7-4
7-4
Applications of Linear Systems
1. Plan
Objectives
1
To write systems of linear
equations
Examples
1
2
3
Real-World Problem Solving
Finding a Break-Even Point
Real-World Problem Solving
What You’ll Learn
GO for Help
Check Skills You’ll Need
Lesson 3-6
1
2
• To write systems of linear
1. Two trains run on parallel tracks. The first train leaves a city hour before
the second train. The first train travels at 55 mi/h. The second train travels
at 65 mi/h. How long does it take for the second train to pass the first train?
2.75 h
2. Carl drives to the beach at an average speed of 50 mi/h. He returns home
on the same road at an average speed of 55 mi/h. The trip home takes
30 min less. What is the distance from his home to the beach? 275 mi
equations
. . . And Why
To find average wind speed
during an airplane flight, as
in Example 3
Math Background
The Japanese mathematician
Seki Kowa (1683) solved
simultaneous linear equations
by using bamboo rods placed in
squares on a table, similar to the
modern way of using matrices
and determinants.
1
Equations
1 1 Writing Systems of Linear
Part
Equations
Below is a summary of the methods you have used to solve systems of equations.
You must choose a method before you solve a word problem.
Key Concepts
Summary
Methods for Solving Systems of Linear Equations
Graphing
Lesson Planning and
Resources
Use graphing for solving systems that are easily graphed. If the
point of intersection does not have integers for coordinates, find
the exact solution by using one of the methods below or by
using a graphing calculator.
Substitution
See p. 372E for a list of the
resources that support this lesson.
Use substitution for solving systems when one variable has a
coefficient of 1 or -1.
Elimination
Use elimination for solving any system.
More Math Background: p. 372D
PowerPoint
1
Bell Ringer Practice
Check Skills You’ll Need
EXAMPLE
Real-World
Problem Solving
Metallurgy A metalworker has some ingots of metal alloy that are 20% copper
and others that are 60% copper. How many kilograms of each type of ingot should
the metalworker combine to create 80 kg of a 52% copper alloy?
For intervention, direct students to:
Equations and Problem Solving
Lesson 3-6: Example 3
Extra Skills and Word
Problem Practice, Ch. 3
Define
Let g = the mass of the 20% alloy.
Let h = the mass of the 60% alloy.
Relate
mass of alloys
mass of copper
Write
g + h = 80
0.2g + 0.6h = 0.52(80)
Solve using substitution.
Real-World
Connection
The melting point of copper is
10838C.
396
Step 1
Choose one of the equations and solve for a variable.
g + h = 80
Subtract h from each side.
Chapter 7 Systems of Equations and Inequalities
Special Needs
Below Level
L1
Encourage students to draw diagrams wherever
possible to describe word problems. Have them label
the diagrams to show what they know and what they
want to find out. Then have them write and solve the
equations.
396
Solve for g.
g = 80 - h
learning style: visual
L2
Before attempting to solve a word problem, students
should list the information given, what they need to
find, and any relationships that may help solve the
problem.
learning style: verbal
2. Teach
Step 2 Find h.
0.2g + 0.6h = 0.52(80)
0.2(80 - h) + 0.6h = 0.52(80)
16 - 0.2h + 0.6h = 0.52(80)
16 + 0.4h = 41.6
Substitute 80 – h for g. Use parentheses.
Guided Instruction
Use the Distributive Property.
Simplify. Then solve for h.
0.4h = 25.6
2
h = 64
Step 3 Find g. Substitute 64 for h in either equation.
g = 80 - 64
g = 16
To make 80 kg of 52% copper alloy, you need 16 kg of 20% copper alloy and 64 kg
of 60% copper alloy.
Quick Check
1 Suppose you combine ingots of 25% copper alloy and 50% copper alloy to create
40 kg of 45% copper alloy. How many kilograms of each do you need?
