Standard Form of a Line

Question

When finding the line between two points, I've been using InterpolatingPolynomial (which I know is probably overkill). I get an answer like the following:

Expand[InterpolatingPolynomial[{{4, -6}, {10, -7}}, x]]

-(44/7) + x/14

I would like this to be reformatted into standard form of a line, e.g. as the expression x-14y-88.

In particular, my initial points are always integer pairs, so the output needed is Ax+By+C where A, B, and C are integers, (A and B relatively prime).

Is there a straightforward way to do this?

(Yes, I could do the math to figure out A, B, and C explicitly, but the heart of my question is wondering if Mathematica has a built in way to convert its default explicit answers into an implicit format.)

Answer 1

For example using your code (which do not work for lines parallel with y axis):

InterpolatingPolynomial[{{x1, y1}, {x2, y2}}, x] - y /. x1 -> 1 /. 
 y1 -> 2 /. x2 -> 3 /. y2 -> 5 // FactorTermsList // Last

(* 1 + 3 x - 2 y *)

Answer 2

eq=Expand[InterpolatingPolynomial[{{4, -6}, {10, -7}}, x]];
(LCM@@Denominator[List@@eq])(eq-y)//Expand