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Slope of a Line - Day 2
Recall, the slope, m, is a measure of
the steepness of a line.
slope = rise
run
The rise is the vertical distance
between two points.
The run is the horizontal distance between two points.
slope = difference of y­coordinates
difference of x­coordinates
slope = Δy
Δx
Key
Idea
Take Note:
The Greek letter "Δ" (delta) represents change,
so Δy represents the difference between two y­values.
The Slope Formula
If A(x1, y1) and B(x2, y2) are the coordinates of
two points on a line then,
slope = y2 ­ y1
x2 ­ x 1
Example 1: Calculate the slope of the line through the following
points, using the slope formula.
a) A(2, ­5) and B(0, 3)
m = y2 ­ y1
x2 ­ x1
c) P
m = y2 ­ y1
x2 ­ x1
b) C(­3, 7) and D(9, ­2)
m = y 2 ­ y1
x 2 ­ x1
,Q
d) M(­4, ­5) and N(­4, 3)
m = y2 ­ y 1
x2 ­ x1
Take Note: The x­coordinates are the same.
There is no change in x.
The line is vertical. The slope is undefined.
e) X(5, 4) and Y(­6, 4)
m = y2 ­ y 1
x2 ­ x1
Take Note: The y­coordinates are the same.
There is no change in y.
The line is horizontal. The slope is zero.
Example 3: A line has a slope
A(5, 3) and B(x, 12). Calculate x.
and contains the points
Example 4: Given the points A(0, 1), B(3, 3) and C(9, 7), show
that these points are collinear.
If points are collinear, it means that they lie on the same line.
Therefore, the slope calculated would be the same for each pair
of points.
Using Slope in Word Problems
In word problems, slope represents a rate of change.
Some common examples are speed (km/h), hourly wage ($/h),
or taxicab fare ($/km).
In word problems, it is important to be able to identify your
independent and dependent variables.
Recall, the independent variable is the variable whose values you
choose and the dependent variable is the variable whose values
you calculate.
Example 5: A taxi charges $12.25 for a 15 km trip and $16.75 for
a 25 km trip.
a) What is the independent variable? dependent variable?
b) Represent the given information as ordered pairs.
c) What rate does the taxi charge per kilometre?
Example 6: The LeBlanc family is driving home. They are
using cruise control and their speed is constant. After 3 h,
they are 350 km from home. After 5h, they are 130 km from home.
a) Which variable is independent and which is dependent?
b) Determine the speed at which they are travelling.