Linear Equations in Two Variables: 2.1 The Rectangular Coordinate System and Midpoint
Linear Equations in Two Variables: 2.1 The Rectangular Coordinate System and Midpoint
Linear Equations in Two Variables: 2.1 The Rectangular Coordinate System and Midpoint
Linear Equations
in Two Variables
Key Terms
W M V E P O L S W W M M R E N E F
coordinate midpoint V D P D I D F K O O R Z J J P P Q
origin x-intercept Z E R G N V R P W L X J C K A W J
x-axis y-intercept O H R I F F X P P E B P F P Y X S
y-axis horizontal J M V T I M T P E C R E T N I X C
quadrant vertical U R P N I W Y I N T E R C E P T R
slope E W K N L C J V D Y Z R N G X D I
T T D I V Y A Z L X N R T K A K V
A N P G T E L L A Z T Q E H X V E
N I R I N Y D K T T K N B T I Z Z
I O W R A E H A N Y L K N A S Y X
D P Q O R C E S O T X Q T S X K Q
R D A V D L T O Z C U N F R Y O J
O I E F A O U G I T D A I D A P X
O M C H U L T K R R Z C E K X T T
C Y K I Q R U R O D W L N B I S U
Z W J U R R P S H E C X M N S N N
103
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40
30
20
10
0
0 10 20 30 40 50 60 70 80
Age of Participant (years)
Figure 2-1
y-axis
To picture two variables simultaneously, we
6
use a graph with two number lines drawn at 5
right angles to each other (Figure 2-2). This 4
forms a rectangular coordinate system. The hor- Quadrant II 3 Quadrant I
2
izontal line is called the x-axis, and the vertical 1 Origin x-axis
line is called the y-axis. The point where the lines
6 5 4 3 2 1 0 1 2 3 4 5 6
intersect is called the origin. On the x-axis, the 1
2
numbers to the right of the origin are positive, Quadrant III 3 Quadrant IV
4
5
6
Figure 2-2
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Section 2.1 The Rectangular Coordinate System and Midpoint Formula 105
and the numbers to the left are negative. On the y-axis, the numbers above the
origin are positive, and the numbers below are negative. The x- and y-axes divide
the graphing area into four regions called quadrants.
2. Plotting Points
Points graphed in a rectangular coordinate system are defined by two numbers as
an ordered pair (x, y). The first number (called the first coordinate or abscissa) is
the horizontal position from the origin. The second number (called the second
coordinate or ordinate) is the vertical position from the origin. Example 1 shows
how points are plotted in a rectangular coordinate system.
Solution:
y a. The point (4, 1) is in quadrant I.
5
(3, 4)
4 b. The point (3, 4) is in quadrant . TIP: Notice that the
3 (0, 3) points (3, 4) and (4, 3)
2 c. The point (4, 3) is in quadrant IV.
(4, 1) are in different quadrants.
1
(4, 0)
x d. The point (52, 2) can also be written as Changing the order of
5 4 3 2 1 0 1
(2.5, 2). This point is in quadrant III.
2 3 4 5
1 the coordinates changes
2
the location of the point.
5
2 , 2 3
(4, 3)
e. The point (0, 3) is on the y-axis.
That is why points are
4
5 f. The point (4, 0) is located on the x-axis. represented by ordered
Figure 2-3 pairs (Figure 2-3).
Skill Practice Plot the point and state the quadrant or axis where it is located.
1a. (3, 5) b. (4, 0) c. (2, 1)
d. (0, 3) e. (2, 2) f. (5, 2)
The effective use of graphs for mathematical models requires skill in identi-
fying points and interpreting graphs. Skill Practice Answers
1a. (3, 5); quadrant I
b. (4, 0); x-axis
Example 2 Interpreting a Graph c. (2, 1); quadrant IV
d. (0, 3); y-axis
Kristine started a savings plan at the beginning of the year and plotted the e. (2, 2); quadrant III
amount of money she deposited in her savings account each month. The graph f. (5, 2); quadrant II
y
of her savings is shown in Figure 2-4. The values on the x-axis represent the first
5
6 months of the year, and the values on the y-axis represent the amount of money 4 (3, 5)
in dollars that she saved. Refer to Figure 2-4 to answer the questions. Let x 1 3 (0, 3)
represent January on the horizontal axis. (5, 2) 2
(4, 0) 1
x
5 4 3 2 1 0 1 2 3 4 5
1
2 (2, 1)
(2, 2) 3
4
5
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y
100
TIP: The scale on the
80
Savings ($)
Solution:
a. When x is 6, the y-coordinate is 40. This means that in June, Kristine
saved $40.
b. The point with the greatest y-coordinate occurs when x is 2. She saved the
most money, $90, in February.
c. The point with the lowest y-coordinate occurs when x is 4. She saved the
least amount, $10, in April.
d. In March, the x-coordinate is 3 and the y-coordinate is 80. She saved $80
in March.
e. The two points with the same y-coordinate occur when x 1 and when
x 5. She saved $60 in both January and May.
Section 2.1 The Rectangular Coordinate System and Midpoint Formula 107
TIP: The midpoint of a line segment is found by taking the average of the
x-coordinates and the average of the y-coordinates of the endpoints.
Solution:
a. 14, 62 and (8, 1)
4 8 6 1
a , b Apply the midpoint formula.
2 2
a2, b
7
Simplify.
2
The midpoint of the segment is 12, 72 2.
Skill Practice Find the midpoint of the line segment with the given endpoints.
3. (5, 6) and (10, 4) 4. (2.6, 6.3) and (1.2, 4.1)
y
6
5
4 (2, 3)
3
Hiker 1
2
1
x
6 5 4 3 2 1 0 1 2 3 4 5 6
1
2 Ranger
(5, 2) Station
3
Hiker 2
4
5
6
Solution:
To find the halfway point on the line segment between the two hikers, apply the
midpoint formula:
(2, 3) and 15, 22
1x1, y1 2 and 1x2, y2 2
x1 x2 y1 y2
a , b
2 2
2 152 3 122
a , b Apply the midpoint formula.
2 2
3 1
a , b
2 2 Simplify.
The halfway point between the hikers is located at 132, 12 2 or 11.5, 0.52 .
Skill Practice y
(7, 10)
5. Find the center of the circle in the figure,
given that the endpoints of a diameter are
(3, 2) and (7, 10).
(3, 2)
x
Section 2.1 The Rectangular Coordinate System and Midpoint Formula 109
y
Concept 2: Plotting Points 5
4
5. Plot the points on a rectangular coordinate system. 3
a. 12, 12 b. 10, 42
2
1
c. 10, 02 d. 13, 02
x
5 4 3 2 1 1 2 3 4 5
1
2
e. a , b f. 14.1, 2.72
3 7 3
2 3 4
5
y
6. Plot the points on a rectangular coordinate system. 6
5
a. 12, 52 b. a , 0b
5 4
2 3
For Exercises 9–12, give the coordinates of the labeled points, and state the quadrant or axis where the point is
located.
9. y 10. y
5 5
A B
4 4
3
A
3
2 2
C
1 E 1
B
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2 C
D
3 3
E
4 4
5 5 D
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11. y 12. y
5 5
4 4
A A
3 3
2 2
B
1 1 C
B
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 0 1 2 3 4 5
1 1
E 2 2
3 C D D 3
4 4
E
5 5
y
35
30
Body Mass Index
25
20
15
10
5
0 x
142 150 158 166 174 182 190 198 206
Weight (lb)
a. What is the body mass index for a 5¿6– person who weighs 154 lb?
b. What is the weight of a 5¿6– person whose body mass index is 29?
14. The graph shows the number of cases of West Nile virus reported in Colorado during the months of May
through October 2005. The month of May is represented by x 1 on the x-axis. (Source: Centers for Disease
Control.)
40
30
20
10
0 x
0 1 2 3 4 5 6
Month (x 1 corresponds to May)
a. Which month had the greatest number of cases reported? Approximately how many cases were
reported?
b. Which month had the fewest cases reported? Approximately how many cases were reported?
c. Which months had fewer than 10 cases of the virus reported?
d. Approximately how many cases of the virus were reported in August?
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Section 2.1 The Rectangular Coordinate System and Midpoint Formula 111
17. y 18. y
5 5
4 4
3 (2, 2) (1, 3) 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2 (3, 1)
(4, 2) 3 3
4 4
5 5
For Exercises 19–26, find the midpoint of the line segment between the two given points.
19. (4, 0) and (6, 12) 20. (7, 2) and 13, 22 21. 13, 82 and 13, 22
22. 10, 52 and 14, 52 23. 15, 22 and 16, 12 24. 19, 32 and 10, 42
25. 12.4, 3.12 and 11.6, 1.12 26. 10.8, 5.32 and 14.2, 7.12
27. Two courier trucks leave the warehouse to make deliveries. One travels 20 mi north and 30 mi east.
The other truck travels 5 mi south and 50 mi east. If the two drivers want to meet for lunch at a
restaurant at a point halfway between them, where should they meet relative to the warehouse? (Hint: Label
the warehouse as the origin, and find the coordinates of the restaurant. See the figure.)
