Systems of Equations and Inequalities

Transcription

7
Systems of Equations
and Inequalities
7.1 Solve Linear Systems by Graphing
7.2 Solve Linear Systems by Substitution
7.3 Solve Linear Systems by Adding or Subtracting
7.4 Solve Linear Systems by Multiplying First
7.5 Solve Special Types of Linear Systems
7.6 Solve Systems of Linear Inequalities
Before
In previous chapters, you learned the following skills, which you’ll use in
Chapter 7: graphing linear equations, solving equations, determining whether
lines are parallel, and graphing linear inequalities in two variables.
Prerequisite Skills
VOCABULARY CHECK
Copy and complete the statement.
1. The least common multiple of 10 and 15 is ? .
2. Two lines in the same plane are ? if they do not intersect.
SKILLS CHECK
Graph the equation. (Review p. 225 for 7.1.)
3. x 2 y 5 4
4. 6x 2 y 5 21
5. 4x 1 5y 5 20
6. 3x 2 2y 5 212
Solve the equation. (Review p. 148 for 7.2–7.4.)
7. 5m 1 4 2 m 5 20
8. 10(z 1 5) 1 z 5 6
Tell whether the graphs of the two equations are parallel lines. Explain your
reasoning. (Review p. 244 for 7.5.)
9. y 5 2x 2 3, y 1 2x 5 23
11. y 5 x 1 10, x 2 y 5 29
10. y 2 5x 5 21, y 2 5x 5 1
12. 6x 2 y 5 4, 4x 2 y 5 6
Graph the inequality. (Review p. 405 for 7.6.)
13. y ≤ 22x 1 1
14. x 2 y < 5
15. x ≥ 24
1SFSFRVJTJUFTLJMMTQSBDUJDFBUDMBTT[POFDPN
424
16. y > 3
Now
In Chapter 7, you will apply the big ideas listed below and reviewed in the
Chapter Summary on page 474. You will also use the key vocabulary listed below.
Big Ideas
1 Solving linear systems by graphing
2 Solving linear systems using algebra
3 Solving systems of linear inequalities
KEY VOCABULARY
• system of linear equations,
p. 427
• solution of a system of
linear equations, p. 427
• consistent independent
system, p. 427
• inconsistent system, p. 459
• consistent dependent
system, p. 459
• system of linear
inequalities, p. 466
• solution of a system of
linear inequalities, p. 466
• graph of a system of
linear inequalities, p. 466
Why?
You can use a system of linear equations to solve problems about traveling
with and against a current. For example, you can write and solve a system of
linear equations to find the average speed of a kayak in still water.
Algebra
The animation illustrated below for Example 4 on page 446 helps you answer
this question: What is the average speed of the kayak in still water?
You have to find the speed of the kayak
in still water.
Click the button that will produce an
equation in one variable.
Algebra at classzone.com
Other animations for Chapter 7: pages 428, 435, 441, 446, 452, 459,
and 466
425
Investigating
g
Algebra
Algebr
ra
ACTIVITY
Use before Lesson 7.1
7.1 Solving Linear Systems Using Tables
M AT E R I A L S • pencil and paper
QUESTION
How can you use a table to solve a linear system?
A system of linear equations, or linear system, consists of two or more linear
equations in the same variables. A solution of a linear system is an ordered
pair that satisfies each equation in the system. You can use a table to fi nd a
solution to a linear system.
EXPLORE
Solve a linear system
Bill and his brother collect comic books. Bill currently has 15 books and adds
2 books to his collection every month. His brother currently has 7 books and
adds 4 books to his collection every month. Use the equations below to fi nd
the number x of months after which Bill and his brother will have the same
number y of comic books in their collections.
y 5 2x 1 15
Number of comic books in Bill’s collection
y 5 4x 1 7
Number of comic books in his brother’s collection
STEP 1 Make a table
Copy and complete the table of values shown.
STEP 2 Find a solution
Find an x-value that gives the same y-value for
both equations.
STEP 3 Interpret the solution
Use your answer to Step 2 to find the number of
months after which Bill and his brother have the
same number of comic books.
DR AW CONCLUSIONS
x
y 5 2x 1 15
y 5 4x 1 7
0
15
7
1
?
?
2
?
?
3
?
?
4
?
?
5
?
?
Use your observations to complete these exercises
1. When Bill and his brother have the same number of books in their
collections, how many books will each of them have?
2. Graph the equations above on the same coordinate plane. What do
you notice about the graphs and the solution you found above?
Use a table to solve the linear system.
3. y 5 2x 1 3
y 5 23x 1 18
426
4. y 5 2x 1 1
y 5 2x 2 5
Chapter 7 Systems of Equations and Inequalities
5. y 5 23x 1 1
y 5 5x 2 31
7.1
Before
Now
Why?
