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I am studying (self learner) linear equations/equation of line and my idea is to discover the equations myself rather than try and understand ready-made equations available in text books. I am using X-Y Cartesian plane, with 2 points A(x1,y1) and B(x2,y2). I know linear equations describe the relation between two points in my case Point A and Point B. So I was wondering if I come up with a definition in words and more Importantly as an Algebraic Equation; How can I make sure that my equation describes the relation Fully/Completely?! Is there a method/procedure to do this?! Also other than "completeness", are there other things/parameters a definition/description/equation is supposed to fulfill?

In addition to explaining it, any pointers, or links to relevant content are also hugely welcome.

Thanks much in Advance...

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    $\begingroup$ You know that between any 2 points, there is only one line, right? So if you find it, you've got it. There might be some weird cases. What might those be? You'll have more idea once you find answers for a few cases. $\endgroup$
    – Sue VanHattum
    Apr 27, 2023 at 23:12
  • $\begingroup$ @SueVanHattum Thanks for the reply. I was thinking since X-Y plane has 2 degrees of freedom or 2 dimensions namely X and Y, if my definition contains both of those then it would be complete. Am I correct in thinking that? $\endgroup$ Apr 28, 2023 at 3:34
  • $\begingroup$ Maybe prove that, for every $m\in R$ and point $(a,b)\in R^2$ there exists a unique a set of points $L\subset R^2$ such that $(a,b)\in L$ and the slope between any two distinct points in $L$ is $m$. That would take care of the "non-vertical" lines. $\endgroup$ Apr 28, 2023 at 10:40
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    $\begingroup$ I fear that this isn't a math education question, it's a math content question, right? E.g. this development is given in NYS Common Core Lessons 19 and 20. $\endgroup$ Apr 29, 2023 at 1:45
  • $\begingroup$ @DanielR.Collins Yes you are correct it isn't a math education question but the educator's answers are better. I am aboe to understand them better than the Mathematics site answers.... $\endgroup$ Apr 29, 2023 at 2:08

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Not sure this is what you are getting at, but here goes.

Theorem: A straight line in $R^2$ is uniquely deteremined by two distinct points on that line.

Proof Sketch: Suppose $(x_1, y_1)\in R^2$ and $(x_2, y_2)\in R^2$ are distinct. There are two cases to consider:

Case 1: $x_1\neq x_2$ (non-vertical lines)

Let $~m=\frac{y_2-y_1}{x_2-x_1}$ (the slope between these two points). Prove that there exists a unique a set of points $L\subset R^2$ such that $(x_1,~y_1)\in L$ and the the slope between any two distinct points in $L$ is $m$. That is, prove that all such subsets of $R^2$ are the same set.

Case 2: $~x_1=x_2$ (vertical lines)

Prove that there exists a unique a set of points $L\subset R^2$ such that $(a,~b)\in L$ if and only if $a=x_1$. That is, prove that all such subsets of $R^2$ are the same set.

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