30 mi
20 mi Restaurant
5 mi
Warehouse
(0, 0) 50 mi
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28. A map of a hiking area is drawn so that the Visitor Center is at the origin of a rectangular grid. Two hikers
are located at positions 11, 12 and 13, 22 with respect to the Visitor Center where all units are in miles. A
campground is located exactly halfway between the hikers. What are the coordinates of the campground?
y
5
4
3
Hiker 1 2 Visitor Center
(1, 1) 1 (0, 0)
x
5 4 3 2 1 0 1 2 3 4 5
1
Hiker 2 2
(3, 2)
3
4
5
29. Find the center of a circle if a diameter of the circle has endpoints 11, 22 and (3, 4).
30. Find the center of a circle if a diameter of the circle has endpoints 13, 32 and 17, 12 .
Solution:
a. 2x 3y 8 The ordered pair 14, 02 indicates that
x 4 and y 0.
?
2142 3102 8 Substitute x 4 and y 0 into the
equation.
8 0 8 ✔ (true) The ordered pair 14, 02 makes the
equation a true statement. The ordered
pair is a solution to the equation.
b8
10 ?
2112 3a Substitute x 1 and y 103.
3
Skill Practice Determine whether each ordered pair is a solution for the
equation x 4y 8.
1a. 12, 12 b. 14, 32 c. 114, 1.52
Solution Check
(x, y) x y3
(3, 0) 3 03✔
(4, 1) 4 13✔
(0, 3) 0 (3) 3 ✔
Solution:
We will find three ordered pairs that are solutions to the equation. In the table,
we have selected arbitrary values for x or y and must complete the ordered
pairs.
x y
0 (0, )
2 ( , 2)
5 (5, )
From the first row, From the second row, From the third row,
substitute x 0. substitute y 2. substitute x 5.
3x 5y 15 3x 5y 15 3x 5y 15
3102 5y 15 3x 5122 15 3152 5y 15
5y 15 3x 10 15 15 5y 15
y3 3x 5 5y 0
5
x y0
3
The completed list of ordered pairs is shown as follows. To graph the equation,
plot the three solutions and draw the line through the points (Figure 2-7). Arrows
on the ends of the line indicate that points on the line extend infinitely in both
directions.
y
5
4
x y (0, 3)
3
3x 5y 15 ( 53 , 2)
0 3 (0, 3) 2
(5, 0)
Q3 , 2 R
5 5 1
3 2 x
5 4 3 2 1 1 2 3 4 5
5 0 (5, 0) 1
2
3
4
5
Figure 2-7
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Skill Practice
2. Given 2x y 1, complete the table and graph the line through the points.
x y
0
5
1
Solution:
Because the y-variable is isolated in the equation, it is easy to substitute a value
for x and simplify the right-hand side to find y. Since any number for x can be
used, choose numbers that are multiples of 2 that will simplify easily when mul-
tiplied by 12.
y
x y
5
0 2 10, 22 4
12, 12
3 1
2 1 2
y 2x 2
Figure 2-9.)
(a, 0)
x
A y-intercept is a point 10, b2 where a graph intersects the y-axis. (see
Figure 2-9.)
*In some applications, an x-intercept is defined as the x-coordinate of a point of intersection
that a graph makes with the x-axis. For example, if an x-intercept is at the point (3, 0), it is sometimes
Figure 2-9 stated simply as 3 (the y-coordinate is understood to be zero). Similarly, a y-intercept is sometimes
defined as the y-coordinate of a point of intersection that a graph makes with the y-axis. For example,
if a y-intercept is at the point (0, 7), it may be stated simply as 7 (the x-coordinate is understood to
be zero).
To find the x- and y-intercepts from an equation in x and y, follow these steps:
Solution:
To find the x-intercept, substitute To find the y-intercept, substitute
y 0. x 0.
2x 4y 8 2x 4y 8
2x 4102 8 2102 4y 8
2x 8 4y 8
x4 y2
The x-intercept is (4, 0). The y-intercept is (0, 2).
In this case, the intercepts are two distinct points and may be used to graph
the line. A third point can be found to verify that the points all fall on the
same line (points that lie on the same line are said to be collinear). Choose a
different value for either x or y, such as y 4.
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2x 4y 8 y
5
2x 4142 8 Substitute y 4. (4, 4) 4
3
(0, 2)
2x 16 8 Solve for x. 2
1 (4, 0)
2x 8 5 4 3 2 1 1 2 3 4 5
x
1
2x 4y 8
x 4 The point (4, 4) lines up 2
3
with the other two points 4
(Figure 2-10). 5
Figure 2-10
Skill Practice
4. Given y 2x 4, find the x- and y-intercepts. Then graph the line.
Solution:
To find the x-intercept, substitute To find the y-intercept, substitute
y 0. x 0.
1 1
y x y x
4 4
102 x 102
1 1
y
4 4
0x y0
The x-intercept is (0, 0). The y-intercept is (0, 0).
Notice that the x- and y-intercepts are both located at the origin (0, 0). In this
case, the intercepts do not yield two distinct points. Therefore, another point
is necessary to draw the line. We may pick any value for either x or y. How-
ever, for this equation, it would be particularly convenient to pick a value for
x that is a multiple of 4 such as x 4. Skill Practice Answers
4. y
1 y 5
y x 4
4 5 y 2x 4
3
4
2
142
1 3 1
y 4x
y Substitute x 4. 2 (4, 1)
1 (2, 0)
4 1 5 4 3 2 1 1 2 3 4 5
x
1
x
y1 5 4 3 2 1
1
1 2 3 4 5 2
(0, 0) 3
2
The point (4, 1) is a solution to the equation 4 (0, 4)
3
5
(Figure 2-11). 4
y
5 5.
5
Figure 2-11
4
3
Skill Practice 2 y 5x
(0, 0) 1
5. Given y 5x, find the x- and y-intercepts. Then graph the line. 5 4 3 2 1 1 2 3 4 5
x
1
2
3
4
5
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Solution:
a. To find the x-intercept, substitute To find the y-intercept, substitute
y 0. x 0.
0 65,000 13,000x y 65,000 13,000102
13,000x 65,000 y 65,000
x5
The x-intercept is (5, 0). The y-intercept is (0, 65,000).
Taxable Value of a Truck Versus
y the Age of the Vehicle
70,000
Taxable Value ($)
60,000
50,000
40,000
30,000
20,000
10,000
0 x
0 1 2 3 4 5 6
Age (years)
b. The x-intercept (5, 0) indicates that when the truck is 5 years old, the
taxable value of the truck will be $0.
c. The y-intercept (0, 65,000) indicates that when the truck was new (0 years
old), its taxable value was $65,000.
Solution:
Because this equation is in the form x k, the line is vertical and must cross the
x-axis at x 6. We can also construct a table of solutions to the equation x 6.
The choice for the x-coordinate must be 6, but y can be any real number
(Figure 2-12).
y
x6
8
6
x y 4
6 –8 2
x
6 1 8 6 4 2 2 4 6 8
2
6 4 4
6
6 8 8
Figure 2-12
Skill Practice
7. Graph the line x 4.
Solution:
Skill Practice Answers
The equation 4y 7 is equivalent to y 74. Because the line is in the form
7. y
y k, the line must be horizontal and must pass through the y-axis at y 74 5
(Figure 2-13). 4
x 4
3
2
1
x
5 4 3 2 1 1 2 3 4 5
1
2
3
4
5
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We can also construct a table of solutions to the equation 4y 7. The choice
for the y-coordinate must be 74 , but x can be any real number.
y
4
3
x y 2
1 y 74
0 74 x
4 3 2 1 1 2 3 4
1
3 74
2
2 74 3
4
Figure 2-13
Skill Practice
8. Graph the line –2y 9.
Calculator Connections
A viewing window of a graphing calculator shows a portion of a rectangular
coordinate system. The standard viewing window for most calculators shows
both the x- and y-axes between 10 and 10. Furthermore, the scale defined
by the tick marks on both axes is usually set to 1.
To see the x- and y-intercepts of this line, we can change the viewing win-
dow to accommodate larger values of x and y. Most calculators have a Range
or Window feature that enables the user to change the minimum and maxi-
mum x- and y-values. In this case, we changed the values of x to range
between 5 and 20, and the values of y to range between 10 and 20.
Review Exercises
3. Plot each point on a rectangular coordinate system, and identify the quadrant or axis where it is located.
a. A12, 32 b. B11, 12 c. C(4, 2) d. D10, 42
y
5
4
3
2
1
x
5 4 3 2 1 1 2 3 4 5
1
2
3
4
5
For Exercises 4–6, find the midpoint of the line segment between the given points. Check your answer by
graphing the line segment and midpoint.