Key Vocabulary
equations
• solution of a
system of linear
equations
• consistent
independent
system
Solve Linear Systems
by Graphing
You graphed linear equations.
You will graph and solve systems of linear equations.
So you can analyze craft fair sales, as in Ex. 33.
A system of linear equations, or simply a linear system, consists of two or
more linear equations in the same variables. An example is shown below.
x 1 2y 5 7
3x 2 2y 5 5
Equation 1
Equation 2
A solution of a system of linear equations in two variables is an ordered pair
that satisfies each equation in the system.
One way to find the solution of a linear system is by graphing. If the lines
intersect in a single point, then the coordinates of the point are the solution
of the linear system. A solution found using graphical methods should be
checked algebraically.
EXAMPLE 1
Check the intersection point
Use the graph to solve the system. Then
check your solution algebraically.
x 1 2y 5 7
3x 2 2y 5 5
y
Equation 1
x 1 2y 5 7
1
Equation 2
2
x
3x 2 2y 5 5
Solution
The lines appear to intersect at the point (3, 2).
CHECK Substitute 3 for x and 2 for y in each equation.
x 1 2y 5 7
3x 2 2y 5 5
3 1 2(2) 0 7
3(3) 2 2(2) 0 5
757✓
555✓
c Because the ordered pair (3, 2) is a solution of each equation, it is a
solution of the system.
TYPES OF LINEAR SYSTEMS In Example 1, the linear system has exactly one
solution. A linear system that has exactly one solution is called a consistent
independent system because the lines are distinct (are independent) and
intersect (are consistent). You will solve consistent independent systems in
Lessons 7.1–7.4. In Lesson 7.5 you will consider other types of systems.
427
For Your Notebook
KEY CONCEPT
Solving a Linear System Using the Graph-and-Check Method
STEP 1 Graph both equations in the same coordinate plane. For
ease of graphing, you may want to write each equation in
slope-intercept form.
STEP 2 Estimate the coordinates of the point of intersection.
STEP 3 Check the coordinates algebraically by substituting into each
equation of the original linear system.
EXAMPLE 2
Use the graph-and-check method
2x 1 y 5 27
x 1 4y 5 28
Solve the linear system:
Equation 1
Equation 2
Solution
STEP 1 Graph both equations.
y
21
1
x
x 1 4y 5 28
2x 1 y 5 27
STEP 2 Estimate the point of intersection. The two lines appear to intersect
at (4, 23).
STEP 3 Check whether (4, 23) is a solution by substituting 4 for x and
23 for y in each of the original equations.
Equation 1
Equation 2
2x 1 y 5 27
x 1 4y 5 28
2(4) + (23) 0 27
4 1 4(23) 0 28
27 5 27 ✓
28 5 28 ✓
c Because (4, 23) is a solution of each equation, it is a solution of
the linear system.
"MHFCSB
✓
GUIDED PRACTICE
at classzone.com
for Examples 1 and 2
Solve the linear system by graphing. Check your solution.
1. 25x 1 y 5 0
5x 1 y 5 10
428
2. 2x 1 2y 5 3
2x 1 y 5 4
3. x 2 y 5 5
3x 1 y 5 3
★
EXAMPLE 3
Standardized Test Practice
The parks and recreation department in your town
offers a season pass for $90.
• As a season pass holder, you pay $4 per session
to use the town’s tennis courts.
• Without the season pass, you pay $13 per
session to use the tennis courts.
Which system of equations can be used to find
the number x of sessions of tennis after which
the total cost y with a season pass, including
the cost of the pass, is the same as the total
cost without a season pass?
ELIMINATE CHOICES
You can eliminate
choice A because
neither of the equations
include the cost of a
season pass.
A y 5 4x
y 5 13x
B y 5 4x
y 5 90 1 13x
C y 5 13x
y 5 90 1 4x
D y 5 90 1 4x
y 5 90 1 13x
Solution
Write a system of equations where y is the total cost (in dollars) for
x sessions.
EQUATION 1
Total
cost
(dollars)
y
5
5
Cost per
session
p
(dollars/session)
Number of
sessions
(sessions)
p
13
x
EQUATION 2
Total
cost
(dollars)
y
5
Cost for
season pass
5
90
(dollars)
c The correct answer is C.
✓
GUIDED PRACTICE
1
Cost per
session
(dollars/session)
1
4
p
p
Number of
sessions
(sessions)
x
A B C D
for Example 3
4. Solve the linear system in Example 3 to find the number of sessions
after which the total cost with a season pass, including the cost of the
pass, is the same as the total cost without a season pass.
5. WHAT IF? In Example 3, suppose a season pass costs $135. After how
many sessions is the total cost with a season pass, including the cost of
the pass, the same as the total cost without a season pass?
