I am going to send you my Ashford University Math 221 material.

09 Jun I am going to send you my Ashford University Math 221 material.

Question
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Chapter

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Linear Equations in Two

Variables and Their Graphs

If you pick up any package of food and read the label, you will nd a long list that
usually ends with some mysterious looking names. Many of these strange
elements are food additives. A food additive is a substance or a mixture of
substances other than basic foodstuffs that is present in food as a result of
production, processing, storage, or packaging. They can be natural or synthetic
and are categorized in many ways: preservatives, coloring agents, processing aids,
and nutritional supplements, to name a few.
Food additives have been around since prehistoric humans discovered that
salt would help to preserve meat. Today, food additives can include simple

3.1

Graphing Lines in the
Coordinate Plane

ingredients such as red color from Concord grape skins, calcium, or an enzyme.

3.3

Equations of Lines in
Slope-Intercept Form

3.4

The Point-Slope Form

3.5

Variation

3.6

Graphing Linear
Inequalities in Two
Variables

0.50

what is healthy to eat. At the

0.40

present time the food industry
is working to develop foods that
have less cholesterol, fats, and
other unhealthy ingredients.
Although they frequently

Absorption

Slope

a

have been lively discussions on

3.2

Throughout the centuries there

0.30
0.20
0.10

have different viewpoints, the
food industry and the Food and
Drug Administration (FDA) are

0

1

2
3
4
5
Concentration (mg/ml)

6

c

working to provide consumers
with information on a healthier
diet. Recent developments such
as the synthetically engineered
tomato stirred great controversy,
even though the FDA declared
the tomato safe to eat.

In Exercise 87 of Section 3.4
you will see how a food chemist
uses a linear equation in testing
the concentration of an enzyme
in a fruit juice.

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Chapter 3 Linear Equations in Two Variables and Their Graphs

3.1
In This Section
U1V Graphing Ordered Pairs
U2V Ordered Pairs as Solutions to
Equations
3V Graphing a Linear Equation
U
in Two Variables
U4V Graphing a Line Using
Intercepts
5V Function Notation and
U
Applications

Graphing Lines in the Coordinate Plane

In Chapter 1 you learned to graph numbers on a number line. We also used number
lines to illustrate the solution to inequalities in Chapter 2. In this section, you will
learn to graph pairs of numbers in a coordinate system made up of a pair of number
lines. We will use this coordinate system to illustrate the solution to equations and
inequalities in two variables.

U1V Graphing Ordered Pairs
A GPS unit uses longitude and latitude to locate points on the earth. In mathematics
we also use pairs of real numbers to describe the locations of points in a plane. We
position two number lines at a right angle as shown in Fig. 3.1. The horizontal number line is the x-axis and the vertical number line is the y-axis. The point at which the
axes intersect is the origin. The axes divide the coordinate plane or xy-plane into
four quadrants The quadrants are numbered as shown in Fig. 3.1. The quadrants do not
include any points on the axes. The system is called the rectangular coordinate system or the Cartesian coordinate system. It is named after the French mathematician
René Descartes (15961650).
y-axis

Quadrant II

5
4
3

Quadrant I

2
1
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
Ϫ1
Ϫ2
Ϫ3
Quadrant III
Ϫ4
Ϫ5

Origin
1

2

3

4

5

x-axis

Quadrant IV

Figure 3.1

y

Origin
Ϫ4

(3, 2)

1

Ϫ2 Ϫ1
Ϫ1
Ϫ2
(Ϫ3, Ϫ2) Ϫ3

Figure 3.2

(2, 3)

3
2

1

2

3

4

x

Every point in the plane in Fig. 3.1 corresponds to a pair of numbers. For example, the point corresponding to the pair (2, 3) is found by starting at the origin and
moving 2 units to the right (in the x direction) and then 3 units up (in the y direction).
To locate (3, 2) start at the origin and go 3 units to the right and 2 units up. To locate
(Ϫ3, Ϫ2) start at the origin and go 3 units to the left and then 2 units down. All three
points are shown in Fig. 3.2.
Note that (3, 2) and (2, 3) correspond to different points in Fig. 3.2. Since the
order of the numbers in the pair makes a difference, a pair of numbers in parentheses is called an ordered pair. The first number in an ordered pair is the
x-coordinate, and the second number is the y-coordinate. Locating a point in
the xy-plane that corresponds to an ordered pair is called plotting or graphing the
point.

