Answer
(a) $(x,y,z) = (-1,-9,20)$
(b) The point $(6,5,-12)$ does not lie on the line.
Work Step by Step
(a) We can find the point on the line when $r=-3$:
$(x,y,z) = (2+r,-3+2r,5-5r)$
$(x,y,z) = (2+(-3),-3+2(-3),5-5(-3))$
$(x,y,z) = (-1,-9,20)$
(b) If the point $(6,5,-12)$ lies on the line, there is a real number $r$ such that
$(x,y,z) = (2+r,-3+2r,5-5r) = (6,5,-12)$.
x coordinate: If $2+r = 6$, then $r = 4$
y coordinate: If $-3+2r = 5$, then $r = 4$
z coordinate: If $5-5r = -12$, then $r = \frac{17}{5}$
Since the required value of $r$ is not the same for all three coordinates, the point $(6,5,-12)$ does not lie on the line.
