Exercise 3.5.1 in Gathmann's lecture notes:
Let $L_1$ and $L_2$ be two disjoint lines in $\mathbb{P}^3$, and let $P\in \mathbb{P}^3 \setminus (L_1 \cup L_2)$ be a point. Show that there is a unique line L meeting $L_1$, $L_2$, and $P$ (i. e. such that $P \in L$ and $L \cap L_i \neq \emptyset$ for $i=1,2$.
And is it true that a line in $\mathbb{P}^3$ is given by a system of two homegeneous linear equations?