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E X A M P L E

3.1

1

Graphing Lines in the Coordinate Plane

171

Plotting points
Plot the points (2, 5), (Ϫ1, 4), (Ϫ3, Ϫ4), and (3, Ϫ2).

Solution
To locate (2, 5), start at the origin, move two units to the right, and then move up ve units.
To locate (Ϫ1, 4), start at the origin, move one unit to the left, and then move up four units.
All four points are shown in Fig. 3.3.
y
(Ϫ1, 4)

U Helpful Hint V
In this chapter, you will be doing a lot
of graphing. Using graph paper will
help you understand the concepts
and help you recognize errors. For
your convenience, a page of graph
paper can be found on page 250 of
this text. Make as many copies of it
as you wish.

5
4
3

(2, 5)

2
1
Ϫ4 Ϫ3 Ϫ2 Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
(Ϫ3, Ϫ4) Ϫ5

1

2

3

4

x

(3, Ϫ2)

Figure 3.3

Now do Exercises 128

CAUTION In Chapter 1 the notation (2, 5) was used to represent an interval of real

numbers. Now it represents an ordered pair of real numbers. The context
should always make it clear what we are referring to.

U2V Ordered Pairs as Solutions to Equations

An equation in two variables such as y ϭ 2x Ϫ 1 is satised if we choose a value for
x and a value for y that make it true. If x ϭ 2 and y ϭ 3, then y ϭ 2x Ϫ 1 becomes
y

x

3 ϭ 2(2) Ϫ 1
3 ϭ 3.
Because the last statement is true, the ordered pair (2, 3) satises the equation or is
a solution to the equation. The x-value is always written rst and the y-value second.
CAUTION The ordered pair (3, 2) does not satisfy y ϭ 2x 1, because for x ϭ 3 and

y ϭ 2, we have

2

2(3) Ϫ 1.

In Section 2.4 we said that an equation such as y ϭ 2x Ϫ 1 expresses y as a
function of x because it uniquely determines y from any chosen x-value. For this reason we call x the independent variable and y the dependent variable. We usually
use a function to determine the value of the dependent variable from the value of the

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Chapter 3 Linear Equations in Two Variables and Their Graphs

independent variable. However, for a function of the form y ϭ mx ϩ b we can nd
either coordinate when given the other, as shown in Example 2.

E X A M P L E

2

Finding solutions to an equation
Each of the following ordered pairs is missing one coordinate. Complete each ordered pair
so that it satises the equation y ϭ Ϫ3x ϩ 4.
a) (2, )

b) ( , Ϫ5)

c) (0, )

Solution
a) The x-coordinate of (2, ) is 2. Let x ϭ 2 in the equation y ϭ Ϫ3x ϩ 4:
y ϭ Ϫ3 и 2 ϩ 4
ϭ Ϫ6 ϩ 4
ϭ Ϫ2
The ordered pair (2, Ϫ2) satises the equation.
b) The y-coordinate of ( , Ϫ5) is Ϫ5. Let y ϭ Ϫ5 in the equation y ϭ Ϫ3x ϩ 4:
Ϫ5 ϭ Ϫ3x ϩ 4
Ϫ9 ϭ Ϫ3x
3ϭx
The ordered pair (3, Ϫ5) satises the equation.
c) Replace x by 0 in the equation y ϭ Ϫ3x ϩ 4:
y ϭ Ϫ3 и 0 ϩ 4 ϭ 4
So (0, 4) satises the equation.

