3.3 The Slope of a Line
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 yvalues, 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
CLASSROOM
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-subtwo”.
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
CLASSROOM
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
3

2
y1  y2
yield the same slope. Make sure to start with the
x1  x2
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
CLASSROOM
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
CLASSROOM
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
CLASSROOM
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, the other is . The table to
1
 several examples.
the right shows
a
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
CLASSROOM
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