3.3 The Slope of a Line
1
Find the slope of a line, given two points.
2
Find the slope from the equation of a line.
3
Use slopes to determine whether two lines are parallel,
perpendicular, or neither.
The Slope of a Line
An important characteristic of the lines we graphed in Section 3.2 is
their slant, or “steepness.”
One way to measure the steepness of a line is to compare the vertical
change in the line with the horizontal change while moving along the
line from one fixed point to another. This measure of steepness is
called the slope of the line.
Slide 3.3-3
Objective 1
Find the slope of a line, given two
points.
Slide 3.3-4
Find the slope of a line, given two points.
To find the steepness, or slope, of the line in the figure below, begin at
point Q and move to point P. The vertical change, or rise, is the
change in the y-values, which is the difference 6 − 1 = 5 units. The
horizontal change, or run, is the change in the x-values, which is the
difference 5 − 2 = 3 units.
The slope is the ratio of the vertical change in y to the horizontal
change in x.
vertical change in y  rise 
5
slope (m) 

horizontal change in x  run  3
Count squares on the grid to find the
change. Upward and rightward movements
are positive. Downward and leftward
movements are negative.
Slide 3.3-5
EXAMPLE 1 Finding the Slope of a Line
Find the slope of the line.
Solution:
6
m
1
m  6
Slide 3.3-6
Find the slope of a line, given two points. (cont’d)
The slope of a line can be found through two nonspecific points. This
notation is called subscript notation, read x1 as “x-sub-one” and x2 as
“x-sub-two”.
Moving along the line from the point (x1, y1) to the point (x2, y2), we see
that y changes by y2 − y1 units. This is the vertical change (rise).
Similarly, x changes by x2 − x1 units, which is the horizontal change
(run). The slope of the line is the ratio of y2 − y1 to x2 − x1.
Slope Formula
The slope m of a line through the points (x1, y1) and (x2, y2) is
vertical change in y  rise 
y y
m
 2 1
horizontal change in x  run  x2  x1
(where x1  x2 ).
The slope of a line is the same for any two points on the line.
Slide 3.3-7
EXAMPLE 2 Finding Slopes of Lines
Find the slope of the line through (6, −8) and (−2, 4).
Solution:
4   8
m
2  6
y2  y1
x2  x1
and
12

8
y1  y2
x1  x2
3

2
yield the same slope. Make sure to start with the
x- and y-values of the same point and subtract the x- and y-values of the other
point.
Slide 3.3-8
Find the slope of a line, given two points. (cont’d)
Orientation of Lines with Positive and Negative Slopes
A line with a positive slope rises (slants up) from left to right.
A line with a negative slope falls (slants down) from left to right.
Slopes of Horizontal and Vertical Lines
Horizontal lines, with equations of the form y = k, have slope 0.
Vertical lines, with equations of the form x = k, have undefined
slopes.
Slide 3.3-9
EXAMPLE 3 Finding the Slope of a Horizontal Line
Find the slope of the line through (2, 5) and (−1, 5).
Solution:
55
m
1  2
0

3
0
Slide 3.3-10
EXAMPLE 4 Finding the Slope of a Vertical Line
Find the slope of the line through (3, 1) and (3,−4).
Solution:
4  1
m
33
5

0
undefined slope
Slide 3.3-11
Objective 2
Find the slope from the equation
of a line.
Slide 3.3-12
Find the slope from the equation of a line.
Consider the equation y = −3x + 5.
The slope of the line can be found by choosing two different points for
value x and then solving for the corresponding values of y. We choose
x = −2 and x = 4.
y  3 x  5
y  3 x  5
y  3  2  5
y  65
y  11
y  3  4   5
y  12  5
y  7
The ordered pairs are (−2,11) and (4, −7). Now we use the slope
formula.
11   7  18
m

 3
2  4
6
Slide 3.3-13
Find the slope from the equation of a line. (cont’d)
The slope, −3 is found, which is the same number as the
coefficient of x in the given equation y = −3x + 5. It can be shown
that this always happens, as long as the equation is solved for y.
Finding the Slope of a Line from Its Equation
Step 1: Solve the equation for y.
Step 2: The slope is given by the coefficient of x.
Slide 3.3-14
EXAMPLE 5 Finding Slopes from Equations
Find the slope of the line 3x + 2y = 9.
Solution:
3x  2 y  3x  9  3x
2 y 3 x  9

2
2
3
9
y   x
2
2
3
m
2
Slide 3.3-15
Objective 3
Use slopes to determine whether
two lines are parallel,
perpendicular, or neither.
Slide 3.3-16
Use slopes to determine whether two lines are parallel,
perpendicular, or neither.
Two lines in a plane that never intersect are parallel. We use slopes
to tell whether two lines are parallel. Nonvertical parallel lines
always have equal slopes.
Lines are perpendicular if they
intersect at a 90° angle. The product
of the slopes of two perpendicular
lines, neither of which is vertical, is
always − 1. This means that the
slopes of perpendicular lines are
negative (or opposite) reciprocals — if
one slope is the nonzero number a,
1
the other is  . The table to the right
a
shows several examples.
Slide 3.3-17
Use slopes to determine whether two lines are
parallel, perpendicular, or neither. (cont’d)
Slopes of Parallel and Perpendicular Lines
Two lines with the same slope are parallel.
Two lines whose slopes have a product of − 1 are perpendicular.
Slide 3.3-18
EXAMPLE 6 Deciding Whether Two Lines Are Parallel or Perpendicular
Decide whether the pair of lines is parallel, perpendicular, or
neither.
3x  y  4
x  3y  9
Solution:
3x  y  3x  4  3x
 y 4  3x

1
1
y  4  3x
m3
1
  3   1
3
x  3y  x  9  x
3y 9 x
 
3 3 3
1
y  3 x
3
1
m
3
The product of their slopes is − 1, so they are
perpendicular
Slide 3.3-19