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Thread: Telling the slope of line

  1. #1 Telling the slope of line 
    Forum Freshman
    Join Date
    Oct 2008
    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?

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  3. #2  
    Forum Freshman
    Join Date
    Oct 2012
    Provence (South east of France)
    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

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  4. #3  
    Ots is offline
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    Join Date
    Jun 2009
    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.
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