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Section 3.3 Equations of Lines
271
Version: Fall 2007
3.3 Equations of Lines
In this section we will develop the
slopeintercept form
of a line. When you have
completed the work in this section, you should be able to look at the graph of a line
and determine its equation in slopeintercept form.
The SlopeIntercept Form
In the previous section, we developed the formula for the slope of a line. Let’s assume
that the dependent variable is
y
and the independent variable is
x
and we have a line
passing through the points
P
(
x
1
,y
1
) and
Q
(
x
2
,y
2
), as shown in
Figure 1
.
x
y
P
(
x
1
,y
1
)
Q
(
x
2
,y
2
)
Δ
x
=
x
2

x
1
Δ
y
=
y
2

y
1
Figure 1.
Determining the slope of a line
through two points.
As we sweep our eyes from left to right, note that the change in
x
is ∆
x
=
x
2
−
x
1
and the change in
y
is ∆
y
=
y
2
−
y
1
. Thus, the slope of the line is determined by the
formula
Slope =
∆
y
∆
x
=
y
2
−
y
1
x
2
−
x
1
.
(1)
Now consider the line in
Figure 2
. Suppose that we are given two facts about this
line:
1. The point where the line crosses the
y
axis (the
y
intercept) is (0
,b
).
2. The “slope” of the line is some number
m
.
To ﬁnd the equation of the line pictured in
Figure 2
, select an arbitrary point
Q
(
x,y
) on the line, then compute the slope of the line using (
x
1
,y
1
) =
P
(0
,b
) and
(
x
2
,y
2
) =
Q
(
x,y
) in the slope formula (
1
).
Slope =
y
2
−
y
1
x
2
−
x
1
=
y
−
b
x
−
0
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Chapter 3 Linear Functions
Version: Fall 2007
x
y
P
(0
,b
)
Q
(
x,y
)
Slope
=
m
Figure 2.
Find the equation of the line
in slopeintercept form.
Simplify.
Slope =
y
−
b
x
We’re given that the slope is the number
m
, so substitute this number for the word
“Slope” in the last result.
m
=
y
−
b
x
Multiply both sides of the last equation by
x
.
mx
=
y
−
b
Add
b
to both sides of the last equation to obtain
mx
+
b
=
y,
or upon exchanging sides of the equation,
y
=
mx
+
b.
The above discussion leads to the following result.
The SlopeIntercept Form of a Line
. If the line
L
intercepts the
y
axis at the
point (0
,b
) and has slope
m
, then the equation of the line is
y
=
mx
+
b.
(2)
This form of the equation of a line is called the
slopeintercept form
. The
function deﬁned by the equation
f
(
x
) =
mx
+
b
is called a
linear function
.
Section 3.3 Equations of Lines
273
Version: Fall 2007
It is important to note two key facts about the slopeintercept form
y
=
mx
+
b
.
•
The coeﬃcient of
x
(the
m
in
y
=
mx
+
b
) is the slope of the line.
•
The constant term (the
b
in
y
=
mx
+
b
) is the
y
coordinate of the
y
intercept (0
,b
).
Procedure for Using the SlopeIntercept Form of a Line
. When given
the slope of a line and the
y
intercept of the line, use the slopeintercept form as
follows:
1. Substitute the given slope for
m
in the formula
y
=
mx
+
b
.
2. Substitute the
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