429
EXAMPLE 4
Solve a multi-step problem
RENTAL BUSINESS A business rents in-line skates and
bicycles. During one day, the business has a total of
25 rentals and collects $450 for the rentals. Find the
number of pairs of skates rented and the number
of bicycles rented.
Solution
STEP 1 Write a linear system. Let x be the number of pairs of skates
rented, and let y be the number of bicycles rented.
x 1 y 5 25
15x 1 30y 5 450
Equation for number of rentals
Equation for money collected from rentals
STEP 2 Graph both equations.
y
STEP 3 Estimate the point of intersection. The
x 1 y 5 25
two lines appear to intersect at (20, 5).
STEP 4 Check whether (20, 5) is a solution.
20 1 5 0 25
25 5 25 ✓
15x 1 30y 5 450
5
15(20) 1 30(5) 0 450
5
450 5 450 ✓
c The business rented 20 pairs of skates and 5 bicycles.
✓
GUIDED PRACTICE
for Example 4
6. WHAT IF? In Example 4, suppose the business has a total of 20 rentals
and collects $420. Find the number of bicycles rented.
7.1
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 15 and 31
★ 5 STANDARDIZED TEST PRACTICE
Exs. 2, 6, 7, 27, 28, 29, and 32
5 MULTIPLE REPRESENTATIONS
Ex. 35
SKILL PRACTICE
1. VOCABULARY Copy and complete: A(n)
? of a system of linear
equations in two variables is an ordered pair that satisfies each equation
in the system.
2.
★ WRITING Explain how to use the graph-and-check method to solve a
linear system of two equations in two variables.
CHECKING SOLUTIONS Tell whether the ordered pair is a solution of the
linear system.
3. (23, 1);
4. (5, 2);
x 1 y 5 22
x 1 5y 5 2
430
2x 2 3y 5 4
2x 1 8y 5 11
5. (22, 1);
6x 1 5y 5 27
x 2 2y 5 0
x
EXAMPLE 1
6.
on p. 427
for Exs. 6–11
★ MULTIPLE CHOICE Which ordered pair is a solution of the linear system
x 1 y 5 22 and 7x 2 4y 5 8?
A (22, 0)
7.
B (0, 22)
C (2, 0)
D (0, 2)
★ MULTIPLE CHOICE Which ordered pair is a solution of the linear
system 2x 1 3y 5 12 and 10x 1 3y 5 212?
A (23, 3)
B (23, 6)
C (3, 3)
D (3, 6)
SOLVING SYSTEMS GRAPHICALLY Use the graph to solve the linear system.
Check your solution.
8. x 2 y 5 4
9. 2x 1 y 5 22
2x 2 y 5 6
4x 1 y 5 1
10. x 1 y 5 5
22x 1 y 5 24
y
y
y
1
1
x
1
1
1
11. ERROR ANALYSIS Describe
and correct the error in solving
y
the linear system below.
x 2 3y 5 6
2x 2 3y 5 3
1
x
Equation 1
Equation 2
x – 3y = 6 x
1
–1
x
The solution is
(3, 21).
2x – 3y = 3
EXAMPLE 2
GRAPH-AND-CHECK METHOD Solve the linear system by graphing. Check
on p. 428
for Exs. 12–26
your solution.
12. y 5 2x 1 3
y5x11
13. y 5 2x 1 4
y 5 2x 2 8
14. y 5 2x 1 2
15. x 2 y 5 2
x 1 y 5 28
16. x 1 2y 5 1
22x 1 y 5 24
17. 3x 1 y 5 15
18. 2x 2 3y 5 21
5x 1 2y 5 26
19. 6x 1 y 5 37
4x 1 2y 5 18
20. 7x 1 5y 5 23
21. 6x 1 12y 5 26
2x 1 5y 5 0
22. 2x 1 y 5 9
2x 1 3y 5 15
23. 25x 1 3y 5 3
13
1
3
24. }
x1}
y5}
8
2
1
25. }
x2}
y 5 2}
26. 21.6x 2 3.2y 5 224
4
2
3
13
x 2 }y 5 }
4
2
4
5
5
y 5 4x 1 6
5
3
2}
x1y53
y 5 215
29x 1 y 5 211
4x 1 3y 5 30
2.6x 1 2.6y 5 26
4
3
Find values for m and b so that the system y 5 }
x21
5
and y 5 mx 1 b has (5, 2) as a solution.
27.
★ OPEN – ENDED
28.
★ WRITING Solve the linear system shown
by graphing. Explain why it is important to
check your solution.
y 5 4x 2 1.5
y 5 22x 1 1.5
Equation 1
Equation 2
431
29.