32 kg 50% alloy; 8 kg 25% alloy
EXAMPLE
Auditory Learners
Encourage students to read the
question quietly to themselves.
Then have students ask themselves
the following questions and write
their answers mathematically.
• What do I know?
• What am I trying to find?
• What can I find using the facts
I have?
PowerPoint
Lose money
Make money
Income
Expenses
y
Dollars
When starting a business, people want to
know the break-even point, the point at
which their income equals their expenses.
The graph at the right shows the
break-even point for one business.
Additional Examples
Break-even
point
x
0 Number of Items Sold
Notice that the values of y on the red line
represent dollars spent on expenses, and
the values of y on the blue line represent dollars received
as income. So y is used to represent both expenses and income.
For: Linear System Activity
Use: Interactive Textbook, 7-4
2
EXAMPLE
Finding a Break-Even Point
Publishing Suppose a model airplane club publishes a newsletter. Expenses are
$.90 for printing and mailing each copy, plus $600 total for research and writing.
The price of the newsletter is $1.50 per copy. How many copies of the newsletter
must the club sell to break even?
Define Let x = the number of copies.
Let y = the amount of dollars of expenses or income.
Relate Expenses are printing costs
plus research and writing.
Write
y = 0.9x + 600
1 A chemist has one solution
that is 50% acid. She has another
solution that is 25% acid. How
many liters of each type of acid
solution should she combine to
get 10 liters of a 40% acid
solution? 6 L of 50% solution,
4 L of 25% solution
2 Suppose you have a typing
service. You buy a personal
computer for $1750 on which to
do your typing. You charge $5.50
per page for typing. Expenses are
$.50 per page for ink, paper,
electricity, and other expenses.
How many pages must you type to
break even? 350 pages
Income is price
times copies sold.
y = 1.5x
Choose a method to solve this system. Use substitution since it is easy to substitute
for y with these equations.
y = 0.9x + 600
Start with one equation.
1.5x = 0.9x + 600
Substitute 1.5x for y.
0.6x = 600
Solve for x.
x = 1000
To break even, the model airplane club must sell 1000 copies.
Lesson 7-4 Applications of Linear Systems
Advanced Learners
397
English Language Learners ELL
L4
Have students write inequalities that illustrate when
the model airplane club in Example 2 is making a
profit and when it is losing money.
learning style: verbal
Some students may have difficulty translating the
word problems. Pair these students with students who
are proficient in English to help them interpret the
word problems.
learning style: verbal
397
3
EXAMPLE
Teaching Tip
Quick Check
Explain to students that A is the
speed of the plane with no wind
at all. Since a tailwind makes you
go faster, you add the speed of
the wind to the speed of the
plane, using A + W when
traveling wih the wind. Since
head winds decrease the speed of
the plane, you subtract the speed
of the wind from the speed of the
plane, using A - W when
traveling against the wind.
2 Suppose an antique car club publishes a newsletter. Expenses are $.35 for printing
and mailing each copy, plus $770 total for research and writing. The price of the
newsletter is $.55 per copy. How many copies of the newsletter must the club sell
to break even? 3850 copies
In Lesson 3-6, you modeled rate-time-distance problems using one variable.
You can also model rate-time-distance problems using two variables. The steady
west-to-east winds across the United States act as tail winds for planes traveling
from west to east. The tail winds increase a plane’s groundspeed. For planes
traveling east to west, the head winds decrease a plane’s groundspeed.
PowerPoint
Additional Examples
From West to East
From East to West
airspeed + windspeed
= groundspeed
airspeed – windspeed
= groundspeed
a
3 Suppose it takes you 6.8 hours
w
a
g
to fly about 2800 miles from
Miami, Florida to Seattle,
Washington. At the same time,
your friend flies from Seattle to
Miami. His plane travels with the
same average airspeed, but his
flight takes only 5.6 hours. Find
the average airspeed of the
planes. Find the average wind
speed. airspeed: 455.9 mi/h;
wind speed: 44.1 mi/h
3
EXAMPLE
w
Real-World
g
Problem Solving
Travel Suppose you fly from Miami, Florida, to San Francisco, California. It takes
6.5 hours to fly 2600 miles against a head wind. At the same time, your friend flies
from San Francisco to Miami. Her plane travels at the same average airspeed, but
her flight only takes 5.2 hours. Find the average airspeed of the planes. Find the
average wind speed.
