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This is something I've been wondering about. Namely, I've always accepted "on intuition" that the equation
$$ax + by = c$$
is, when graphed, a line. You can plot the points $(x, y)$ satisfying the equation and see that yeah, they do indeed form a line.
But when I came across this, I realized was that I had accepted this idea without proof:
There is enough material in the text to convince the students empirically that a line in the plane is represented by a linear equation, and that the graph of a linear equation is a line. However, these two fundamental theorems on linear functions are not justified mathematically. (pg. 5)
Important theorems on linear functions are not proved. Relevant to the above two standards are two fundamental theorems: A line in the plane is represented by a linear equation and the graph of a linear equation is a line. Neither of these theorems is proved. (pg. 16)
Which made me wonder (I haven't seen the texts) -- just how would you not only "justify" this mathematically, but in a way a high-schooler would understand? And not only that, but in a manner which is actually enlightening? In addition, this criticism is leveled against all four high school geometry/algebra books.
For example, consider if we were using, say, Hilbert's axioms as our axiom set for Euclidean geometry. Then we could show that the equation is a line by something like this: find three points A, B, and C satisfying it so that $A * B * C$ (the "betweenness" relation), then show that for any point $D$ which is not $A$ or $B$, then one of $D * A * B$, $A * D * B$, or $A * B * D$ must hold. Going the other way (the converse), to show the line is given by the equation, you'd show that for any points A, B, C with $A * B * C$, then every point $D$ with $D * A * B$, $A * D * B$, or $A * B * D$ satisfies some equation of the form $ax + by = c$. However, it seems this kind of proof is fairly tedious (you have to check three cases in both implications), and relies on quadratic functions and radicals since you have to use the distance formula as that's how you'd define the "betweenness" relation for three points.
I suppose the details would vary with regard to the axiom set we use (I don't know if Hilbert's would necessarily be the best for "high school geometry") -- but it seems no matter which one we use, we need some way to determine that three points A, B, and C "lie on the same line" (which is what the "betweenness" relation does, although it does more, since it also orders the points), and a way to express this with regards to coordinatized points as well as points in the axiomatic geometry which the coordinate plane models. It seems that any formula I've seen for that fact using Cartesian coordinates requires a quadratic polynomial expression, for one. Any proof along these lines seems like it would be tedious, or require additional motivation, and so might not be enlightening at this lower level of the person's mathematical development. The "message" seems to easily get lost as one gets bogged down in mechanics.
How would you solve this problem? What's a good way to justify this at such a level?