★ EXTENDED RESPONSE Consider the equation 2}14x 1 6 5 }12x 1 3.
a. Solve the equation using algebra.
b. Solve the linear system below using a graph.
1
y 5 2}
x16
Equation 1
1
y5}
x13
Equation 2
4
2
c. How is the linear system in part (b) related to the original equation?
2
1
d. Explain how to use a graph to solve the equation 2}
x155}
x 1 2.
5
5
30. CHALLENGE The three lines given below form a triangle. Find the
coordinates of the vertices of the triangle.
Line 1: 23x 1 2y 5 1
Line 2: 2x 1 y 5 11
Line 3: x 1 4y 5 9
PROBLEM SOLVING
31. TELEVISION The graph shows a projection, from
1990 on, of the percent of eighth graders who
watch 1 hour or less of television on a weekday
and the percent of eighth graders who watch more
than 1 hour of television on a weekday. Use the
graph to predict the year when the percent of
eighth graders who watch 1 hour or less will equal
the percent who watch more than 1 hour.
on pp. 429–430
for Exs. 31–33
p
Percent of
eighth graders
EXAMPLES
3 and 4
1 hour or less
80
40
0
more than 1 hour
0
20
40
60
80
Years since 1990
t
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
32.
★ MULTIPLE CHOICE A car dealership is offering interest-free car loans
for one day only. During this day, a salesperson at the dealership
sells two cars. One of his clients decides to pay off his $17,424 car in
36 monthly payments of $484. His other client decides to pay off his
$15,840 car in 48 monthly payments of $330. Which system of equations
can be used to determine the number x of months after which both
clients will have the same loan balance y?
A y 5 2484x
y 5 2330x
B y 5 2484x 1 17,424
y 5 2330x 1 15,840
C y 5 2484x 1 15,840
y 5 2330x 1 17,424
D y 5 484x 1 17,424
y 5 330x 1 15,840
33. CRAFTS Kirigami is the Japanese art of making paper designs
by folding and cutting paper. A student sells small and large
greeting cards decorated with kirigami at a craft fair.
The small cards cost $3 per card, and the large cards cost
$5 per card. The student collects $95 for selling
a total of 25 cards. How many of each type
of card did the student sell?
432
on p. WS1
★ 5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
34. FITNESS You want to burn 225 calories while exercising at a gym. The
number of calories that you burn per minute on different machines at
the gym is shown below.
Stair machine
Elliptical trainer
Stationary bike
You burn 5 Cal/min.
You burn 8 Cal/min.
You burn 6 Cal/min.
a. Suppose you have 40 minutes to exercise at the gym and you want
to use the stair machine and stationary bike. How many minutes
should you spend on each machine so that you burn 225 calories?
b. Suppose you have 30 minutes to exercise at the gym and you want
to use the stair machine and the elliptical trainer. How many minutes
should you spend on each machine so that you burn 225 calories?
35.
MULTIPLE REPRESENTATIONS It costs $15 for a yearly membership to
a movie club at a movie theater. A movie ticket costs $5 for club members
and $8 for nonmembers.
a. Writing a System of Equations Write a system of equations that you
can use to find the number x of movies viewed after which the total
cost y for a club member, including the membership fee, is the same
as the cost for a nonmember.
b. Making a Table Make a table of values that shows the total cost for a
club member and a nonmember after paying to see 1, 2, 3, 4, 5, and
6 movies.
c. Drawing a Graph Use the table to graph the system of equations.
Under what circumstances does it make sense to become a movie
club member? Explain your answer by using the graph.
36. CHALLENGE With a minimum purchase of $25, you can open a credit
account with a clothing store. The store is offering either $25 or 20% off
of your purchase if you open a credit account. You decide to open a credit
account. Should you choose $25 or 20% off of your purchase? Explain.
MIXED REVIEW
Solve the equation.
37. x 1 5 5 214 (p. 134)
38. 25x 1 6 5 21 (p. 141)
39. 3(x 1 2) 5 26 (p. 148)
40. 11x 1 9 5 13x 2 3 (p. 154)
41. 3x 2 8 5 11x 1 12 (p. 154)
42. 4(x 1 1) 5 22x 2 18 (p. 154)
PREVIEW
Write the equation so that y is a function of x. (p. 184)
Prepare for
Lesson 7.2
in Exs. 43–48.
43. 5y 1 25 5 3x
44. 7x 5 y 1 9
45. 4y 1 11 5 3y 1 4x
46. 6x 1 3y 5 2x 1 3
47. y 1 2x 1 6 5 21
48. 4x 2 12 5 5x 1 2y
EXTRA PRACTICE for Lesson 7.1, p. 944
ONLINE
QUIZ at classzone.com
7.1
433
Graphing
p
g
Calculator
ACTIVITY
ACTIVITY
Use after Lesson 7.1
classzone.com
Keystrokes
7.1 Solving Linear Systems
by Graphing
QUESTION
How can you use a graphing calculator to solve a
linear system?
