finding the intersection line of two planes when they can't be equated

Question

An exam from a couple years ago has the following question: Line $l$ is the intersection between planes $U$ and $V$. $U: y=2$ and $V: 3x-\sqrt{3}z=0$. How can line $l$ be found? The method of expressing both equations in terms of the same variable won't work here.

Answer 1

The line certain satisfies the $y$-component is equal to $2$ and you can express $z$ in terms of $x$.

Hence you should be able to express it in a function of one parameter $\begin{bmatrix} x\\ 2 \\ z\end{bmatrix}$, I will leave the task of expressing $z$ in terms of $x$ to you.

Answer 2

On the opposite, it still works. For instance, as functions of $z$:

$$\begin{cases}x(z)=\dfrac z{\sqrt 3},\\y(z)=2.\end{cases}$$

(From $z=0$ and $z=\sqrt3$, you find the two points given by Vasya in a comment.)