Resources
• Daily Notetaking Guide 7-4 L3
• Daily Notetaking Guide 7-4—
L1
Adapted Instruction
Define
Let A = the airspeed. Let W = the wind speed.
Relate
with tail wind
(rate)(time) = distance
(A + W)(time) = distance
with head wind
(rate)(time) = distance
(A - W)(time) = distance
Write
(A + W)5.2 = 2600
(A - W)6.5 = 2600
Step 1 Divide to get the variables of each equation with coefficients of 1 or -1.
Closure
Ask students to tell what they
found most difficult about
writing a system of equations to
solve a word problem.
(A + W)5.2 = 2600 S A + W = 500
Divide each side by 5.2.
(A - W)6.5 = 2600 S A - W = 400
Divide each side by 6.5.
Step 2 Eliminate W.
A + W = 500
A 2 W 5 400
Add the equations to eliminate W.
2A + 0 = 900
Step 3 Solve for A.
A = 450
Divide each side by 2.
Step 4 Solve for W using either of the original equations.
A + W = 500
450 + W = 500
W = 50
Use the first equation.
Substitute 450 for A.
Solve for W.
The average airspeed of the planes is 450 mi/h. The average wind speed is 50 mi/h.
398
398
Chapter 7 Systems of Equations and Inequalities
Quick Check
3 A plane takes about 6 hours to fly 2400 miles from New York City to Seattle,
Washington. At the same time, your friend flies from Seattle to New York City.
His plane travels with the same average airspeed, but his flight takes 5 hours.
Find the average airspeed of the planes. Find the average wind speed.
440 mi/h; 40 mi/h
EXERCISES
For more exercises, see Extra Skill and Word Problem Practice.
Practiceand
andProblem
ProblemSolving
Solving
Practice
Practice by Example
Example 1
(page 396)
GO for
Help
1. Tyrel and Dalia bought some pens and pencils. Tyrel bought 4 pens and
5 pencils, which cost him $6.71. Dalia bought 5 pens and 3 pencils, which cost
her $7.12. Let a equal the price of a pen. Let b equal the price of a pencil.
a. Write an equation that relates the number of pens and pencils Tyrel bought
to the amount he paid for them. 4a ± 5b ≠ 6.71
b. Write an equation that relates the number of pens and pencils Dalia bought
to the amount she paid for them. 5a ± 3b ≠ 7.12
c. Solve the system you wrote for parts (a) and (b) to find the price of a pen
and the price of a pencil. pen: $1.19, pencil: $.39
1 A B 1-22
C Challenge
23, 24
Test Prep
Mixed Review
25-28
29-43
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 4, 8, 15, 20, 21.
Exercise 2 Tell students they can
avoid decimal errors by writing
the amounts in terms of cents,
such as 195 instead of 1.95.
Alternative Method
2. Suppose you have just enough money, in coins, to pay for a loaf of bread
priced at $1.95. You have 12 coins, all quarter and dimes. Let q equal the
number of quarters and d equal the number of dimes. Which system
models the given information? D
A. q + d = 12
B. 25q + 10d = 195
q + d = 1.95
q + 12 = d
C. 10q + 25d = 12
q + d = 1.95
Assignment Guide
Exercise 4 You can use a graphing
calculator to determine when the
balances are the same. Input each
equation into the Y= function,
and then press
. Use the
arrows to scroll to find the row in
which Y1 and Y2 are the same.