EXAMPLE
Solve a linear system
Solve the linear system using a graphing calculator.
5x 1 2y 5 6
x 2 3y 5 25
Equation 1
Equation 2
STEP 1 Rewrite equations
STEP 2 Enter equations
Solve each equation for y.
Equation 1
5x 1 2y 5 6
Press
Equation 2
Y1=-(5/2)X+3
Y2=(1/3)X+(5/3)
Y3=
Y4=
Y5=
Y6=
Y7=
x 2 3y 5 25
2y 5 25x 1 6
23y 5 2x 2 5
5
y 5 2}
x13
2
5
y 5 }1 x 1 }
3
and enter the equations.
3
STEP 3 Display graph
STEP 4 Find point of intersection
Graph the equations using a standard
viewing window.
Use the intersect feature to find the
point where the graphs intersect.
X=.47058824 Y=1.8235294
The solution is about (0.47, 1.8).
PRACTICE
Solve the linear system using a graphing calculator.
1. y 5 x 1 4
y 5 23x 2 2
434
2. 5x 1 y 5 24
x 2 y 5 22
3. 20.45x 2 y 5 1.35
21.8x 1 y 5 21.8
4. 20.4x 1 0.8y 5 216
1.2x 1 0.4y 5 1
7.2
Before
Now
Why?
Key Vocabulary
equations, p. 427
by Substitution
You solved systems of linear equations by graphing.
You will solve systems of linear equations by substitution.
So you can find tubing costs, as in Ex. 32.
For Your Notebook
KEY CONCEPT
Solving a Linear System Using the Substitution Method
STEP 1 Solve one of the equations for one of its variables. When possible,
solve for a variable that has a coefficient of 1 or 21.
STEP 2 Substitute the expression from Step 1 into the other equation and
solve for the other variable.
STEP 3 Substitute the value from Step 2 into the revised equation from
Step 1 and solve.
EXAMPLE 1
Use the substitution method
Solve the linear system: y 5 3x 1 2
x 1 2y 5 11
Equation 1
Equation 2
Solution
STEP 1 Solve for y. Equation 1 is already solved for y.
STEP 2 Substitute 3x 1 2 for y in Equation 2 and solve for x.
x 1 2y 5 11
x 1 2(3x 1 2) 5 11
7x 1 4 5 11
7x 5 7
x51
Write Equation 2.
Substitute 3x 1 2 for y.
Simplify.
Subtract 4 from each side.
Divide each side by 7.
STEP 3 Substitute 1 for x in the original Equation 1 to find the value of y.
y 5 3x 1 2 5 3(1) 1 2 5 3 1 2 5 5
c The solution is (1, 5).
CHECK Substitute 1 for x and 5 for y in each of the original equations.
y 5 3x 1 2
5 0 3(1) 1 2
555✓
x 1 2y 5 11
1 1 2(5) 0 11
11 5 11 ✓
"MHFCSB
at classzone.com
435
EXAMPLE 2
Use the substitution method
Solve the linear system: x 2 2y 5 26
4x 1 6y 5 4
Equation 1
Equation 2
Solution
CHOOSE AN
EQUATION
Equation 1 was chosen
in Step 1 because x has
a coefficient of 1. So,
only one step is needed
to solve Equation 1
for x.
STEP 1 Solve Equation 1 for x.
x 2 2y 5 26
Write original Equation 1.
x 5 2y 2 6
Revised Equation 1
STEP 2 Substitute 2y 2 6 for x in Equation 2 and solve for y.
4x 1 6y 5 4
4(2y 2 6) 1 6y 5 4
8y 2 24 1 6y 5 4
14y 2 24 5 4
14y 5 28
y52
Write Equation 2.
Substitute 2y – 6 for x.
Distributive property
Simplify.
Add 24 to each side.
STEP 3 Substitute 2 for y in the revised Equation 1 to find the value of x.
x 5 2y 2 6
Revised Equation 1
x 5 2(2) 2 6
Substitute 2 for y.
x 5 22
Simplify.
Equation 1
Equation 2
x 2 2y 5 26
4x 1 6y 5 4
22 2 2(2) 0 26
4(22) 1 6(2) 0 4
26 5 26 ✓
454✓
CHECK REASONABLENESS When solving
a linear system using the substitution
method, you can use a graph to check
the reasonableness of your solution. For
example, the graph at the right verifies that
(22, 2) is a solution of the linear system in
Example 2.
✓
GUIDED PRACTICE
y
4x 1 6y 5 4
2
x 2 2y 5 26
x
21
Solve the linear system using the substitution method.