4. 13, 12 and 115, 12 5. (7, 8) and 14, 12 6. 12, 102 and 12, 02
1 2 1 2
3 3
4 4
5 5
y y
1 1
13. y x 5 14. y x 5
5 4 3 4
3 3
2 x y 2
x y 1 1
0
x x
0 5 4 3 2 1 1 2 3 4 5 43 2 1 1 2 3 4 5 6
1 3 1
5 2 2
3
6 3
5 4 4
5 5
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y y y
7 2 5
6 1 4
5 x 3
8 7 6 5 4 3 2 1 1 2
4 1 2
3 2 1
2 3 x
5 4 3 2 1 1 2 3 4 5
1 4 1
x 5 2
3 2 1 1 2 3 4 5 6 7
1 6 3
2 7 4
3 8 5
y y y
6 7 5
5 6 4
4 5 3
3 4 2
2 3 1
1 2 x
5 4 3 2 1 1 2 3 4 5
x 1 1
5 4 3 2 1 1 2 3 4 5
1 x 2
5 4 3 2 1 1 2 3 4 5
2 1 3
3 2 4
4 3 5
2 5
21. y x 1 22. y x 1 23. x 5y 5
5 3
y y y
5 7 5
4 6 4
3 5 3
2 4 2
1 3 1
x 2 x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 x 2
5 4 3 2 1 1 2 3 4 5
3 1 3
4 2 4
5 3 5
y y y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
x x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
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y y
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
30. Can the point (4, 1) be an x- or y-intercept? Why or why not?
For Exercises 31–42, a. find the x-intercept, b. find the y-intercept, and c. graph the line.
31. 2x 3y 18 32. 2x 5y 10 33. x 2y 4
y y y
8 5
5
7 4 4
6 3 3
5 2 2
4 1 1
3 x x
3 2 1 1 2 3 4 5 6 7 5 4 3 2 1 1 2 3 4 5
2 1 1
1 2 2
x 3
1 1 2 3 4 5 6 7 8 9 3
1 4 4
2 5 5
y y y
9 5 5
8 4 4
7 3 3
6 2 2
5 1 1
4 x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
3 1 1
2 2 2
1 3 3
x 4 4
1 1 2 3 4 5 6 7 8 9
1 5 5
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4
37. y 2x 4 38. y 3x 1 39. y x 2
3
y y y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
x x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
2 1 2
40. y x 1 41. x y 42. x y
5 4 3
y y y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
x x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
43. A salesperson makes a base salary of $10,000 a year plus a 5% commission on the total sales for the year.
The yearly salary can be expressed as a linear equation as
y 10,000 0.05x
where y represents the yearly salary and x represents the total yearly sales.
40,000
($)
20,000
0 x
0 200,000 400,000 600,000 800,000
Total Yearly Sales ($)
a. What is the salesperson’s salary for a year in which his sales total $500,000?
b. What is the salesperson’s salary for a year in which his sales total $300,000?
c. What does the y-intercept mean in the context of this problem?
d. Why is it unreasonable to use negative values for x in this equation?
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44. A taxi company in Miami charges $2.00 for any distance up y Cost of Cab Ride Versus Number of Miles
to the first mile and $1.10 for every mile thereafter. The cost 6.00
of a cab ride can be modeled graphically. 5.00
4.00
Cost ($)
a. Explain why the first part of the model is represented by a 3.00
horizontal line. 2.00
b. What does the y-intercept mean in the context of this 1.00
problem? 0 x
0 1 2 3 4
Number of Miles
c. Explain why the line representing the cost of traveling
more than 1 mi is not horizontal.
d. How much would it cost to take a cab 312 mi?
y y y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
x x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
y y y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
x x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
y y
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
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58. 3y 9 59. 2y 10 0
For Exercises 60–63, use a graphing calculator to graph the lines on the suggested viewing window.
1 1
60. y x 10 61. y x 12
2 3
30 x 10 10 x 40
15 y 5 10 y 20
For Exercises 64–65, graph the lines in parts (a)–(c) on the same viewing window. Compare the graphs. Are the
lines exactly the same?
64. a. y x 3 65. a. y 2x 1
b. y x 3.1 b. y 1.9x 1
c. y x 2.9 c. y 2.1x 1
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3 ft 3 ft
18 ft
4 ft
Figure 2-14
To measure the slope of a line quantitatively, consider two points on the line. The
slope is the ratio of the vertical change between the two points to the horizontal
change. That is, the slope is the ratio of the change in y to the change in x. As a
memory device, we might think of the slope of a line as “rise over run.”
Change in x (run)
change in y rise Change in y
Slope (rise)
change in x run
To move from point A to point B on the stairs, rise 3 ft and move to the right
4 ft (Figure 2-15).
B
3-ft
change in y
change in y 3 ft 3
Slope
A change in x 4 ft 4
4-ft
change in x
Figure 2-15
To move from point A to point B on the wheelchair ramp, rise 3 ft and move
to the right 18 ft (Figure 2-16).
B
3-ft
change in y
A
18-ft
change in x
Figure 2-16
change in y 3 ft 1
Slope
change in x 18 ft 6
3
The slope of the stairs is 4 which is greater than the slope of the ramp, which
is 16.
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Solution:
change in y
Slope
change in x
15 ft
15 ft
5 ft
3
or 3 5 ft
1
The slope is 31 which indicates that a person climbs 3 ft for every 1 ft traveled
horizontally.
Skill Practice
1. Find the slope of the roof.
8 ft
20 ft
y
(x2, y2)
Change in y
y2 y1
x
(x1, y1)
x2 x1
Change in x
The slope of a line is often symbolized by the letter m and is given by the
following formula.
Solution:
To use the slope formula, first label the coordinates of each point, and then
substitute their values into the slope formula.
11, 12 and 17, 22
1x1, y1 2 1x2, y2 2 Label the points.
y2 y1 2 112
m Apply the slope formula.
x2 x1 71
3
Simplify.
6
1
2
The slope of the line can be verified from the graph (Figure 2-17).
y
5
4
(7, 2)
3
6
2
1 3
x
2 1 1 2 3 4 5 6 7 8 9 10
1
(1, 1)
2
3
4
5
Figure 2-17
TIP: The slope formula does not depend on which point is labeled (x1, y1)
and which point is labeled (x2, y2). For example, reversing the order in which
the points are labeled in Example 2 results in the same slope:
11, 12 and 17, 22
1x2, y2 2 1x1, y1 2
1 2 3 1
then m
17 6 2
Skill Practice
2. Find the slope of the line that passes through the points (–4, 5) and (6, 8).
When you apply the slope formula, you will see that the slope of a line
may be positive, negative, zero, or undefined.
• Lines that “increase,” or “rise,” from left to right have a positive slope.
• Lines that “decrease,” or “fall,” from left to right have a negative slope.
• Horizontal lines have a zero slope.
• Vertical lines have an undefined slope.
Solution:
13, 42 and 15, 12
1x1, y1 2 1x2, y2 2 Label points.
y2 y1 1 142
m Apply the slope formula.
x2 x1 5 3
3 3
Simplify.
8 8
The two points can be graphed to verify that 38 is the correct slope
(Figure 2-18).
y
5
4
3
2
1
x
5 4 3 2 8 1 2 3 4 5
1
(5, 1) 2
3
3
4
5 (3, 4)
The line slopes m 38
downward from left to right.
Figure 2-18
Skill Practice
3. Find the slope of the line that passes through the given points.
(1, –8) and (–5, –4)
Solution:
a. 13, 42 13, 22
y
and
5
(3, 4)
3
1x1, y1 2 1x2, y2 2 Label points.
2
y2 y1 2 4
m
3 132
1 Apply slope formula.
x x2 x1
5 4 3 2 1 1 2 3 4 5
1
(3, 2) 2 6
3
4
3 3
5
6
Figure 2-19 Undefined
0
The slope is undefined. The points define a vertical line (Figure 2-19).
Skill Practice Find the slope of the line that passes through the given points.
4. (5, –2) and (5, 5) 5. (1, 6) and (–7, 6)
3 ft
3 ft
Figure 2-21 Figure 2-22
Solution:
a. The slope of a line parallel to the given line is m 5 (same slope).
b. The slope of a line perpendicular to the given line is m 15 (the opposite
of the reciprocal of 5).
Skill Practice
4
6. The slope of line L1 is .
3
a. Find the slope of a line parallel to L1.
b. Find the slope of a line perpendicular to L1.
Solution:
First determine the slope of each line. Then compare the values of the slopes
to determine if the lines are parallel or perpendicular.
For line 1: For line 2:
L1: 12, 32 and (4, 1) L2: 15, 62 and 13, 22
1x1, y1 2 1x2, y2 2 1x1, y1 2 1x2, y2 2 Label the points.
1 132 2 162
m m
3 152
Apply the slope
42
formula.
4 4
2 8
1
2
2
The slope of L1 is 2. The slope of L2 is 12. The slope of L1 is the opposite of the
reciprocal of L2. By comparing the slopes, the lines must be perpendicular.
Skill Practice
7. Two points are given for lines L1 and L2. Determine if the lines are parallel,
perpendicular, or neither.