D. q + d = 12
25q + 10d = 195
3. Suppose you want to combine two types of fruit drink to create 24 kilograms of
a drink that will be 5% sugar by weight. Fruit drink A is 4% sugar by weight,
and fruit drink B is 8% sugar by weight.
a. Copy and complete the table below.
Fruit Drink A
4% Sugar
Fruit Drink B
8% Sugar
Mixed Fruit Drink
5% Sugar
Fruit Drink (kg)
■ a
■ b
■ 24
Sugar (kg)
■ 0.04a
■ 0.08b
■ 0.05(24)
GPS Guided Problem Solving
b. Write a system of equations that relates the amounts of fruit drink A and
fruit drink B to the total amount of drink needed and to the total amount
of sugar needed. a ± b ≠ 24; 0.04a ± 0.08b ≠ 1.2
c. Solve the system to find how much of each type of fruit drink you need to use.
18 kg A, 6 kg B
4. You have $22 in your bank account and deposit $11.50 each week. At the same
time your cousin has $218 but is withdrawing $13 each week.
a. When will your accounts have the same balance? at 8 wk
b. How much money will each of you have after 12 weeks? $160; $62
Example 2
(page 397)
5. Business Suppose you invest $10,410 in equipment to manufacture a new
board game. Each game costs $2.65 to manufacture and sells for $20. How
many games must you make and sell before your business breaks even?
600 games
Lesson 7-4 Applications of Linear Systems
399
L2
Reteaching
L1
Adapted Practice
Practice
Name
L3
L4
Enrichment
Class
Practice 7-4
Date
L3
Applications of Linear Systems
Use a system of linear equations to solve each problem.
1. Your teacher is giving you a test worth 100 points containing
40 questions. There are two-point and four-point questions on the
test. How many of each type of question are on the test?
2. Suppose you are starting an office-cleaning service. You have spent
$315 on equipment. To clean an office, you use $4 worth of supplies.
You charge $25 per office. How many offices must you clean to break
even?
3. The math club and the science club had fundraisers to buy supplies for
a hospice. The math club spent $135 buying six cases of juice and one
case of bottled water. The science club spent $110 buying four cases
of juice and two cases of bottled water. How much did a case of juice
cost? How much did a case of bottled water cost?
4. On a canoe trip, Rita paddled upstream (against the current) at an
average speed of 2 mi/h relative to the riverbank. On the return trip
downstream (with the current), her average speed was 3 mi/h. Find
Rita’s paddling speed in still water and the speed of the river’s current.
5. Kay spends 250 min/wk exercising. Her ratio of time spent on aerobics
to time spent on weight training is 3 to 2. How many minutes per week
does she spend on aerobics? How many minutes per week does she
spend on weight training?
6. Suppose you invest $1500 in equipment to put pictures on T-shirts. You
buy each T-shirt for $3. After you have placed the picture on a shirt,
you sell it for $20. How many T-shirts must you sell to break even?
© Pearson Education, Inc. All rights reserved.
A
3. Practice
7. A light plane flew from its home base to an airport 255 miles away.
With a head wind, the trip took 1.7 hours. The return trip with a tail
wind took 1.5 hours. Find the average airspeed of the plane and the
average windspeed.
8. Suppose you bought supplies for a party. Three rolls of streamers and
15 party hats cost $30. Later, you bought 2 rolls of streamers and
4 party hats for $11. How much did each roll of streamers cost?
How much did each party hat cost?
9. A new parking lot has spaces for 450 cars. The ratio of spaces for fullsized cars to compact cars is 11 to 4. How many spaces are for full-sized
cars? How many spaces are for compact cars?
10. While on vacation, Kevin went for a swim in a nearby lake. Swimming
against the current, it took him 8 minutes to swim 200 meters.
Swimming back to shore with the current took half as long. Find
Kevin’s average swimming speed and the speed of the lake’s current.
399
Error Prevention!
