# Telling the slope of line

• November 18th, 2012, 07:59 PM
Jarwulf
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?
• November 19th, 2012, 05:21 AM
caKus
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
• November 19th, 2012, 12:03 PM
Ots
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.