L1: (4, 1) and (3, 6)
L2: (1, 3) and (2, 0)
60
50
40
(1970, 43.0)
30
20
10
0
1965 1970 1975 1980 1985 1990 1995 2000 2005
Figure 2-23
Source: Current population survey.
a. Find the slope of the line, using the points (1970, 43.0) and (2005, 65.4).
b. Interpret the meaning of the slope in the context of this problem.
Solution:
a. 11970, 43.02 and 12005, 65.42
1x1, y1 2 1x2, y2 2 Label the points.
y2 y1 65.4 43.0
m Apply the slope formula.
x2 x1 2005 1970
22.4
m or m 0.64
35
b. The slope is approximately 0.64, meaning that the full-time workforce has
increased by approximately 0.64 million men (or 640,000 men) per year
between 1970 and 2005.
Skill Practice The number of people per square mile in Alaska was 0.96 in
1990. This number increased to 1.17 in 2005.
8a. Find the slope of the line that represents the population growth of Alaska.
Use the points (1990, 0.96) and (2005, 1.17). Skill Practice Answers
8a. 0.014
b. Interpret the meaning of the slope in the context of this problem. b. The population increased by 0.014
person per square mile per year.
Review Exercises
3. Find the missing coordinate so that the ordered pairs are solutions to the equation 12x y 4.
a. 1 0, 2 b. 1 , 0 2 c. 14, 2
For Exercises 4–7, find the x- and y-intercepts (if possible) for each equation, and sketch the graph.
4. 2x 8 0 5. 4 2y 0
y
y
5
5
4
4
3
3
2
2
1
1
x
6 5 4 3 2 1 1 2 3 4 x
1 5 4 3 2 1 1 2 3 4 5
1
2
2
3
3
4
4
5
5
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1
6. 2x 2y 6 0 7. x y 6
3
y y
5 21
4 18
3 15
2 12
1 9
x 6
5 4 3 2 1 1 2 3 4 5
1 3
2 x
9 6 3 3 6 9 12 15 18 21
3 3
4 6
5 9
19 ft
15 ft
24 ft
20 ft
7 ft
10. Find the slope of the treadmill. 11. Find the average slope of the hill.
150 yd
500 yd
8 in.
72 in.
12. The road sign shown in the figure indicates the percent grade of
a hill. This gives the slope of the road as the change in elevation per
100 horizontal ft. Given a 4% grade, write this as a slope in
fractional form.
13. If a plane gains 1000 ft in altitude over a distance of 12,000 horizontal ft,
what is the slope? Explain what this value means in the context of the 4% Grade
problem.
17. (4, 5) and 11, 02 18. 12, 52 and 17, 12 19. 14, 22 and 13, 12
20. 10.3, 1.12 and 10.1, 0.82 21. 10.4, 0.22 and 10.3, 0.12 22. (2, 3) and (2, 7)
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23. 11, 52 and 11, 02 24. 15, 12 and 13, 12 25. 18, 42 and (1, 4)
26. 14.6, 4.12 and (0, 6.4) 27. (1.1, 4) and 13.2, 0.32 28. a , b and a , 1b
3 4 7
2 3 2
29. a , b and a , b
2 1 1 3
3 2 6 2
30. Explain how to use the graph of a line to determine whether the slope of a line is positive, negative, zero,
or undefined.
31. If the slope of a line is 43, how many units of change in y will be produced by 6 units of change in x?
For Exercises 32–37, estimate the slope of the line from its graph.
y y
32. 33.
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
y y
34. 35.
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
y y
36. 37.
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
2
42. m 43. m 0 44. m is undefined.
11
In Exercises 45–52, two points are given from each of two lines L1 and L2. Without graphing the points,
determine if the lines are perpendicular, parallel, or neither.
45. L1: 12, 52 and 14, 92 46. L1: 13, 52 and 11, 22 47. L1: 14, 22 and 13, 12
L2: 11, 42 and 13, 22 L2: 10, 42 and 17, 22 L2: 15, 12 and 110, 162
48. L1: 10, 02 and 12, 32 49. L1: 15, 32 and 15, 92 50. L1: 13, 52 and 12, 52
L2: 12, 52 and 10, 22 L2: 14, 22 and 10, 22 L2: 12, 42 and 10, 42
51. L1: 13, 22 and 12, 32 52. L1: 17, 12 and 10, 02
L2: 14, 12 and 10, 52 L2: 110, 82 and 14, 62
150
a. Use the coordinates of the given points to find the slope of 100
the line, and express the answer in decimal form.
50 (1998, 70)
b. Interpret the meaning of the slope in the context of this 0
problem. 1996 1998 2000 2002 2004 2006
54. The number of SUVs (in millions) sold in the United States Number of SUVs Sold in the
grew approximately linearly between 1990 and 2002. United States
400
(2000, 341)
a. Find the slope of the line defined by the two given points. 350
300
b. Interpret the meaning of the slope in the context of this
Millions
250
problem. 200
150
(1994, 155)
100
50
0
1990 1992 1994 1996 1998 2000 2002 2004
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55. The data in the graph show the average weight for boys Average Weight for Boys by Age
based on age. 100
80
a. Use the coordinates of the given points to find the
Weight (lb)
60 (10, 74.5)
slope of the line. (5, 44.5)
40
b. Interpret the meaning of the slope in the context of
this problem. 20
0
0 1 2 3 4 5 6 7 8 9 10 11 12
Age (yr)
56. The data in the graph show the average weight for girls based on age.
a. Use the coordinates of the given points to find the slope of the line, and write the answer in decimal form.
b. Interpret the meaning of the slope in the context of this problem.
70
60
50
40 (5, 42.5)
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11 12
Age (yr)
2 4
60. P12, 42 and m 0 61. P11, 22 and m 62. P11, 42 and m
3 5
y y y
7 5 5
6 4 4
5 3 3
4 2 2
3 1 1
2 x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
x 2 2
5 4 3 2 1 1 2 3 4 5
1 3 3
2 4 4
3 5 5
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Solution:
Write the equation in slope-intercept form, y mx b, by solving for y.
3x 4y 4
4y 3x 4
4y 3x 4
4 4 4
3 3
y x1 The slope is and the y-intercept is (0, 1).
4 4
Skill Practice
Skill Practice Answers 1. Write the equation in slope-intercept form. Determine the slope and the
1 3 y-intercept.
1. y x
2 4
2x 4y 3
Slope: ; y-intercept: a0, b
1 3
2 4
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To find a second point on the line, start at the y-intercept and move up
3 units and to the left 4 units. Then draw the line through the two points (see
Figure 2-24).
Skill Practice
2. Graph the line y 15 x 2 by using the slope and y-intercept.
Two lines are parallel if they have the same slope and different y-intercepts.
Two lines are perpendicular if the slope of one line is the opposite of the recip-
rocal of the slope of the other line. Otherwise, the lines are neither parallel nor
perpendicular.
Solution:
a. The equations are written in slope-intercept form.
L1: y 2x 7 The slope is 2 and the y-intercept is (0, 7).
L2: y 2x 1 The slope is 2 and the y-intercept is 10, 12.
Because the slopes are the same and the y-intercepts are different, the
lines are parallel.
b. Write each equation in slope-intercept form by solving for y.
L1: 2y 3x 2 L2: 4x 6y 12
2y 3x 2 Add 4x to
Divide by 2. 6y 4x 12
2 2 2 both sides.
3 6y 4 12
y x1 x Divide by 6.
2 6 6 6
2
y x2
3
3 2
The slope of L1 is . The slope of L2 is .
2 3
The value 32 is the opposite of the reciprocal of 23 . Therefore, the lines
are perpendicular.
c. L1: x y 6 is equivalent to y x 6. The slope is 1.
L2: y 6 is a horizontal line, and the slope is 0.
The slopes are not the same. Therefore, the lines are not parallel. The slope
of one line is not the opposite of the reciprocal of the other slope.
Therefore, the lines are not perpendicular. The lines are neither parallel
nor perpendicular.
Skill Practice Given the pair of equations, determine if the lines are parallel,
perpendicular, or neither.
3
3. y x 1 4. 3x y 4 5. xy7
4 6x 6 2y x1
4
y x3
3
Solution:
To find an equation of a line in slope-intercept form, y mx b, it is necessary
to find the slope, m, and the y-intercept, b. The slope is given in the problem as
m 3. Therefore, the slope-intercept form becomes
y mx b
Skill Practice Answers
3. Perpendicular 4. Parallel
5. Neither
y 3x b
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Furthermore, because the point (1, 4) is on the line, it is a solution to the equa-
tion. Therefore, if we substitute (1, 4) for x and y in the equation, we can solve
for b.
4 3112 b
4 3 b
1 b
Skill Practice
6. Use slope-intercept form to find an equation of a line with slope 2 and
passing through 13, 52 .
y
TIP: We can check the answer to Example 4, 5
by graphing the line. Notice that the line appears 4
to pass through (1, 4) as desired. 3
2
1
x
5 4 3 2 1 1 2 3 4 5
1
2
3
(1, 4)
4
5
Solution:
m 3 and (x1, y1) (1, 4)
y y1 m1x x1 2
y 142 31x 12 Apply the point-slope formula.
y 4 31x 12 Simplify.