6. Business Several students decide to start a T-shirt company. After initial
expenses of $280, they purchase each T-shirt wholesale for $3.99. They sell each
T-shirt for $10.99. How many must they sell to break even? 40 T-shirts
Exercise 8 Some students may
stop after solving for the first
variable—the airspeed—and
assume it is the complete answer
to the problem. Remind students
they should always go back and
read the question again after
they have done the algebraic
work to make sure their answers
are reasonable. It would not be
reasonable for the wind’s speed
to be about 406 mi/h.
Example 3
(page 398)
8. Travel John flies from Atlanta, Georgia, to San Francisco, California. It takes
5.6 hours to travel 2100 miles against the head wind. At the same time Debby
flies from San Francisco to Atlanta. Her plane travels with the same average
airspeed but, with a tail wind, her flight takes only 4.8 hours.
a. Write a system of equations that relates time, airspeed, and wind speed to
distance for each traveler. (A ± W)4.8 ≠ 2100, (A – W)5.6 ≠ 2100
b. Solve the system to find the airspeed. 406.25 mi/h
c. Find the wind speed. 31.25 mi/h
Exercises 9–14 Suggest to
students that they read the
key concepts at the beginning
of this lesson and quickly review
Lessons 7-1 through 7-3 to remind
them when it is best to use each
solution method.
pages 399–402
B
Exercises
Apply Your Skills
9 –14. Answers may vary.
Samples are given.
9. Substitution; one eq. is
solved for t.
12. Substitution; both eqs.
are solved for y.
13. Elimination; mult. first
eq. by 3 and add to
elim. y.
Real-World
Connection
Glass can be drawn into
optical fibers 16 km long.
One fiber can carry 20 times
as many phone calls as
500 copper wires.
400
14. u = 4v
3u - 2v = 7
16. Geometry The perimeter of the rectangle is 34 cm. The perimeter of the
triangle is 30 cm. Find the values of m and n. 5 cm; 12 cm
n+1
m
m
n
b. After 16.5 min, the
temp. of either piece
will be 41.25C.
17. Answers may vary.
Sample: You have
10 coins, all dimes and
quarters. The value of
the coins is $1.75. How
many dimes do you
have? How many
quarters do you have?
q ± d ≠ 10
0.25q ± 0.10d ≠ 1.75
You have 5 dimes and
5 quarters.
13. 2x - y = 4
x + 3y = 16
15. Chemistry A piece of glass with an initial temperature of 998C is cooled at a
rate of 3.5 degrees Celsius per minute (8C/min). At the same time, a piece of
copper with an initial temperature of 08C is heated at a rate of 2.58C/min.
Let m = the number of minutes, and t = the temperature in degrees Celsius
after m minutes.
a. Write a system of equations that relates the temperature t of each material
to the time m. Solve the system. a–b. See margin.
b. Writing Explain what the solution means in this situation.
11. Elimination; subtract to
eliminate m.
15a. t ≠ 99 – 3.5m;
t ≠ 0 ± 2.5m;
t ≠ 41.25°,
m ≠ 16.5 min
Open-Ended Without solving, what method would you choose to solve each
system: graphing, substitution, or elimination? Explain your reasoning.
9–14. See margin.
9. 4s - 3t = 8
10. y = 3x - 1
11. 3m - 4n = 1
t = -2s - 1
y = 4x
3m - 2n = -1
12. y = -2x
y = 2 12 x + 3
10. Substitution; both eqs.
are solved for y.
14. Substitution; one eq. is
solved for u.
7. Travel A family is canoeing downstream (with the current). Their speed
relative to the banks of the river averages 2.75 mi/h. During the return trip,
they paddle upstream (against the current), averaging 1.5 mi/h relative to
the riverbank.
a. Write an equation for the rate of the canoe downstream. s ± c ≠ 2.75
b. Write an equation for the rate of the canoe upstream. s – c ≠ 1.5
c. Solve the system to find the family’s paddling speed in still water. 2.125 mi/h
d. Find the speed of the current of the river. 0.625 mi/h
n
17. Open-Ended Write a problem for the total of two types of coins. Then solve
the problem. See margin.