1. y 5 2x 1 5
3x 1 y 5 10
436
2. x 2 y 5 3
x 1 2y 5 26
3. 3x 1 y 5 27
22x 1 4y 5 0
EXAMPLE 3
Solve a multi-step problem
ANOTHER WAY
WEBSITES Many businesses pay website hosting companies to store and
For an alternative
method for solving the
problem in Example 3,
turn to page 442 for
the Problem Solving
Workshop.
maintain the computer files that make up their websites. Internet service
providers also offer website hosting. The costs for website hosting offered by
a website hosting company and an Internet service provider are shown in
the table. Find the number of months after which the total cost for website
hosting will be the same for both companies.
Company
Set-up fee
(dollars)
Cost per month
(dollars)
10
21.95
None
22.45
Internet service provider
Website hosting company
Solution
STEP 1 Write a system of equations. Let y be the total cost after x months.
Equation 1: Internet service provider
Total
cost
5
Set-up
fee
1
Cost per
month
p
Number
of months
y
5
10
1
21.95
p
x
Equation 2: Website hosting company
Total
cost
5
Cost per
month
p
Number
of months
y
5
22.45
p
x
The system of equations is: y 5 10 1 21.95x
y 5 22.45x
Equation 1
Equation 2
STEP 2 Substitute 22.45x for y in Equation 1 and solve for x.
y 5 10 1 21.95x
22.45x 5 10 1 21.95x
0.5x 5 10
x 5 20
Write Equation 1.
Substitute 22.45x for y.
Subtract 21.95x from each side.
Divide each side by 0.5.
c The total cost will be the same for both companies after 20 months.
✓
GUIDED PRACTICE
for Example 3
4. In Example 3, what is the total cost for website hosting for each company
after 20 months?
5. WHAT IF? In Example 3, suppose the Internet service provider offers
$5 off the set-up fee. After how many months will the total cost for website
hosting be the same for both companies?
437
EXAMPLE 4
Solve a mixture problem
ANTIFREEZE For extremely cold temperatures, an automobile manufacturer
recommends that a 70% antifreeze and 30% water mix be used in the
cooling system of a car. How many quarts of pure (100%) antifreeze and a
50% antifreeze and 50% water mix should be combined to make 11 quarts of
a 70% antifreeze and 30% water mix?
Solution
STEP 1 Write an equation for the total number of quarts and an equation
for the number of quarts of antifreeze. Let x be the number of
quarts of 100% antifreeze, and let y be the number of quarts a
50% antifreeze and 50% water mix.
Equation 1: Total number of quarts
x 1 y 5 11
Equation 2: Number of quarts of antifreeze
x quarts of
100% antifreeze
DRAW A DIAGRAM
Each bar shows the
liquid in each mix. The
green portion shows the
percent of the mix that
is antifreeze.
y quarts of
50%–50% mix
1
100%
11 quarts of
70%–30% mix
5
70%
50%
1px
1
0.5 p y
5
0.7(11)
x 1 0.5y 5 7.7
The system of equations is: x 1 y 5 11
x 1 0.5y 5 7.7
Equation 1
Equation 2
STEP 2 Solve Equation 1 for x.
x 1 y 5 11
x 5 11 2 y
Write Equation 1.
Revised Equation 1
STEP 3 Substitute 11 2 y for x in Equation 2 and solve for y.
x 1 0.5y 5 7.7
(11 – y) 1 0.5y 5 7.7
y 5 6.6
Write Equation 2.
Substitute 11 2 y for x.
Solve for y.
STEP 4 Substitute 6.6 for y in the revised Equation 1 to find the value of x.
x 5 11 2 y 5 11 2 6.6 5 4.4
c Mix 4.4 quarts of 100% antifreeze and 6.6 quarts of a 50% antifreeze and
50% water mix to get 11 quarts of a 70% antifreeze and 30% water mix.
✓
GUIDED PRACTICE
for Example 4
6. WHAT IF? How many quarts of 100% antifreeze and a 50% antifreeze and
50% water mix should be combined to make 16 quarts of a 70% antifreeze
and 30% water mix?
438
7.2
EXERCISES
HOMEWORK
KEY
Exs. 2, 18, 29, 33, and 37
SKILL PRACTICE
1. VOCABULARY Give an example of a system of linear equations.
2.
★ WRITING If you are solving the linear system
shown using the substitution method, which
equation would you solve for which variable?
Explain.
EXAMPLE 1
on p. 435
for Exs. 3–8
on p. 436
for Exs. 9–19
Equation 1
2x 1 y 5 8
Equation 2
SOLVING LINEAR SYSTEMS Solve the linear system using substitution.