To write the answer in slope-intercept form, clear parentheses and solve for y.
y 4 3x 3 Clear parentheses.
y 3x 1 Solve for y. The answer is written in slope-
intercept form. Notice that this is the same
equation as in Example 4.
Skill Practice
7. Use the point-slope formula to write an equation for a line passing through
the point 12, 62 and with a slope of 5. Write the answer in slope-
intercept form.
Solution:
The slope formula can be used to compute the slope of the line between two
points. Once the slope is known, the point-slope formula can be used to find
TIP: In Example 6, the an equation of the line.
point (3, 1) was used for
(x1, y1) in the point-slope First find the slope.
1 112
formula. However, either
y2 y1 2
point could have been m 1 Hence, m 1.
used. Using the point x2 x1 35 2
(5, 1) for (x1, y1) produces Next, apply the point-slope formula.
y y1 m1x x1 2
the same final equation:
y 112 11x 52
y 1 x 5 y 1 11x 32 Substitute m 1 and use either point for
(x1, y1). We will use (3, 1) for (x1, y1).
y x 4
y 1 x 3 Clear parentheses.
y x 4 Solve for y. The final answer is in slope-intercept
Skill Practice Answers form.
7. y 5x 16
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Skill Practice
8. Use the point-slope formula to write an equation of the line that passes
through the points 15, 22 and 11, 12. Write the answer in slope-intercept
form.
Solution:
To find an equation of a line, we must know a point on the line and the slope.
The known point is 12, 32. Because the line is parallel to 4x y 8, the
two lines must have the same slope. Writing the equation 4x y 8 in slope-
intercept form, we have y 4x 8. Therefore, the slope of both lines must
be 4.
We must now find an equation of the line passing through 12, 32 having
a slope of 4.
y y1 m1x x1 2 Apply the point-slope formula.
y 132 4 3x 122 4 Substitute m 4 and 12, 32 for 1x1, y1 2.
y 3 41x 22
y 3 4x 8 Clear parentheses.
y 4x 11 Write the answer in slope-intercept form.
Skill Practice
We can verify the answer to Example 7 by graphing both lines. We see that the
line y 4x 11 passes through the point 12, 32 and is parallel to the line
y 4x 8. See Figure 2-25.
y
12
10
8
6
y 4x 11
4
2
x
12108 6 4 2 2 4 6 8 10 12
2
(2, 3)
4
y 4x 8
6
8
10
12
Figure 2-25
Skill Practice Answers
3 7
8. y x
4 4
9. y 2x 9
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Solution:
The slope of the given line can be found from its slope-intercept form.
2x 3y 3
3y 2x 3 Solve for y.
3y 2x 3
3 3 3
2 2
y x1 The slope is .
3 3
The slope of a line perpendicular to this line must be the opposite of the
reciprocal of 23; hence, m 32. Using m 32 and the known point (4, 3), we
can apply the point-slope formula to find an equation of the line.
y y1 m1x x1 2 Apply the point-slope formula.
y 3 1x 42
3
Substitute m 32 and (4, 3) for (x1, y1).
2
3
y3 x6 Clear parentheses.
2
3
y x3 Solve for y.
2
Skill Practice
10. Find an equation of the line passing through the point 11,62 and perpendicular
to the line x 2y 8. Write the answer in slope-intercept form.
Calculator Connections
Solution:
Any line perpendicular to the x-axis must be vertical. Recall that all vertical
lines can be written in the form x k, where k is constant. A quick sketch
can help find the value of the constant (Figure 2-26).
Because the line must pass through a point whose x-coordinate is 4, the
equation of the line is x 4.
y
5
4
3
2
(4, 1) 1
x
5 4 3 2 1 1 2 3 4 5
1
2
3
4
5
x 4
Figure 2-26
Skill Practice
11. Write an equation of the line through the point (20, 50) and having a Skill Practice Answers
slope of 0. 11. y 50
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Review Exercises
x y y
3. Given 1 5
2 3 4
3
a. Find the x-intercept. b. Find the y-intercept. c. Sketch the graph. 2
1
4. Using slopes, how do you determine whether two lines are parallel? 5 4 3 2 1 1 2 3 4 5
x
1
2
5. Using the slopes of two lines, how do you determine whether the lines are 3
perpendicular? 4
5
6. Write the formula to find the slope of a line given two points 1x1, y1 2 and 1x2, y2 2 .
1
12. 18 2y 13. 7 y 14. 8x 12y 9 15. 9x 10y 4
2
1 1
21. y x 22. x 2 23. y x 2
2 2
a. y b. y c. y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
x x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
d. y e. y f. y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
x x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
For Exercises 24–31, write the equations in slope-intercept form (if possible). Then graph each line, using the
slope and y-intercept.
24. y 2 4x 25. 3x 5 y 26. 3x 2y 6
y y y
8 7 5
7 6 4
6 5 3
5 4 2
4 3 1
3 2 x
5 4 3 2 1 1 2 3 4 5
2 1 1
1 x 2
5 4 3 2 1 1 2 3 4 5
x 1 3
5 4 3 2 1 1 2 3 4 5
1 2 4
2 3 5
30. 2x 5y 0 31. 3x y 0
y y
5 5
4 4
3 3
2 2
1 1
x x
3 2 1 1 2 3 4 5 6 7 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
32. Given the standard form of a linear equation Ax By C, B 0, solve for y and write the equation in
slope-intercept form. What is the slope of the line? What is the y-intercept?
33. Use the result of Exercise 32 to determine the slope and y-intercept of the line 3x 7y 9.
For Exercises 34–39, determine if the lines are parallel, perpendicular, or neither.
34. 3y 5x 1 35. x 6y 3 36. 3x 4y 12
1 1 2
6x 10y 12 3x y 0 x y1
2 2 3
For Exercises 40–51, use the slope-intercept form of a line to find an equation of the line having the given slope
and passing through the given point.
40. m 2, 14, 32 41. m 3, 11, 52 42. m 4, 11, 22
53. The line passes through the point (0, 5) and has a slope of 12.
54. The line passes through the point (2, 7) and has a slope of 2.
55. The line passes through the point (3, 10) and has a slope of 2.
56. The line passes through the point 12, 52 and has a slope of 3.
57. The line passes through the point 11, 62 and has a slope of 4.
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58. The line passes through the point 16, 32 and 59. The line passes through the point 17, 22
has a slope of 45. and has a slope of 72.
60. The line passes through (0, 4) and (3, 0). 61. The line passes through (1, 1) and (3, 7).
62. The line passes through (6, 12) and (4, 10). 63. The line passes through 12, 12 and 13, 42.
64. The line passes through 15, 22 and 11, 22. 65. The line passes through 14, 12 and 12, 12.
66. The line contains the point (3, 2) and is parallel to a line with a slope of 34.
67. The line contains the point 11, 42 and is parallel to a line with a slope of 12.
68. The line contains the point (3, 2) and is perpendicular to a line with a slope of 34.
69. The line contains the point 12, 52 and is perpendicular to a line with a slope of 12.
70. The line contains the point 12, 52 and is parallel to y 34x 74.
71. The line contains the point 16, 12 and is parallel to y 23x 4.
72. The line contains the point 18, 12 and is parallel to x 5y 8.
73. The line contains the point 14, 22 and is parallel to 3x 4y 8.
74. The line contains the point (4, 0) and is parallel to the line defined by 3x 2y.
75. The line contains the point 13, 02 and is parallel to the line defined by 5x 6y.
76. The line is perpendicular to the line defined by 3y 2x 21 and passes through the point (2, 4).
77. The line is perpendicular to 7y x 21 and passes through the point 114, 82.
78. The line is perpendicular to 12y x and passes through 13, 52.
79. The line is perpendicular to 14y x and passes through 11, 52.
82. The line contains the point 12, 32 and has an 83. The line contains the point 1 52, 02 and has a
undefined slope. zero slope.
84. The line is parallel to the x-axis and passes 85. The line is perpendicular to the x-axis and passes
through (4, 5). through (4, 5).
86. The line is parallel to the line x 4 and passes 87. The line is parallel to the line y 2 and passes
through (5, 1). through 13, 42.
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89. Is the equation x 1 in slope-intercept form? Identify the slope and y-intercept.
90. Is the equation y 3 in slope-intercept form? Identify the slope and the y-intercept.
91. Is the equation y 5 in slope-intercept form? Identify the slope and the y-intercept.
94. Use a graphing calculator to graph the lines on 95. Use a graphing calculator to graph the lines
the same viewing window. Then explain how the on the same viewing window. Then explain how
lines are related. the lines are related.
y1 x 2 y1 2x 1
y2 2x 2 y2 3x 1
y3 3x 2 y3 4x 1
96. Use a graphing calculator to graph the lines on 97. Use a graphing calculator to graph the lines on a
a square viewing window. Then explain how square viewing window. Then explain how the lines
the lines are related. are related.
y1 4x 1 1
y1 x 3
2
1 y2 2x 3
y2 x 1
4
98. Use a graphing calculator to graph the equation 99. Use a graphing calculator to graph the equation
from Exercise 60. Use an Eval feature to verify from Exercise 61. Use an Eval feature to verify
that the line passes through the points (0, 4) that the line passes through the points (1, 1)
and (3, 0). and (3, 7).