Now do Exercises 2944

U3V Graphing a Linear Equation in Two Variables
In Chapter 2 we dened a linear equation in one variable as an equation of the form
ax ϭ b, where a ϶ 0. A linear equation in two variables is dened similarly:
Linear Equation in Two Variables
A linear equation in two variables is an equation of the form
Ax ϩ By ϭ C,
where A and B are not both zero.
Consider the linear equation Ϫ2x ϩ y ϭ Ϫ1. If we solve it for y, we get y ϭ 2x Ϫ 1.
If we choose any real number for x, we can use y ϭ 2x Ϫ1 to compute a corresponding y-value. So there are innitely many ordered pairs that satisfy the equation. To get
a better understanding of the solution set to a linear equation in two variables, we often
graph all of the ordered pairs in the solution set. The graph of the solution set to a linear equation in two variables is a straight line, as shown in Example 3.

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E X A M P L E

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3

Graphing Lines in the Coordinate Plane

173

Graphing an equation
Graph the equation y ϭ 2x Ϫ 1 in the coordinate plane.

Solution
U Calculator Close-Up V
You can make a table of values for x
and y with a graphing calculator.
Enter the equation y ϭ 2x Ϫ 1 using
Y ϭ and then press TABLE.

To nd ordered pairs that satisfy y ϭ 2x Ϫ 1, we arbitrarily select some x-coordinates and
calculate the corresponding y-coordinates:
If x ϭ Ϫ3,

then y ϭ 2(Ϫ3) Ϫ 1 ϭ Ϫ7.

If x ϭ Ϫ2,

then y ϭ 2(Ϫ2) Ϫ 1 ϭ Ϫ5.

If x ϭ Ϫ1,

then y ϭ 2(Ϫ1) Ϫ 1 ϭ Ϫ3.

If x ϭ 0,

then y ϭ 2(0) Ϫ 1 ϭ Ϫ1.

If x ϭ 1,

then y ϭ 2(1) Ϫ 1 ϭ 1.

If x ϭ 2,

then y ϭ 2(2) Ϫ 1 ϭ 3.

If x ϭ 3,

then y ϭ 2(3) Ϫ 1 ϭ 5.

We can make a table for these results as follows:
x

The graph of a linear equation in one
variable consists of a single point on a
number line. The graph of a linear
equation in two variables consists of
a line in a coordinate plane.

Ϫ2

Ϫ1

0

1

2

3

y ϭ 2x Ϫ 1

U Helpful Hint V

Ϫ3
Ϫ7

Ϫ5

Ϫ3

Ϫ1

1

3

5

The ordered pairs (Ϫ3, Ϫ7), (Ϫ2, Ϫ5), (Ϫ1, Ϫ3), (0, Ϫ1), (1, 1), (2, 3), and (3, 5) are
graphed in Fig. 3.4. Draw a straight line through these points, as shown in Fig. 3.5. The
line in Fig. 3.5 is the graph of the solution set to y ϭ 2x Ϫ 1. The arrows on the ends of
the line indicate that it goes indenitely in both directions.
y
5
4
3

y

2
1
Ϫ3 Ϫ2 Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
Ϫ6
Ϫ7
Figure 3.4

1 2

3

4
3
2
1

x

Ϫ4 Ϫ3 Ϫ2 Ϫ1
Ϫ1

y ϭ 2x Ϫ 1
1

2 3

4

x

Ϫ3
Ϫ4
Figure 3.5

Now do Exercises 4552

A linear equation in two variables is an equation of the form Ax ϩ By ϭ C, where
A and B are not both zero. Note that we can have A ϭ 0 if B 0, and we can have
B ϭ 0 with A
0. So equations such as x ϭ 8 and y ϭ 2 are linear equations.

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Chapter 3 Linear Equations in Two Variables and Their Graphs

Equations such as x Ϫ y Ϫ 5 ϭ 0 and y ϭ 2x ϩ 3 are also called linear equations
because they could be rewritten in the form Ax ϩ By ϭ C. Equations such as
5
y ϭ 2×2 or y ϭ x are not linear equations.

