Prove that a line is within a plane

Question

Prove that

$x = -1+t\\ y = 2 +3t \\ z = 5t$

is within the plane

$2x+y-z = 0$

I tried to substitute $t$ in the plane but when I do it I get $t=0$ and it would be only a point in the plane

Answer 1

$x = -1+t\\ y = 2 +3t \\ z = 5t$

Plug into the plane $2x+y-z = 0$

$$2(-1+t) + (2+3t) - 5t =0$$ This is true for whatever value of $t$. Notice that one value of $t$ corresponds to a point on the line, and since it also suffice the plane equation, it means that point is in the plane as well.

So all points of the line is in the plane, we conclude the line is in the plane.

Answer 2

Just plug it in. $$2\cdot (-1+t)+(2+3t)-5t = (-2+2)+(2+3-5)\cdot t = 0 + 0\cdot t= 0$$ Hence, it's on the plane.