3. x 5 17 2 4y
4. y 5 2x 2 1
5. x 5 y 1 3
6. 4x 2 7y 5 10
7. x 5 16 2 4y
8. 25x 1 3y 5 51
y5x22
y5x27
EXAMPLE 2
2x 2 3y 5 24
9. 2x 5 12
2x 1 y 5 3
2x 2 y 5 5
3x 1 4y 5 8
y 5 10x 2 8
10. 2x 2 y 5 23
11. x 1 y 5 0
12. 2x 1 y 5 9
13. 5x 1 2y 5 9
14. 5x 1 4y 5 32
15. 11x 2 7y 5 214
16. 20x 2 30y 5 250
17. 6x 1 y 5 4
x 2 5y 5 229
4x 2 y 5 215
x 2 2y 5 24
18.
x 2 9 5 21
x 2 2y 5 6
x 1 y 5 23
9x 2 y 5 33
x 1 2y 5 1
x 2 4y 5 19
4x 2 y 5 17 and 29x 1 8y 5 2?
A (6, 7)
B (7, 6)
C (7, 11)
D (11, 7)
19. ERROR ANALYSIS Describe and correct the error in solving the linear
system 4x 1 2y 5 6 and 3x 1 y 5 9.
Step 1
Step 2
3x 1 y 5 9
4x 1 2(9 2 3x) 5 6
y 5 9 2 3x
4x 1 18 2 6x 5 6
6 5 9 2 3x
y 5 9 2 3x
Step 3
22x 5 212
23 5 23x
x56
15x
The solution
is (6, 1).
SOLVING LINEAR SYSTEMS Solve the linear system using substitution.
20. 4.5x 1 1.5y 5 24
21. 35x 1 y 5 20
22. 3x 2 2y 5 8
23. 0.5x 1 0.6y 5 5.7
24. x 2 9 5 0.5y
25. 0.2x 1 y 5 21.8
x2y54
2x 2 y 5 21
1
1
26. }
x1}
y55
2
4
1
x2}
y51
2
1.5x 2 0.1y 5 18
2.2x 2 3.1y 5 20.2
0.5x 1 y 5 17
1.8y 1 5.5x 5 27.6
3
3
28. }
x1}
y 5 12
1
27. x 1 }
y 5 22
3
2
28x 2 }
y54
3
4
8
2 x 1 1 y 5 13
}
}
3
2
439
29.
★ WRITING Suppose you solve a linear system using substitution.
Explain how you can use a graph to check your solution.
30. CHALLENGE Find values of a and b so that the
linear system shown has a solution of (29, 4).
ax 1 by 5 216
Equation 1
ax 2 by 5 256
Equation 2
PROBLEM SOLVING
EXAMPLE 3
31. FUNDRAISING During a football game, the parents of the football players
sell pretzels and popcorn to raise money for new uniforms. They charge
$2.50 for a bag of popcorn and $2 for a pretzel. The parents collect $336
in sales during the game. They sell twice as many bags of popcorn as
pretzels. How many bags of popcorn do they sell? How many pretzels do
they sell?
on p. 437
for Exs. 31–33
32. TUBING COSTS The members of an outing club take a day-long tubing trip
down a river. The company that offers the tubing trip charges $15 to rent a
tube for a person to use and $7.50 to rent a “cooler” tube, which is used to
carry food and water in a cooler. The club members spend $360 to rent a
total of 26 tubes. How many of each type of tube do they rent?
33.
★ SHORT RESPONSE In the mobile shown, objects are
attached to each end of a dowel. For the dowel to balance,
the following must be true:
xp
Weight hanging
from point A
5yp
!
X
Weight hanging
from point B
The weight of the objects hanging from point A is 1.5 pounds,
and the weight of the objects hanging from point B is
1.2 pounds. The length of the dowel is 9 inches. How far from
point A should the string be placed? Explain.
34. MULTI-STEP PROBLEM Two swimming teams are competing in a
400 meter medley relay. During the last leg of the race, the swimmer in
lane 1 has a 1.2 second head start on the swimmer in lane 2, as shown.
1
Swimming at 1.8 m/sec
with a 1.2 sec head start
2
Swimming at 1.9 m/sec
a. Let t be the time since the swimmer in lane 2 started the last leg. After
how many seconds into the leg will the swimmer in lane 2 catch up to
the swimmer in lane 1?
b. Does the swimmer in lane 2 catch up to the swimmer in lane 1 before
the race ends? Explain.
440
on p. WS1
★ 5 STANDARDIZED
TEST PRACTICE
Y
DOWEL
"
EXAMPLE 4
35. CHEMISTRY In your chemistry lab, you have a bottle of 1% hydrochloric
acid solution and a bottle of 5% hydrochloric acid solution. You need
100 milliliters of a 3% hydrochloric acid solution for an experiment. How
many milliliters of each solution do you need to mix together?
on p. 438
for Ex. 35
36. MONEY Laura has $4.50 in dimes and quarters. She has 3 more dimes
than quarters. How many quarters does she have?
