.