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Solution:
a. The constant or base amount of snow before the storm began is 24 in. The
variable amount is given by 58 in. of snow per hour. If m is replaced by 58 and
b is replaced by 24, we have the linear equation
y mx b
5
y x 24
8
b. The equation is in slope-intercept form, and the corresponding graph is
shown in Figure 2-27.
35
30
25 5 in.
20 8 hr
15
10
5
0 x
0 2 4 6 8 10 12 14 16 18 20
Time (hr)
Figure 2-27
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5
c. y x 24
8
182 24
5
y Substitute x 8.
8
y 5 24 Solve for y.
y 29 in.
The snow depth was 29 in. after 8 hr. The corresponding ordered pair is
(8, 29) and can be confirmed from the graph.
5
d. y x 24
8
5
31.5 x 24 Substitute y 31.5.
8
8131.52 8 a x 24b
5
Multiply by 8 to clear fractions.
8
252 5x 192 Clear parentheses.
60 5x Solve for x.
12 x
The storm lasted for 12 hr. The corresponding ordered pair is (12, 31.5)
and can be confirmed from the graph.
Skill Practice
1. When Joe graduated from college, he had $1000 in his savings account.
When he began working, he decided he would add $120 per month to his
savings account.
a. Write a linear equation to compute the amount of money y in Joe’s
account after x months of saving.
b. Use the equation to compute the amount of money in Joe’s account
after 6 months.
c. Joe needs $3160 for a down payment for a car. How long will it take for
Joe’s account to reach this amount?
5
4
3 y 0.10x 0.82
2
1
0 x
0 5 10 15 20 25 30 35 40 45 50
Year (x 0 corresponds to 1960)
Figure 2-28
Solution:
a. The equation y 0.10x 0.82 is written in slope-intercept form. The slope
is 0.10 and indicates that minimum hourly wage rose an average of $0.10
per year between 1960 and 2005.
b. The y-intercept is (0, 0.82). The y-intercept indicates that the minimum
wage in the year 1960 1x 02 was approximately $0.82 per hour. (The
actual value of minimum wage in 1960 was $1.00 per hour.)
c. The year 1985 is 25 years after the year 1960. Substitute x 25 into the
linear equation.
y 0.10x 0.82
y 0.101252 0.82 Substitute x 25.
y 2.50 0.82
y 3.32
According to the linear model, the minimum wage in 1985 was approximately
$3.32 per hour. (The actual minimum wage in 1985 was $3.35 per hour.)
d. The year 2010 is 50 years after the year 1960. Substitute x 50 into the
linear equation.
y 0.10x 0.82
y 0.101502 0.82 Substitute x 50.
y 5.82
According to the linear model, minimum wage in 2010 will be approximately
$5.82 per hour provided the linear trend continues. (How does this com-
pare with the current value for minimum wage?)
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Skill Practice
2. The cost of long-distance service with a certain phone company is given by
the equation y 0.12x 6.95, where y represents the monthly cost in
dollars and x represents the number of minutes of long distance.
a. Find the slope of the line, and interpret the meaning of the slope in the
context of this problem.
b. Find the y-intercept and interpret the meaning of the y-intercept in the
context of this problem.
c. Use the equation to determine the cost of using 45 min of long-distance
service in a month.
50
40
30
20
10
0 x
0 10 20 30 40 50 60 70 80 90 100 110
Year (x 0 corresponds to 1900)
Figure 2-29
In 1924, the winning time was 72.4 sec. This corresponds to the ordered pair
(24, 72.4). In 1972, the winning time was 58.6 sec, yielding the ordered pair
(72, 58.6).
Skill Practice Answers a. Use these ordered pairs to find a linear equation to model the winning
2a. The slope is 0.12. This means time versus the year.
that the monthly cost increases by
12 cents per minute. b. What is the slope of the line, and what does it mean in the context of this
b. The y-intercept is (0, 6.95). The problem?
cost of the long-distance service is
$6.95 if 0 min is used.
c. $12.35
*The strength of a linear correlation can be measured mathematically by using techniques
often covered in statistics courses.
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c. Use the linear equation to approximate the winning time for the 1964
Olympics.
d. Would it be practical to use the linear model to predict the winning time
in the year 2048?
Solution:
a. The slope formula can be used to compute the slope of the line between
the two points. (Round the slope to 2 decimal places.)
(24, 72.4) and (72, 58.6)
1x1, y1 2 and 1x2, y2 2
y2 y1 58.6 72.4
m 0.2875 Hence, m L 0.29.
x2 x1 72 24
y y1 m1x x1 2 Apply the point-slope formula,
using m 0.29 and the point
(24, 72.4).
y 72.4 0.291x 242
y 72.4 0.29x 6.96 Clear parentheses.
y 0.29x 6.96 72.4 Solve for y.
y 0.29x 79.36 The answer is in slope-intercept
form.
b. The slope is 0.29 and indicates that the winning time in the women’s
100-m Olympic freestyle event has decreased on average by 0.29 sec/yr
during this period.
c. The year 1964 is 64 years after the year 1900. Substitute x 64 into the
linear model.
y 0.29x 79.36
y 0.291642 79.36 Substitute x 64.
y 18.56 79.36
y 60.8
According to the linear model, the winning time in 1964 was approxi-
mately 60.8 sec. (The actual winning time in 1964 was set by Dawn Fraser
from Australia in 59.5 sec. The linear equation can only be used to
approximate the winning time.)
d. It would not be practical to use the linear model y 0.29x 79.36 to
predict the winning time in the year 2048. There is no guarantee that the
linear trend will continue beyond the last observed data point in 2004. In
fact, the linear trend cannot continue indefinitely; otherwise, the swimmers’
times would eventually be negative. The potential for error increases for
predictions made beyond the last observed data value.
Skill Practice
Cost of Textbook versus Number
3. The figure shows data relating the cost of college y of Pages
textbooks in dollars to the number of pages in 120
(400, 107)
100
the book. Let y represent the cost of the book,
Cost ($)
80
(200, 57)
and let x represent the number of pages. 60
40
a. Use the ordered pairs indicated in the figure 20
to write a linear equation to model the cost of 0 x Skill Practice Answers
0 100 200 300 400 500
textbooks versus the number of pages. Number of Pages 3a. y 0.25x 7
b. $97
b.
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Review Exercises
For Exercises 2–5,
a. Find the slope (if possible) of the line passing through the two points.
b. Find an equation of the line passing through the two points. Write the answer in slope-intercept form (if
possible) and in standard form.
c. Graph the line by using the slope and y-intercept. Verify that the line passes through the two given points.
y y
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
y y
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
y
b. Graph the equation. 60
50
40
Cost ($)
30
20
10
x
20 40 60 80 100 120 140 160 180 200
Miles
c. What is the y-intercept and what does it mean in the context of this problem?
d. Using the equation from part (a), find the cost of driving the rental car 50, 100, and 200 mi.
e. Find the total cost of driving the rental car 100 mi if the sales tax is 6%.
f. Is it reasonable to use negative values for x in the equation? Why or why not?
7. Alex is a sales representative and earns a base salary of $1000 per month plus a 4% commission on his
sales for the month.
a. Write a linear equation that expresses Alex’s monthly salary y in terms of his sales x.
y
b. Graph the equation. 3000
2500
Salary ($)
2000
1500
1000
500
x
10,000 20,000 30,000 40,000 50,000
Sales ($)
c. What is the y-intercept and what does it represent in the context of this problem?
d. What is the slope of the line and what does it represent in the context of this problem?
e. How much will Alex make if his sales for a given month are $30,000?
8. Ava recently purchased a home in Crescent Beach, Florida. Her property taxes
for the first year are $2742. Ava estimates that her taxes will increase at a rate
of $52 per year.
a. Write an equation to compute Ava’s yearly property taxes. Let y be the
amount she pays in taxes, and let x be the time in years.
b. Graph the line. y
4000
3500
3000
Taxes ($)
2500
2000
1500
1000
500
x
2 4 6 8 10 12 14 16 18 20
Time (Years)
c. What is the slope of this line? What does the slope of the line represent in the context of this problem?
d. What is the y-intercept? What does the y-intercept represent in the context of this problem?
e. What will Ava’s yearly property tax be in 10 years? In 15 years?
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9. Luigi Luna has started a chain of Italian restaurants called Luna Italiano. He has 19 restaurants in various
locations in the northeast United States and Canada. He plans to open five new restaurants per year.
a. Write a linear equation to express the number of restaurants, y, Luigi opens in terms of the time in years, x.
11. The force y (in pounds) required to stretch a particular spring x inches beyond its rest (or “equilibrium”)
position is given by the equation y 2.5x, where 0 x 20.
a. Use the equation to determine the amount of force necessary to stretch the spring 6 in. from its rest
position. How much force is necessary to stretch the spring twice as far?
b. If 45 lb of force is exerted on the spring, how far will the spring be stretched?
