Key concept: The slope of a line containing two points with coordinates (x ,y ) and (x ,y )is given by the formula
33 Slopes of Lines
Slope of a line
Slope =
Vertical rise
Horizontal run
In a coordinate plane, the slope of a line is the ratio of the change along the yaxis to the change along the xaxis
1
2
2
1
m=
y y
x x
2
2
1
1
where x doesn't equal x .
1
2
The slope of a line indicates whether the line rises to the right, falls to the right, or is horizontal. The slope of a vertical line where x = x .
1
2
.
0
x
y
(
1
, 2)
(
3
, 2)
First use the
rise
run
From (3, 2) to (1, 2), go up 4 units and right 2 units
rise
run
=
4
2
or 2
method.
.
Use the slope method.
(
4
, 0)
(
0
, 1)
Let (4, 0) be (x ,y ) and (0, 1) be (x ,y ).
1
1
2
2
m=
y y
x x
1
1
2
2
m=
1  0
0 (4)
or

1
4
y
x
0
x
0
.
(
3
, 5)
(
1
, 5)
m=
y y
x x
1
1
2
2
m=
55
3 1
m=
0
4
or 0.
(
6
, 4)
.
(
6
, 3)
x
y
0
m=
y y
x x
1
1
2
2
m=
3 (4)
66
m=
7
0
which is undefined
The slope of a line can be used to identify the coordinates of any point on the line. It can also be used to describe a rate of change. The
rate of change
describes how a quantity is changing over time.
Look at the lines l, m, and n. Lines l and m are parallel, and n is perpendicular to l and m. Lets find the slopes of these lines
.
.
.
.
x
y
0
slope of l
slope of m
slope of n
m=

3
5
m=
3

5
m=
5
3
Since lines l and m are parallel, their slopes are the same. Line n is perpendicular to lines l and m, and its slope is the opposite reciprocal of the slopes of l and m; the is, 3/5* 5/3= 1. These results suggest 2 important algebraic properties of parallel and perpendicular lines
Postulates:
3.2 Two nonvertical lines have the same slope if and only if they are parallel.
3.3 Two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Slope Formula
Slope Formula
Slope Formula
m
l
n
(0, 4)
(4, 2)
(5, 1)
(3, 5)
(2, 2)
(1, 3)
.
.
y
.
.
.
.
What you'll learn
Find slopes of different lines.
Use slope to identify perpendicular and parallel lines.
Sarah Prosser
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