GO for Help
For a guide to solving
Exercise 18, see p. 403.
400
18. Sales A garden supply store sells two types of lawn mowers. Total sales of
mowers for the year were $8379.70. The total number of mowers sold was 30.
The small mower costs $249.99. The large mower costs $329.99. Find the
number sold of each type of mower. 19 small mowers, 11 large mowers
Chapter 7 Systems of Equations and Inequalities
4. Assess & Reteach
19. Aviation Suppose you are flying an ultralight aircraft like the one pictured
at the left. You fly to a nearby town, 18 miles away. With a tail wind, the trip
takes 31 hour. Your return flight with a head wind takes 35 hour.
a. Find the average airspeed of the ultralight aircraft. 42 mi/h
b. Find the average wind speed. 12 mi/h
Real-World
Connection
Ultralight aircraft like the one
pictured above can weigh less
than 400 lb.
20b. g ± b ≠ 1908
g ≠ 19
17 b
901 boys, 1007 girls
GO
PowerPoint
Lesson Quiz
20. Suppose the ratio of girls to boys in your school is 19 i 17. There are 1908
students altogether.
g
19
a. Solve the proportion b = 19
17 for g. g ≠ 17 b
b. Write and solve the system of equations to find the total number of boys b
and girls g. See left.
21. Consumer Decisions Suppose you are trying to decide whether to buy ski
GPS equipment. Typically, it costs you $60 a day to rent ski equipment and buy a lift
ticket. You can buy ski equipment for about $400. A lift ticket alone costs $35
for one day.
a. Find the break-even point. 16 days
b. Critical Thinking If you expect to ski five days a year, should you buy the ski
equipment? Explain. See margin.
22. Geometry Find the values of x and y. x ≠ 2, y ≠ 4
nline
perimeter ⫽ 14
Homework Help
Visit: PHSchool.com
Web Code: ate-0704
y
y
5
4y
x
5
4y
3x
C
Challenge
perimeter ⫽ 12
23. You can represent the value of any two-digit number with the expression
10a + b, where a is the tens’ place digit and b is the ones’ place digit. If a is 5
and b is 7, then the value of the number is 10(5) + 7, or 57. 37
Use a system of equations to find the two-digit number described below.
• The ones’ place digit is one more than twice the tens’ place digit.
• The value of the number is two more than five times the ones’ place digit.
24a. 2.50s ± 4.00< ≠ 10,000
< ≠ 52 s
800 small, 2000 large
24. Sales An artist sells original hand-painted greeting cards. He makes $2.50
profit on a small card and $4.00 profit on a large card. He generally sells 5 large
cards for every 2 small cards. He wants a profit of $10,000 from large and small
cards this year. a. See left.
a. Find the quantity of each card the artist needs to sell to reach his goal.
b. The artist can create a card every 12 minutes. How many hours will he need
to make enough to reach his profit target if he sells them all? 560 h
c. What is the artist’s hourly rate of pay? $17.86/h
Test Prep
Standardized
Test Prep
Multiple Choice
25. Which system describes the following situation: The sum of two numbers is
20. The difference between three times the larger and twice the smaller is 40. C
A. x + y = 20
B. x - y = 20
3x + 2y = 40
3x - 2y = 40
C. x + y = 20
D. x - y = 20
3x - 2y = 40
3x + 2y = 40
lesson quiz, PHSchool.com, Web Code: ata-0704
Lesson 7-4 Applications of Linear Systems
1. One antifreeze solution is 10%
alcohol. Another antifreeze
solution is 18% alcohol. How
many liters of each antifreeze
solution should be combined to
create 20 liters of antifreeze
solution that is 15% alcohol?