12. The figure represents the median cost of new privately owned, one-family houses sold in the midwest from
1980 to 2005.
Median Cost of New One-Family Houses
y Sold in the Midwest, 1980–2005
250
y 5.3x 63.4
200
Price ($1000)
150
100
50
0 x
0 5 10 15 20 25 30
Year (x 0 corresponds to 1980)
Source: U.S. Bureau of the Census and U.S.
Department of Housing and Urban Development.
Let y represent the median cost of a new privately owned, one-family house sold in the midwest. Let x
represent the year, where x 0 corresponds to the year 1980, x 1 represents 1981, and so on. Then the
median cost of new privately owned, one-family houses sold in the midwest can be approximated by the
equation y 5.3x 63.4, where 0 x 25.
a. Use the linear equation to approximate the median cost of new privately owned, one-family houses in the
midwest for the year 2005.
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b. Use the linear equation to approximate the median cost for the year 1988, and compare it with the actual
median cost of $101,600.
c. What is the slope of the line and what does it mean in the context of this problem?
d. What is the y-intercept and what does it mean in the context of this problem?
13. Let y represent the average number of miles driven per year Average Yearly Mileage for Passenger
for passenger cars in the United States between 1980 and y Cars, United States, 1980–2005
2005. Let x represent the year where x 0 corresponds to 14,000
12,000
1980, x 1 corresponds to 1981, and so on. The average
Miles Driven
10,000
yearly mileage for passenger cars can be approximated by y 142x 9060
8,000
the equation y 142x 9060, where 0 x 25. 6,000
4,000
a. Use the linear equation to approximate the average 2,000
yearly mileage for passenger cars in the United States in 0 x
the year 2005. 0 5 10 15 20 25
Year (x 0 corresponds to 1980)
b. Use the linear equation to approximate the average
mileage for the year 1985, and compare it with the actual value of 9700 mi.
c. What is the slope of the line and what does it mean in the context of this problem?
d. What is the y-intercept and what does it mean in the context of this problem?
(0, 3.3)
4
0 x
0 20 40 60 80 100 120
Year (x 0 corresponds to 1900)
a. Let y represent the winning height. Let x represent the year, where x 0 corresponds to the year 1900,
x 4 represents 1904, and so on. Use the ordered pairs given in the graph (0, 3.3) and (96, 5.92) to find a
linear equation to estimate the winning pole vault height versus the year. (Round the slope to three
decimal places.)
b. Use the linear equation from part (a) to approximate the winning vault for the 1920 Olympics.
c. Use the linear equation to approximate the winning vault for 1976.
d. The actual winning vault in 1920 was 4.09 m, and the actual winning vault in 1976 was 5.5 m. Are your
answers from parts (b) and (c) different from these? Why?
e. What is the slope of the line? What does the slope of the line mean in the context of this problem?
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15. The figure represents the winning time for the men’s 100-m freestyle swimming event for selected Olympic
games.
40
(48, 48.7)
30
20
10
0 x
0 10 20 30 40 50 60
Year (x 0 corresponds to 1948)
a. Let y represent the winning time. Let x represent the number of years since 1948 (where x 0
corresponds to the year 1948, x 4 represents 1952, and so on). Use the ordered pairs given in the graph
(0, 57.3) and (48, 48.7) to find a linear equation to estimate the winning time for the men’s 100-m
freestyle versus the year. (Round the slope to 2 decimal places.)
b. Use the linear equation from part (a) to approximate the winning 100-m time for the year 1972, and
compare it with the actual winning time of 51.2 sec.
c. Use the linear equation to approximate the winning time for the year 1988.
d. What is the slope of the line and what does it mean in the context of this problem?
e. Interpret the meaning of the x-intercept of this line in the context of this problem. Explain why the men’s
swimming times will never “reach” the x-intercept. Do you think this linear trend will continue for the
next 50 years, or will the men’s swimming times begin to “level off” at some time in the future? Explain
your answer.
16. At a high school football game in Miami, hot dogs were sold for $1.00 each. At the end of the night, it was
determined that 650 hot dogs were sold. The following week, the price of hot dogs was raised to $1.50, and
this resulted in fewer sales. Only 475 hot dogs were sold.
a. Make a graph with the price of hot dogs on the x-axis and the corresponding sales on the y-axis. Graph
the points (1.00, 650) and (1.50, 475), using suitable scaling on the x- and y-axes.
y
1000
Number of Hot Dogs Sold
900
800
700
600
500
400
300
200
100
x
0.50 1.00 1.50 2.00
Price of Hot Dogs ($)
b. Find an equation of the line through the given points. Write the equation in slope-intercept form.
c. Use the equation from part (b) to predict the number of hot dogs that would sell if the price were
changed to $1.70 per hot dog.
17. At a high school football game, soft drinks were sold for $0.50 each. At the end of the night, it was
determined that 1020 drinks were sold. The following week, the price of drinks was raised to $0.75, and this
resulted in fewer sales. Only 820 drinks were sold.
a. Make a graph with the price of drinks on the x-axis and the corresponding sales per night on the y-axis.
Graph the points (0.50, 1020) and (0.75, 820), using suitable scaling on the x- and y-axes.
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y
1600
b. Find an equation of the line through the given points. Write the equation in slope-intercept form.
c. Use the equation from part (b) to predict the number of drinks that would sell if the price were changed
to $0.85 per drink.
y
90
80
70
Test Score (%)
60
50
40
30
20
10
x
10 20 30 40 50 60 70 80 90 100
Minutes
b. Find a linear equation that relates Loraine’s weekly test score y to the amount of time she studied per
day x. (Hint: Pick two ordered pairs from the observed data, and find an equation of the line through the
points.)
c. How many minutes should Loraine study per day in order to score at least 90% on her weekly
examination? Would the equation used to determine the time Loraine needs to study to get 90% work
for other students? Why or why not?
d. If Loraine is only able to spend 12 hr/day studying her math, predict her test score for that week.
Points are collinear if they lie on the same line. For Exercises 19–22, use the slope formula to determine if the
points are collinear.
19. 13, 42 10, 52 19, 22 20. 14, 32 14, 12 12, 22
21. 10, 22 12, 122 11, 62 22. 12, 22 10, 32 14, 12
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25. Graph the line y 800x 1420 on the viewing window defined by 0 x 1 and 0 y 1600. Use
the Trace key to support your answer to Exercise 17 by showing that the line passes through the points
(0.50, 1020) and (0.75, 820).
26. Graph the line y 350x 1000 on the viewing window defined by 0 x 2 and 0 y 1000. Use
the Trace key to support your answer to Exercise 16 by showing that the line passes through the points
(1.00, 650) and (1.50, 475).
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Summary 165
Chapter 2 SUMMARY
Example 2
The midpoint between two points is found by using the Find the midpoint between (3, 1) and (5, 7).
formula
3 5 1 7
x1 x2 y1 y2 a , b 11, 42
a , b 2 2
2 2
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0 3 5
4
4 0 3
9 2
1 1
4 x
5 4 3 2 1 1 2 3 4 5
1
2
3
4
5
Example 2
Given the equation, 2x 3y 8
An x-intercept is a point (a, 0) where a graph in- x-intercept: 2x 3102 8
tersects the x-axis. To find an x-intercept, substitute 0
2x 8
for y in the equation and solve for x.
x4 14, 02
A y-intercept is a point (0, b) where a graph inter- y-intercept: 2102 3y 8
sects the y-axis. To find a y-intercept, substitute 0 for x
3y 8
in the equation and solve for y.
a0, b
8 8
y
3 3
Summary 167
Zero slope
Undefined slope
Summary 169
(20, 22,346)
15,000
10,000
(5, 11,013)
5,000
0 x
0 5 10 15 20 25
Year (x 0 corresponds to 1980)
• Given two points from the data, use the point-slope Write an equation of the line, using the points
formula to find an equation of the line. (5, 11,013) and (20, 22,346).
22,346 11,013 11,333
Slope: 756
20 5 15
y 11,013 7561x 52
y 11,013 756x 3780
y 756x 7233
• Interpret the meaning of the slope and y-intercept in The slope m 756 indicates that the average in-
the context of the problem. come has increased by $756 per year.
The y-intercept (0, 7233) means that the aver-
age income in 1980 1x 02 was $7233.
• Use the equation to predict values. Predict the average income for 2010 1x 302.
y 7561302 7233
y 29,913
According to this model, the average income in 2010
will be approximately $29,913.
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d. Quadrant I 4 1
x
5 4 3 2 1 1 2 3 4 5
1
e. Quadrant II 2
3
f. Quadrant III
7. 6 x 2
g. Quadrant IV
x y y
2. Find the midpoint of the line segment between 5
the two points (13, 12) and (4, 18). 0 4
1 3
2
3. Find the midpoint of the line segment between 2 1
the two points (1.2, 3.7) and (4.1, 8.3). 5 4 3 2 1 1 2 3 4 5
x
1
2
4. Determine the coordinates of the points labeled 3
in the graph. 4
5
y
5
4
For Exercises 8–11, graph the lines. In each case find
3 at least three points and identify the x- and y-intercepts
F
2
B (if possible).