E X A M P L E

4

Graphing an equation
Graph the equation 3x ϩ y ϭ 2. Plot at least ve points.

Solution
It is easier to make a table of ordered pairs if we express y as a function of x. So subtract
3x from each side to get y ϭ Ϫ3x ϩ 2. Now select some values for x and then calculate
the corresponding y-coordinates:
If x ϭ Ϫ2,

(Ϫ2, 8)

(Ϫ1, 5)

5
4

then y ϭ Ϫ3(1) ϩ 2 ϭ Ϫ1.

If x ϭ 2,

then y ϭ Ϫ3(2) ϩ 2 ϭ Ϫ4.

The following table shows these ve ordered pairs:

2 (0, 2)
1
Ϫ3 Ϫ2 Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4

then y ϭ Ϫ3(0) ϩ 2 ϭ 2.

If x ϭ 1,
y ϭ Ϫ3 x ϩ 2

then y ϭ Ϫ3(Ϫ1) ϩ 2 ϭ 5.

If x ϭ 0,

8
7
6

then y ϭ Ϫ3(Ϫ2) ϩ 2 ϭ 8.

If x ϭ Ϫ1,

y

x

2 3 4
(1, Ϫ1)
(2, Ϫ4)

5

Ϫ1

0

1

2

y ϭ Ϫ3x ϩ 2

x

Ϫ2
8

5

2

Ϫ1

Ϫ4

Plot (Ϫ2, 8), (Ϫ1, 5), (0, 2), (1, Ϫ1), and (2, Ϫ4). Draw a line through them, as shown in
Fig. 3.6.

Now do Exercises 5356

Figure 3.6

U Calculator Close-Up V
To graph y ϭ Ϫ3x ϩ 2, enter the equation
using the Y ϭ key:

x-values used for the graph, and likewise for
Ymin and Ymax. Xscl and Yscl (scale) give

Press GRAPH to get the graph:
10

Ϫ10

10

Ϫ10

Next, set the viewing window (WINDOW) to
get the desired view of the graph. Xmin and
Xmax indicate the minimum and maximum

the distance between tick marks on the
respective axes.

Even though the graph is not really
straight, it is consistent with the graph of
y ϭ Ϫ3x ϩ 2 in Fig. 3.6.

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E X A M P L E

3.1

5

Graphing Lines in the Coordinate Plane

175

Horizontal and vertical lines
Graph each linear equation.
a) y ϭ 4

b) x ϭ 3

Solution
a) The equation y ϭ 4 is a simplication of 0 и x ϩ y ϭ 4. So if y is replaced with 4,
then we can use any real number for x. For example, (Ϫ1, 4) satises 0 и x ϩ y ϭ 4
because 0(Ϫ1) ϩ 4 ϭ 4 is correct. The following table shows ve ordered pairs
that satisfy y ϭ 4.
x

Ϫ2

Ϫ1

0

1

2

yϭ4

4

4

4

4

4

Figure 3.7 shows a horizontal line through these points.
b) The equation x ϭ 3 is a simplication of x ϩ 0 и y ϭ 3. So if x is replaced
with 3, then we can use any real number for y. For example, (3, Ϫ2) satises
x ϩ 0 и y ϭ 3 because 3 ϩ 0(Ϫ2) ϭ 3 is correct. The following table shows ve
ordered pairs that satisfy x ϭ 3.
xϭ3

3

3

3

3

3

y

Ϫ2

Ϫ1

0

1

2

Figure 3.8 shows a vertical line through these points.
y
y
3
5
(Ϫ2, 4)

U Calculator Close-Up V
You cannot graph the vertical line
x ϭ 3 on most graphing calculators.
The only equations that can be
graphed are ones in which y is written
in terms of x.

