Chapter 7: Linear Relationships
Example 3
Slope is written as a ratio. If the line is drawn on a set of axes,
a slope triangle can be drawn between any two convenient
points (that is, where grid lines cross and coordinates are
integers), as shown in the graph at right. Count the vertical
distance and the horizontal distance on the dashed sides of the
slope triangle. Write the distances in a ratio:
m =
vertical! !
horizontal!!
means change. The order in the fraction is important: the
numerator (top of the fraction) must be the vertical distance
and the denominator (bottom of the fraction) must be the
horizontal distance.
Parallel lines have the same steepness and direction, so they
have the same slope, as shown in the graph at right.
If
, then the line is horizontal and has a slope of zero,
that is,
, then the line is vertical and its slope is
undefined, so we say that it has no slope.
Example 4
When the vertical and horizontal distances are not easy to determine, you can find the slope by
drawing a generic slope triangle and using it to find the lengths of the vertical (
) segments. The figure at right shows how to find the slope of the line that passes
through the points (-21, 9) and (19, -15). First graph the
points on unscaled axes by approximating where they are
located, then draw a slope triangle. Next find the distance
along the vertical side by noting that it is 9 units from point
B to the x-axis then 15 units from the x-axis to point C, so
is 24. Then find the distance from point A to the y-axis
(21) and the distance from the y-axis to point B (19).
is
40. This slope is negative because the line goes downward
from left to right, so the slope is
A
B
(-21,9)
(19,-15)(19,-15)
C
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