Given two points in a plane, what is the most concise way to express the y-intercept associated with a linear relationship between the two points in terms of their coordinates? Any thoughts would be greatly appreciated. Thank you.

NJZ

You mean like y=mx+b where m is the slope (found by finding the change in x over the change in y) and b is the y-intercept?

--CS

Yes, I'm looking for a simple way to express b in terms of the points (x1, y1) and (x2, y2) (sorry the subscripts don't work in this text). The slope m is (y2-y1)/(x2-x1), so you can write an equation for the line as (y-y1) = m(x-x1), or y = /(x2-x1)+y1, or y = mx-mx1+y1, making b = y1-mx1. I was just wondering if there is a more concice way of expressing b.

NJZ

I very much doubt that there is any better formula.
Personally, I think the formula b = y1 - m*x1 is very nice. While it does require calculation of the slope, you can look at it as saying you start from y1 and you run down a hill of slope m over a distance of x1 horizontal units.

If the two points A, B are on a line through the origin, we have a direct proportionality

yA/xA = yB/xB

and the y-axis intercept of this line is q = 0. If the 2 points are not colinear with the origin, i.e., we do not have a direct proportionality, the y-axis intercept tells us the distance we would have to shift the origin and the x-axis up/down to achive the direct proportionality:

(yA - q)/xA = (yB - q)/xB

However, I do not see it as a much of an improvement