Ϫ4 Ϫ3 Ϫ2 Ϫ1

Figure 3.7

yϭ4

3
2
1

xϭ3

(2, 4)
Ϫ1
Ϫ1

1 2

(3, 2)

2
1

3

4

x

Ϫ2
Ϫ3
Ϫ4

1 2

4

5

x

(3, Ϫ2)

Figure 3.8

Now do Exercises 5768

CAUTION If x ϭ 3 occurs in the context of equations in a single variable, then x ϭ 3

has only one solution, 3. In the context of equations in two variables,
x ϭ 3 is assumed to be a simplied form of x ϩ 0 и y ϭ 3, and it has innitely many solutions (all of the ordered pairs on the line in Fig. 3.8).
All of the equations we have considered so far have involved single-digit
numbers. If an equation involves large numbers, then we must change the scale on the
x-axis, the y-axis, or both to accommodate the numbers involved. The change of scale
is arbitrary, and the graph will look different for different scales.

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Chapter 3 Linear Equations in Two Variables and Their Graphs

E X A M P L E

6

Adjusting the scale
Graph the equation y ϭ 20x ϩ 500. Plot at least ve points.

Solution
The following table shows ve ordered pairs that satisfy the equation.

y
800
600
(0, 500)
(Ϫ20, 100)
Ϫ40

200

(10, 700)

x

Ϫ20
Ϫ200
Ϫ400

10 20 30 40

Ϫ10

0

10

20

y ϭ 20x ϩ 500

y ϭ 20x ϩ 500

Ϫ20
100

300

500

700

900

x

To t these points onto a graph, we change the scale on the x-axis to let each division represent 10 units and change the scale on the y-axis to let each division represent 200 units.
The graph is shown in Fig. 3.9.

Figure 3.9

Now do Exercises 6974

U4V Graphing a Line Using Intercepts
For many lines, the easiest points to locate are the points where the line crosses the
axes.
Intercepts
The x-intercept is the point at which a line crosses the x-axis.
The y-intercept is the point at which a line crosses the y-axis.
The second coordinate of the x-intercept is 0, and the rst coordinate of the y-intercept
is 0. If a line has two distinct intercepts, they can be used as two points that determine
the location of the line.

E X A M P L E

7

U Helpful Hint V
You can nd the intercepts for
2x Ϫ 3y ϭ 6 using the cover-up
method. Cover up Ϫ3y with your
pencil, and then solve 2x ϭ 6
mentally to get x ϭ 3 and an
x-intercept of (3, 0). Now cover up 2x
and solve Ϫ3y ϭ 6 to get y ϭ Ϫ2 and
a y-intercept of (0, Ϫ2).

Graphing a line using intercepts
Graph the equation 2x Ϫ 3y ϭ 6 by using the x- and y-intercepts.

Solution
To nd the x-intercept, let y ϭ 0 in the equation 2x Ϫ 3y ϭ 6:
2x Ϫ 3 и 0 ϭ 6
2x ϭ 6
xϭ3
The x-intercept is (3, 0). To nd the y-intercept, let x ϭ 0 in 2x Ϫ 3y ϭ 6:
2 и 0 Ϫ 3y ϭ 6
Ϫ3y ϭ 6
y ϭ Ϫ2

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3.1

U Calculator Close-Up V
To graph 2x Ϫ 3y ϭ 6 on a calculator
you must solve for y. In this case,
y ϭ (2͞3)x Ϫ 2.

Graphing Lines in the Coordinate Plane

177

The y-intercept is (0, Ϫ2). Locate the intercepts and draw a line through them, as shown
in Fig. 3.10. To check, nd one additional point that satises the equation, say (6, 2), and
see whether the line goes through that point.
y

3

Ϫ3

3
2
1

5

Ϫ3 Ϫ2 Ϫ1
Ϫ1
(0, Ϫ2)

Ϫ4

Check point (6, 2)
(3, 0)
1

Ϫ3
Ϫ4

Since the calculator graph appears to
be the same as the graph in Fig. 3.10,
it supports the conclusion that
Fig. 3.10 is correct.

