Using Coordinates to Find Slope
We can obtain the rise and run from a graph, or we can get them without a graph by
subtracting the ycoordinates to get the rise and the xcoordinates to get the run for
two points on the line. See the figure below.
Slope Using Coordinates
The slope m of the line containing the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by
, provided that x_{2}
 x_{1} ≠ 0.
Example 1
Finding slope from coordinates
Find the slope of each line.
a) The line through (2, 5) and (6, 3)
b) The line through (2, 3) and (5, 1)
c) The line through (6, 4) and the origin
Solution
a) Let (x_{1}, y_{1}) = (2, 5) and (x_{2}, y_{2})
= (6, 3). The assignment of (x_{1}, y_{1}) and (x_{2}, y_{2}) is
arbitrary.
b) Let (x_{1}, y_{1}) = (5, 1) and (x_{2}, y_{2})
= (2, 3):
c) Let (x_{1}, y_{1}) = (0, 0) and (x_{2}, y_{2})
= (6, 4):
Caution
Do not reverse the order of subtraction from numerator to
denominator when finding the slope. If you divide y_{2}  y_{1} by x_{1}
 x_{2}, you will get
the wrong sign for the slope.
Example 3
Slope for horizontal and vertical lines
Find the slope of each line.
Solution
a) Using (3, 2) and (4, 2) to find the slope of the horizontal line, we get
b) Using (1, 4) and (1, 2) to find the slope of the vertical line, we get x_{2}
 x_{1} = 0. Because the definition of slope using coordinates says that
x_{2}  x_{1} must be
nonzero, the slope is undefined for this line.
Helpful Hint
Think about what slope means to skiers. No one skis on cliffs or even refers
to them as slopes.
Since the ycoordinates are equal for any two points on a horizontal line,
y_{2}  y_{1} = 0 and the slope is 0. Since the xcoordinates are equal for any two points
on a vertical line, x_{2}  x_{1} = 0 and the slope is undefined.
Horizontal and Vertical Lines
The slope of any horizontal line is 0.
Slope is undefined for any vertical line.
Caution
Do not say that a vertical line has no slope because â€œno slopeâ€
could be confused with 0 slope, the slope of a horizontal line.
As you move the tip of your pencil from left to right along a line with positive
slope, the ycoordinates are increasing. As you move the tip of your pencil from
left to right along a line with negative slope, the ycoordinates are decreasing. See
the following figure.
