Describe the point or set of points that a plane in $\mathbb{R}^3$ has in common with another line

Question

Describe the point or set of points that the plane in $\mathbb{R}^3$ with equation $wx + 3y - z = 2$ has in common with the line having the given parameterization $(x,y,z) = t(1,1,5) + (0,0,-2)$

I need help understanding the problem. I think that I need to find points $(x,y,z)$ such that they are solutions for both of the equation of the lines shown. The problem I am at is I think I need to convert both of the lines to one form or the other (both of them parametric, or both of them to be equation of lines format). Is that the right way to approach the problem? If so, what steps would I have to take to do that?

Answer 1

You could use the equations in the forms given.

Points on the line have the form $(x,y,z)=t(1,1,5)+(0,0,-2)= (t,t,5t-2)$.

A point $(x,y,z)$ is in the plane if $wx+3y-z=2$.

So a point is on the line and in the plane if $wt+3t-(5t-2)=2$;

i.e., $wt-2t=0;$ i.e., $w=2$ or $t=0$.

When $w=2$, all of the points on the line are in the plane.

Otherwise, only the point when $t=0$ [i.e., $(0,0,-2)$] is on the line and in the plane.