4

5

x

Intercepts
2 x Ϫ 3y ϭ 6

Figure 3.10

Now do Exercises 7582

U5V Function Notation and Applications

An equation of the form y ϭ mx ϩ b expresses y as a function of x, and it is called a
linear function. Linear functions occur in many real-life situations. For example, if the
monthly cost C of a cell phone is $50 plus 10 cents per minute, then C ϭ 50 ϩ 0.10n
where n is the number of minutes used. We may also write C(n) in place of C. This
notation is called function notation. We read C(n) as the cost of n minutes or simply C of n. Using function notation is very convenient for identifying more than one
cost. For example, to express the fact that the cost for 100 minutes is $60, 200 minutes
is $70, and 300 minutes is $80 we can simply write
C(100) ϭ $60, C(200) ϭ $70, and C(300) ϭ $80.

E X A M P L E

8

House plans
An architect uses the function C(x) ϭ 30x ϩ 900 to determine the cost C for drawing house
plans, where x is the number of copies of the plan that the client receives.
a) Find C(5), C(6), and C(7).
b) Find the intercepts and interpret them.
c) Graph the function.
d) Does the cost increase or decrease as x increases?

Solution
a) Replace x by 5, 6, and 7 in the equation C(x) ϭ 30x ϩ 900:
C(5) ϭ 30(5) ϩ 900 ϭ 1050
C(6) ϭ 30(6) ϩ 900 ϭ 1080
C(7) ϭ 30(7) ϩ 900 ϭ 1110
So the cost of 5 plans is $1050, the cost of 6 plans is $1080, and the cost of 7 plans
is $1110.

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Chapter 3 Linear Equations in Two Variables and Their Graphs

b) If x ϭ 0, then C(0) ϭ 30(0) ϩ 900 ϭ 900. The C-intercept is (0, 900). The cost is
$900 for the labor involved in drawing the plans, even if you get no copies of the
plan. If C(x) ϭ 0, then 30x ϩ 900 ϭ 0 or x ϭ Ϫ30. So the x-intercept is (Ϫ30, 0),
but in this situation the x-intercept is meaningless. The number of plans cant be
negative.

C
1200
(8, 1140)

(0, 900)
900

C ϭ 30x ϩ 900

600

c) The graph goes through (0, 900) and (8, 1140) as shown in Fig. 3.11. Since
negative values of x are meaningless, the graph is drawn in the rst quadrant only.

300

d) As x increases, the cost increases.

0

2

4

6

8

Now do Exercises 8388

10 x

Figure 3.11

E X A M P L E

9

Ticket demand
The demand for tickets to see the Ice Gators play hockey can be modeled by the equation
d ϭ 8000 Ϫ 100p, where d is the number of tickets sold and p is the price per ticket in
dollars.
a) How many tickets will be sold at $20 per ticket?
b) Find the intercepts and interpret them.
c) Graph the linear equation.
d) What happens to the demand as the price increases?

Solution
d
8000

a) If tickets are $20 each, then d ϭ 8000 Ϫ 100 и 20 ϭ 6000. So at $20 per ticket, the
demand will be 6000 tickets.

(0, 8000)

b) Replace d with 0 in the equation d ϭ 8000 Ϫ 100p and solve for p:
6000

0 ϭ 8000 Ϫ 100p
4000

100p ϭ 8000 Add 100p to each side.
p ϭ 80

2000
0

20

Figure 3.12

40

(80, 0)
60
80 p

Divide each side by 100.

If p ϭ 0, then d ϭ 8000 Ϫ 100 и 0 ϭ 8000. So the intercepts are (0, 8000) and
(80, 0). If the tickets are free, the demand will be 8000 tickets. At $80 per ticket, no
tickets will be sold.
c) Graph the line using the intercepts (0, 8000) and (80, 0) as shown in Fig. 3.12. The
line is graphed in the rst quadrant only, because negative values for demand or
price are meaningless.
d) When the tickets are free, the demand is high. As the price increases, the demand
goes down. At $80 per ticket, there will be no demand.

