Thread: Telling the slope of line

Problem
Given that Line k does not pass through the origin
Book says statement ( Line k passes through points (a,b) and (r,s) where (a-r)(b-s)<0) is enough to tell whether the slope is negative.

But the book is assuming that r,s is to the right of a,b right?

Assuming that the equation of the line is y = px + q
The line passes thru the point (a, b), thus b = pa + q (1)
The line passes thry the point (r, s), thus s = pr + q (2)

substacting (2) from (1) => b - s = p (a - r)
So if (b - s) and (a - r) are of the same sign, then p is > 0
if (b - s) and (a - r) are of opposite signs, then p is < 0

Either a-r or b-s has to be negative, but not both, to satisfy the inequality. Put a and r on the x axis so r is greater than a, and put b and s on the y axis so b is greater than s. This satisfies the inequality. Draw a line through the points (a,b) and (r,s) and notice that it is of negative slope.
Switch a and r and switch b and s. Now the inequality is still satisfied and draw the line again. It's of negative slope as well. So, both cases where the inequality is satisfied result in negative slopes.