@hexomino's answer is correct and well-reasoned, as always. Here's another approach, which to me feels much more.. "axe-to-the-head" is what I'd call it in my native language, so I thought it might be interesting enough to warrant posting.
Lower bound: (This is what @trolley813's encrypted comment is saying.)
Here is a solution with six lines:
It's difficult to tell how I found this, except that I already knew a solution for a 3 by 3 grid with 4 lines, which can be found e.g. here on our sister site Mathematics Stack Exchange. It's also possible that a solution with 5 lines exists.
(By the way, the puzzle is missing the requirement that the lines must be ...