Now do Exercises 8992

Note that d ϭ 8000 Ϫ 100p is a model for the demand in Example 9. A model
car has only some of the features of a real car, and the same is true here. For instance,
the line in Fig. 3.12 contains innitely many points. But there is really only a nite
number of possibilities for price and demand, because we cannot sell a fraction of
a ticket.

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3.1

179

Fill in the blank.

True or false?

1. The point at the intersection of the x- and y-axis is the
.
2. Every point in the coordinate plane corresponds to an
of real numbers.
3. The point at which a line crosses the x-axis is the
.
4. The point at which a line crosses the y-axis is the
.
5. The graph of y ϭ 5 is a
line.
6. The graph of x ϭ 3 is a
line.
7. A
equation in two variables has the form
Ax ϩ By ϭ C where A and B are not both zero.

8. The point (2, 4) satises 2y Ϫ 3x ϭ Ϫ8.
9. If (1, 5) satises an equation, then (5, 1) does also.
10. The origin is in quadrant I.
11. The point (4, 0) is on the y-axis.
12. The graph of x ϩ 0 и y ϭ 9 is the same as the graph
of x ϭ 9.
13. The y-intercept for x ϩ 2y ϭ 5 is (5, 0).
14. The point (5, Ϫ3) is in quadrant III.

Exercises
U Study Tips V
It is a good idea to work with others, but dont be misled. Working a problem with help is not the same as working a problem on your
own.
Math is personal. Make sure that you can do it.

15. (1.4, 4)

U1V Graphing Ordered Pairs

16. (Ϫ3, 0.4)

Plot the points on a rectangular coordinate system.
See Example 1.
1. (1, 5)

2. (4, 3)

3. (Ϫ2, 1)

4. (Ϫ3, 5)

΂

΃

΂

΃

1
5. 3, Ϫ
2

1
6. 2, Ϫ
3

7. (Ϫ2, Ϫ4)

8. (Ϫ3, Ϫ5)

9. (0, 3)

10. (0, Ϫ2)

11. (Ϫ3, 0)

12. (5, 0)

13. (, 1)

14. (Ϫ2, )

For each point, name the quadrant in which it lies or the axis
on which it lies.
17. (Ϫ3, 45)

18. (Ϫ33, 47)

19. (Ϫ3, 0)

20. (0, Ϫ9)

21. (Ϫ2.36, Ϫ5)

22. (89.6, 0)

3.1

Warm-Ups

Graphing Lines in the Coordinate Plane

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23. (3.4, 8.8)

΂

1
25. Ϫ, 50
2

24. (4.1, 44)

΃

1
41. y ϭ x ϩ 2
3
x

΂

΃

1
26. Ϫ6, Ϫ
2

27. (0, Ϫ99)

28. (, 0)

1
42. y ϭ Ϫ x ϩ 1
2
y

x

Ϫ6

Ϫ2

Ϫ3

Ϫ1
2

1

3

1

2

U2V Ordered Pairs as Solutions to Equations
Complete each ordered pair so that it satises the given
equation. See Example 2.

΂ ΃

44. 200x ϩ y ϭ 50
x

y

0
Ϫ10

100
0

0

, Ϫ5)

y

Ϫ1
2

Ϫ30

30. y ϭ 2x ϩ 5: (8, ), (Ϫ1, ), ( , Ϫ1)
,(

43. y Ϫ 20x ϭ 400
x

29. y ϭ 3x ϩ 9: (0, ), ( , 24), (2, )

1
31. y ϭ Ϫ3x Ϫ 7: (0, ), ,
3

y

0

600
1

2

΂

1
32. y ϭ Ϫ5x Ϫ 3: (Ϫ1, ), Ϫ,
2

΃, (

, Ϫ2)

U3V Graphing a Linear Equation in Two Variables
Graph each equation. Plot at least ve points for each equation.
Use graph paper. See Examples 35. If you have a graphing
calculator, use it to check your graphs when possible.

