Slope

Lines and Slope 

Definition

Let (x1,y1) and (x2,y2) be two points then the slope for those two points is rise/run or
 
                  y2 - y1
         m =                  
                   x2 - x1



Example:
 

For the points (1,2) and (5,-3) we have

                    -3 - 2              5
        m  =                     =             
                    5 - 1               4




Definition

A line through a point P with slope m is the collection of points Q such that the slope between P and Q is m.



Example:
 

Sketch the line through the point (1,-1) with slope  m = 3.  

 

Solution

We plot the point (1,-1) and rise 3 and run 1 from (1,-1) arriving at 

        (1 + 1,-1 + 3)  =  (2,2)  

Then we connect the dots.

       



Exercise:
 

Sketch the line through the point (2,-3) with slope 1/2.


Special Cases

Vertical lines have an undefined slope and horizontal lines have a zero slope.

Example:  

Without graphing, describe the line through

  1. (3,5) and (3,2)

  2. (1,2) and (-3,2)

Solution:  

  1. We compute the slope:  

                      2 - 5            3
            m =               =  -                 
                      3 - 3            0

    which is undefined, hence the line is vertical.

  2. We compute the slope:  

                         2 - 2                0
            m =                    =  -         =  0         
                        -3 - 2              -5
        

    hence the line is horizontal.


Exercise

Determine whether the points (1,1), (2,4), and (7,9) are colinear.

        No, the slope between the first 2 is 3, while the slope from the first to the third is 4/3 and the slope from the second to the third is 1.

 



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