Rediscovering euqation of line

Question

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...

Answer 1

Finally this is what I could work out. Two points belong to a particular non-vertical line if they satisfy this equation:

y1-c/x1 = y2-c/x2

where c is y-intercept. So what I found out was that not only a point should have a same slope with respect to another point to point to belong to a line but even a single point's y and x sould have same ratio with respect to y-intercept for them to belong to a particular line.

I will work on more formal-rigorous definition and proof for the same.