Slope

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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.

$Graph of the line through (1,-1) and (2,2). The rise is shown as 1 and the run is shown as 3$

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

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

Solution:

We compute the slope:

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

which is undefined, hence the line is vertical.
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.

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