1
E A
5 4 3 2 1 1 2 3 4 5
x 8. 2x 3y 6
1
2
3 y
D G
4 5
C
5 4
3
2
Section 2.2 1
x
5 4 3 2 1 1 2 3 4 5
1
For Exercises 5–7, complete the table and graph the 2
line defined by the points. 3
4
5. 3x 2y 6 5
x y y 9. 5x 2y 0
5
0 4 y
0 3
5
2
4
1
1 3
x
5 4 3 2 1 1 2 3 4 5 2
1
1
2
x
3 5 4 3 2 1 1 2 3 4 5
1
4
2
5
3
4
5
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10. 2y 6 c. y
5
y 4
3
6
2
5
1
4
x
3 5 4 3 2 1 1 2 3 4 5
1
2
2
1
3
x
5 4 3 2 1 1 2 3 4 5 4
1
5
2
3
4 13. Draw a line with slope 2 (answers may vary).
11. 3x 6 y
5
4
y
3
5 2
4 1
3 x
2 5 4 3 2 1 1 2 3 4 5
1
1
2
x
5 4 3 2 1 1 2 3 4 5 3
1
4
2 5
3
4
5 14. Draw a line with slope 34 (answers may vary).
Section 2.3 5
4
3
12. Find the slope of the line. 2
y 1
a. x
5
5 4 3 2 1 1 2 3 4 5
4 1
3 2
2 3
1 4
x 5
5 4 3 2 1 1 2 3 4 5
1
2
3 For Exercises 15–18, find the slope of the line that
4
5
passes through each pair of points.
y 15. 12, 62, 11, 02 16. 17, 22, 13, 52
b.
5
For Exercises 20–22, the slopes of two lines are For Exercises 26–30, write your answer in slope-
given. Based on the slopes, are the lines parallel, intercept form or in standard form.
perpendicular, or neither? 1
26. Write an equation of the line that has slope 9
1 5 4 and y-intercept (0, 6).
20. m1 , m2 3 21. m1 , m2
3 4 5
27. Write an equation of the line that has slope 23
22. m1 7, m2 7 and x-intercept (3, 0).
23. The graph indicates that the enrollment for a 28. Write an equation of the line that passes
small college has been increasing linearly through the points (8, 1) and (5, 9).
between 1990 and 2005.
29. Write an equation of the line that passes
a. Use the two data points to find the slope of through the point (6, 2) and is perpendicular
the line. to the line y 13x 2.
b. Interpret the meaning of the slope in the
context of this problem. 30. Write an equation of the line that passes
through the point (0, 3) and is parallel to the
line 4x 3y 1.
College Enrollment by Year
3000
Number of Students
2500
31. For each of the given conditions, find an
(2005, 2815)
equation of the line
2000
(1990, 2020)
1500 a. Passing through the point (3, 2) and
1000 parallel to the x-axis.
500
0
b. Passing through the point (3, 2) and
1985 1990 1995 2000 2005 2010 parallel to the y-axis.
24. Find the slope of the stairway pictured c. Passing through the point (3, 2) and
here. having an undefined slope.
d. Passing through the point (3, 2) and
having a zero slope.
48 in.
Section 2.5
33. Keosha loves the beach and decides to spend
the summer selling various ice cream products
Section 2.4 on the beach. From her accounting course, she
25. Write an equation for each of the following. knows that her total cost is calculated as
a. Horizontal line Total cost fixed cost variable cost
b. Point-slope formula She estimates that her fixed cost for the
summer season is $20 per day. She also knows
c. Standard form that each ice cream product costs her $0.25
d. Vertical line from her distributor.
e. Slope-intercept form
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Test 173
a. Write a relationship for the daily cost y in the number of rushing yards gained by the star
terms of the number of ice cream products running back. The table shows the statistics.
sold per day x.
Margin of
b. Graph the equation from part (a) by letting Yards Rushed Victory
the horizontal axis represent the number of
100 20
ice cream products sold per day and letting
the vertical axis represent the daily cost. 60 10
120 24
y
200 50 7
180
160
140 a. Graph the data to determine if a linear
Cost ($)
f. What does the slope of the line represent in b. Find an equation for the line through the
the context of this problem? points (50, 7) and (100, 20).
34. The margin of victory for a certain college c. Based on the equation, what would be the
football team seems to be linearly related to result of the football game if the star
running back did not play?
Chapter 2 Test
1. Given the equation x 23y 6, complete the 2. Determine whether the following statements
ordered pairs and graph the corresponding are true or false and explain your answer.
points. 10, 2 1 , 02 1 , 32
a. The product of the x- and y-coordinates is
y positive only for points in quadrant I.
41
b. The quotient of the x- and y-coordinates is
x
2 1 1 2 3 4 5 6 7 8 negative only for points in quadrant IV.
1
2
3 c. The point (2, 3) is in quadrant III.
4
5 d. The point (0, 0) lies on the x-axis.
6
7 3. Find the midpoint of the line segment between
8
9
the points (21, 15) and (5, 32).
miL2872X_ch02_103-176 09/14/2006 06:44 PM Page 174 IA CONFIRMING PAGES
4. Explain the process for finding the x- and 10. Describe the relationship of the slopes of
y-intercepts.
a. Two parallel lines
b. Two perpendicular lines
For Exercises 5–8, identify the x- and y-intercepts
(if possible) and graph the line.
11. The slope of a line is 7.
5. 6x 8y 24 y
5 a. Find the slope of a line parallel to the
4 given line.
3
2 b. Find the slope of a line perpendicular to the
1
given line.
x
5 4 3 2 1 1 2 3 4 5
1
2 12. Two points are given for each of two lines.
3
Determine if the lines are parallel,
4
5 perpendicular, or neither.
L1: 14, 42 and 11, 62
6. x 4 y
5 L2: 12, 02 and (0, 3)
4
3
2 13. Given the equation 3x 4y 4,
1
x a. Write the line in slope-intercept form.
7 6 5 4 3 2 1 1 2 3
1
2 b. Determine the slope and y-intercept.
3
4 c. Graph the line, using the slope and
5 y-intercept.
y
7. 3x 5y y
5 5
4 4
3 3
2 2
1 1
x x
5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5
1 1
2 2
3 3
4 4
5 5
c. A line perpendicular to the y-axis. What is c. What does the y-intercept mean in the
the slope of such a line? context of this problem?
d. A slanted line that passes through the origin d. How much will Jack earn in a month if he
and has a negative slope sells 17 automobiles?
16. Write an equation of the line that passes 21. The following graph represents the life
through the point 18, 12 2 with slope 2. Write expectancy for females in the United States
the answer in slope-intercept form. born from 1940 through 2005.
17. Write an equation of the line containing the Life Expectancy for Females in the United
points 12, 32 and (4, 0). y States According to Birth Year
90
Life Expectancy
(30, 75)
80
(0, 66)
(years)
and parallel to 6x 3y 1. 70
6. The formula for the volume of a right circular 12. Find an equation for the vertical line that
cylinder is V pr 2h. passes through the point (7, 11).
a. Solve for h.
13. Find an equation for the horizontal line that
b. Find h if a soda can contains 355 cm3 (which passes through the point (19, 20).
is approximately 12 oz) of soda and the
diameter is 6.6 cm. Round the answer to 14. Find an equation of the line passing through
1 decimal place. (1, 4) and parallel to 2x y 6. Write the
answer in slope-intercept form.
7. Solve the inequalities. Write your answers in
interval notation. 15. Find an equation of the line passing through
(1, 4) and perpendicular to y 14x 2. Write
a. 5x 4 21x 12 b. x 4 7 1
the answer in slope-intercept form.
8. Find the slope of the line that passes through
the points 14, 52 and 16, 32.
16. At the movies, Laquita paid for drinks and
popcorn for herself and her two children. She
spent twice as much on popcorn as on drinks. If
9. Find the midpoint of the line segment with
endpoints 12, 32 and 10, 152.
her total bill came to $17.94, how much did she
spend on drinks and how much did she spend
on popcorn?
For Exercises 10–11, a. find the x- and y-intercept,
b. find the slope, and c. graph the line.
17. Three raffle tickets are represented by three
10. 3x 5y 10 consecutive integers. If the sum of the ticket
numbers is 1776, find the three numbers.
y
5
4 18. A chemist mixes a 20% salt solution with a
3 50% salt solution to get 25 L of a 38% salt
2
solution. How much of the 20% solution and
1
x how much of the 50% solution did she use?
5 4 3 2 1 1 2 3 4 5
1
2
19. The yearly rainfall for Seattle, Washington, is
3
4 0.7 in. less than twice the yearly rainfall for
5 Los Angeles, California. If the total yearly
rainfall for the two cities is 50 in., how much
11. 2y 4 10 rain does each city get per year?