A machine shop makes two parts. At most 3 of the first part can be made, so x ≤ 3, and at most 3 of…

Step 1. Pairs that meet. Five lines make $10$ pairs. The lines $x = 0$ and $x = 3$ are parallel, and so are $y = 0$ and $y = 3$. That removes $2$ pairs, leaving $8$ that meet.

Step 2. The meeting points. The eight pairs meet at $(0, 0)$, $(0, 3)$, $(0, 6)$, $(3, 0)$ and $(6, 0)$, and at $(3, 3)$ three times: from $x = 3$ with $y = 3$, from $x = 3$ with the tool line, and from $y = 3$ with the tool line. The point $(0, 6)$ breaks $y \le 3$, and $(6, 0)$ breaks $x \le 3$.

Step 3. The corners. The feasible ones are $(0, 0)$, $(3, 0)$, $(0, 3)$ and $(3, 3)$. That is $4$ corner points.

Step 4. Lines through $(3, 3)$. $x = 3$, $y = 3$ and $x + y = 6$ all hold at $(3, 3)$. That is $3$ lines.