I'm agreeing with you.

x+y=1, x+y=2 clearly has no solution since two numbers can’t simultaneously add to both one and two.

x+y=1,2x+2y=2 clearly has infinitely many solutions. There’s only one equation here after canceling the 2, so you can plug in x’s and y’s all day long, no end to it.

x+y=1, 2x+y=1 clearly has exactly one solution (0,1) after elimination.

This example stuck with me so I use it even now. The author/Claude/Gemini/whatever could have just used this simple example instead of “trichotomy of curves through space conjoin through the realm of …” math, not Shakespeare.

To explain this I would first and foremost use a picture, where the 3 cases : parallel, identical, intersection can be intuitively seen (using our visual system, rather than our language system), with merely a glance.

The overuse of the $15 synonyms is almost always a bad idea--you want to use them sparingly, where dropping them in for their subtly different meanings enhances the text. But what is extremely sloppy here is that the possibilities of "no solutions, one solution, infinite solutions" is now being described with a different metaphor for solution here. And by the end of the paragraph, I'm not actually sure what point I'm supposed to take away from this text. (As bad as this paragraph is, the next paragraph is actually far worse.)

Mathematics already has a problem for the general audience with a heavy focus on abstraction that can be difficult to intuit on more concrete objects. Adding florid metaphors to spice up your writing makes that problem worse.