33. y ϭ 1.2x ϩ 54.3: (0, ), (10, ), ( , 54.9)

45. y ϭ x ϩ 1
34. y ϭ 1.8x ϩ 22.6: (1, ), (Ϫ10,

), (

46. y ϭ x Ϫ 1

47. y ϭ 2x ϩ 1

48. y ϭ 3x Ϫ 1

49. y ϭ 3x Ϫ 2

50. y ϭ 2x ϩ 3

, 22.6)

35. 2x Ϫ 3y ϭ 6: (3, ), ( , Ϫ2), (12, )
36. 3x ϩ 5y ϭ 0: (Ϫ5, ), ( , Ϫ3), (10, )
37. 0 и y ϩ x ϭ 5: ( , Ϫ3), ( , 5), ( , 0)
38. 0 и x ϩ y ϭ Ϫ6: (3, ), (Ϫ1, ), (4, )

Use the given equations to nd the missing coordinates in the
following tables.
39. y ϭ Ϫ2x ϩ 5
x

40. y ϭ Ϫx ϩ 4
y

x

Ϫ2

Ϫ2

0

0

2

y

2
Ϫ3

0

Ϫ7

Ϫ2

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51. y ϭ x

3.1

52. y ϭ Ϫx

Graphing Lines in the Coordinate Plane

57. y ϭ Ϫ3

66. x ϩ 4y ϭ 5

67. y ϭ 0.36x ϩ 0.4
55. y ϭ Ϫ2x ϩ 3

64. x Ϫ 2y ϭ 6

65. x Ϫ 3y ϭ 6
53. y ϭ 1 Ϫ x

63. x ϩ 2y ϭ 4

68. y ϭ 0.27x Ϫ 0.42

54. y ϭ 2 Ϫ x

56. y ϭ Ϫ3x ϩ 2

58. y ϭ 2

Graph each equation. Plot at least ve points for each
equation. Use graph paper. See Example 6. If you have
a graphing calculator, use it to check your graphs.
69. y ϭ x ϩ 1200

71. y ϭ 50x Ϫ 2000

59. x ϭ 2

61. 2x ϩ y ϭ 5

70. y ϭ 2x Ϫ 3000

72. y ϭ Ϫ300x ϩ 4500

73. y ϭ Ϫ400x ϩ 2000

74. y ϭ 500x ϩ 3

60. x ϭ Ϫ4

62. 3x ϩ y ϭ 5

181

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U4V Graphing a Line Using Intercepts
For each equation, state the x-intercept and y-intercept. Then
graph the equation using the intercepts and a third point. See
Example 7.
75. 3x ϩ 2y ϭ 6

76. 2x ϩ y ϭ 6

a) Find C(0) and C(2).
b) If the cost was $440, then how many hours were spent
on the job?
84. Moving day. The one-day cost of renting a truck for a local
move is a function of the number of miles put on the truck.
The cost C in dollars is determined from the mileage m by
the linear function
C(m) ϭ 0.42m ϩ 39.
a) Find the cost for 66 miles.
b) If the cost of the truck was $54.96, then how many
miles were driven?

77. x Ϫ 4y ϭ 4

78. Ϫ2x ϩ y ϭ 4

85. Social Security. The percentage of full benet that you
receive from Social Security is a function of the age at
which you retire. The linear function
p(a) ϭ 8a 436
determines the percentage of full benet p from the retirement age a for ages 67 through 70.

3
79. y ϭ x Ϫ 9
4

1
80. y ϭ Ϫ x ϩ 5
2

a) Find p(67) and p(68).
b) If a person receives 124% of full benet, then at what
age did the person retire?
c) If full benet is $14,000 per year for Bob Jones, then
how much does he get per year if he retires at age 69?
86. Retiring early. If you retire before the full retirement age
of 67, you get less than full benet. For ages 64 through 67
the linear function

1
1
81. x ϩ y ϭ 1
2
4

1
1
82. x Ϫ y ϭ 3
3
2

1040
20
p(a) ϭ a
3
3
determines the percentage of full benet p for retirement
age a.
a) Find…