its been a long time, but when i was taught this material, i was told there are only 3 cases -
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.
Also, isn't this a great example of "when you have a hammer, everything looks like a nail" ?
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.
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.