7.5 L of 10% solution;
12.5 L of 18% solution
2. A local band is planning to
make a compact disk. It will
cost $12,500 to record and
produce a master copy, and an
additional $2.50 to make each
sale copy. If they plan to sell
the final product for $7.50,
how many disks must they sell
to break even? 2500 disks
3. Suppose it takes you and a
friend 3.2 hours to canoe
12 miles downstream (with the
current). During the return trip,
it takes you and your friend 4.8
hours to paddle upstream
(against the current) to the
original starting point. Find the
average paddling speed in still
water of you and your friend
and the average speed of the
current of the river. Round
answers to the nearest tenth.
still water: 3.1 mi/h;
current: 0.6 mi/h
Alternative Assessment
Organize students in groups of
three. Let each group choose a
word problem from the exercises.
Assign each student a different
method for solving the problem.
401
21b. Answers may vary.
Sample: If you plan to ski
for many years,
you should buy the
equipment, since you will
break even at 16 days.
401
Test Prep
26. The federal tax on a $12,000 salary was 8 times the state tax. If the
combined taxes were $2700, find the state’s share of taxes. H
F. $400
G. $150
H. $300
J. $350
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 425
• Test-Taking Strategies, p. 420
• Test-Taking Strategies with
Transparencies
27. Which system describes the following situation? Craig has 80¢ in nickels n
and dimes d. He has four more nickels than dimes. B
A. d + n = 4
B. n - d = 4
10d + 5n = 80
10d + 5n = 80
D. d + n = 4
C. d - n = 4
10d + 5n = 80
10d - 5n = 80
Exercise 27 Suggest to students
that they first write “four more
nickels than dimes” as n = d + 4.
Short Response
28. A large group of students wants to go to the movies. If the students
take 3 vans and 1 car, they can transport 22 people. If they take 2 vans
and 4 cars, they can transport 28 people. Write and solve a system of
equations to find the number of people that can be transported in
a van. Show your work. See margin.
Mixed Review
GO for
Help
Lesson 7-3
Solve by elimination.
29. 2x + 5y = 13
3x - 5y = 7 (4, 1)
Lesson 6-1
30. 4x + 2y = -10
31. 7x + 6y = 30
-2x + 3y = 33 (–6, 7)
9x - 8y = 15 (3, 32 )
Find the slope of the line that passes through each pair of points.
Lesson 4-5
32. (2, 4), (6, 10) 23
33. (-3, 1), (10, 14) 1
34. (8, -11), (5, -12) 13
35. (1.2, 7), (4.6, 0.2) –2
4
36. Q 5, -12 R , Q -6, 312 R –11
37. (8, 0), (8, 5) undefined
Solve each inequality and graph the solutions. 38–43. See margin.
38. 6 , y , 10
39. -8 , n # 3
40. 2 , k + 1 , 7
41. 4 # 4p # 16
42. -13 , 3c + 2 # 17
43. 21 . 5w - 4 . 1
Algebra at Work
Businessperson
S
Restriction Polygon
y
Advertising
Raw Materials
Transportation
Packaging
Equipment
Labor
x
O
ome of the goals of a business are to
minimize costs and maximize profits. People
in business use systems of linear inequalities
to analyze data in order to achieve these goals.
The illustration lists some of the variables involved in operating a small
manufacturing company. To solve a problem, a businessperson must
identify the variables and restrictions, and then search for the best of many
possible solutions.
PHSchool.com
402
pages 399–402
28. [2]
402
For: Information about a career in business
Web Code: atb-2031
Chapter 7 Systems of Equations and Inequalities
Exercises
3V ± C ≠ 22
12V ± 4C
2V ± 4C ≠ 28 1 2V ± 4C
10V
V
6 people/van
≠
≠
≠
≠
88
28
60
6
[1] 6 people/van, no
work shown
38. 6 R y R 10
0
6
10
39. –8 R n K 3
8
0
3