Sunday, January 16, 2022

Lines with Opposite-Signed Slopes

 Lines with Opposite-Signed Slopes


Take two lines, each with opposite signed slopes.  One slope has a positive slope, the other has a negative slope.   In general, the pair of lines will always intersect at one point.


Define two lines as such:


y = m1 ∙ x  + b1,  where m is the slope and b is the y-intercept.   


y = m2 ∙ x + b2


Assume that m1 and m2 are not zero, and m1 > 0 and m2 < 0.


If m2 < 0, -|m2| < 0  (see Aside) 

and as a result, m2 = -|m2|   


Also,  since m1 > 0, |m1| > 0, and m1 = |m1|.   


Equating both lines and solving for x:   


m1 ∙ x + b1 = m2 ∙ x + b2

|m1| ∙ x + b1 = -|m2| ∙ x + b2

|m1| ∙ x + |m2| ∙ x = b2 - b1

x = (b2 - b1) ÷ (|m1| + |m2|)


Since m1 and m2 are not zero, the above solution is defined.   


QED


- - - - - - - - - --  - 

Aside:    If x < 0, then  -|x| < 0  


Assume x is not zero. 


By definition, the absolute value of x, denoted as |x|, is the defined as the distance x is from 0 and is always positive. 


Then:

|x| > 0 


Multiply both sides by -1:

-|x| < 0

- - - - - - - - - --  - 


Eddie


All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.