37.
★ SHORT RESPONSE A gazelle can run 73 feet per second for several
minutes. A cheetah can run 88 feet per second, but it can sustain this
speed for only 20 seconds. A gazelle is 350 feet from a cheetah when
both animals start running. Can the gazelle stay ahead of the cheetah?
Explain.
"MHFCSB
at classzone.com
38. CHALLENGE A gardener needs 6 bushels of a potting medium of 40% peat
moss and 60% vermiculite. He decides to add 100% vermiculite to his
current potting medium that is 50% peat moss and 50% vermiculite. The
gardener has 5 bushels of the 50% peat moss and 50% vermiculite mix.
Does he have enough of the 50% peat moss and 50% vermiculite mix to
make 6 bushels of the 40% peat moss and 60% vermiculite mix? Explain.
MIXED REVIEW
Solve the proportion. Check your solution.
40. } 5 } (p. 162)
3z
12
41. }
5}
(p. 162)
5
35
42. }
5}
(p. 168)
q
26
3
43. }
5}
(p. 168)
s
4
44. }
5}
(p. 168)
7
21
3
PREVIEW
Prepare for
Lesson 7.3
in Exs. 45–50.
y
30
x
3
39. }
5}
(p. 162)
2r
9
10
16
16
3
4
s22
Write two equations in standard form that are equivalent to the
given equation. (p. 311)
45. x 2 4y 5 0
46. 23x 1 9y 5 6
47. 27x 2 y 5 1
48. 5x 2 10y 5 5
49. 22x 2 12y 5 8
50. 6x 1 15y 5 23
QUIZ for Lessons 7.1–7.2
Solve the linear system by graphing. Check your solution. (p. 427)
1. x 1 y 5 22
2x 1 y 5 6
2. x 2 y 5 0
3. x 2 2y 5 12
5x 1 2y 5 27
23x 1 y 5 21
Solve the linear system using substitution. (p. 435)
4. y 5 x 2 4
5. y 5 4 2 3x
6. x 5 y 1 9
7. 2y 1 x 5 24
8. 5x 2 4y 5 27
9. 3x 2 5y 5 13
22x 1 y 5 18
y 2 x 5 25
5x 2 y 5 22
22x 1 y 5 3
5x 2 3y 5 7
x 1 4y 5 10
ONLINE QUIZ at classzone.com
441
Using
ALTERNATIVE METHODS
LESSON 7.2
Another Way to Solve Example 3, page 437
MULTIPLE REPRESENTATIONS In Example 3 on page 437, you saw how
to solve the problem about website hosting by solving a linear system
algebraically. You can also solve the problem using a table.
PROBLEM
WEBSITES Many businesses pay website hosting companies to store
and maintain the computer files that make up their websites. Internet
service providers also offer website hosting. The costs for website
hosting offered by a website hosting company and an Internet service
provider are shown in the table. Find the number of months after which
the total cost for website hosting will be the same for both companies.
Company
Internet service provider
Website hosting company
METHOD
Set-up fee
Cost per month
$10
$21.95
None
$22.45
Making a Table An alternative approach is to make a table.
STEP 1 Make a table for the total
cost of website hosting for
both companies.
Include the set-up fee in the
cost for the first month.
STEP 2 Look for the month in
which the total cost of the
service from the Internet
service provider and the
website hosting company
is the same. This happens
after 20 months.
Months
Internet
service
provider
Website
hosting
company
1
$31.95
$22.45
2
$53.90
$44.90
3
$75.85
$67.35
A
A
A
19
$427.05
$426.55
20
$449.00
$449.00
21
$470.95
$471.45
P R AC T I C E
1. TAXIS A taxi company charges $2.80 for the
first mile and $1.60 for each additional mile.
Another taxi company charges $3.20 for the
first mile and $1.50 for each additional mile.
After how many miles will each taxi cost the
same? Use a table to solve the problem.
442
2. SCHOOL PLAY An adult ticket to a school
play costs $5 and a student ticket costs $3. A
total of $460 was collected from the sale of
120 tickets. How many student tickets were
purchased? Solve the problem using algebra.
Then use a table to check your answer.
Investigating
g
g
Algebra
ACTIVITY
Use before Lesson 7.3
7.3 Linear Systems and Elimination
M AT E R I A L S • algebra tiles
QUESTION
How can you solve a linear system using algebra tiles?
You can use the following algebra tiles to model equations.
1-tiles
x-tiles
1 2
EXPLORE
1
y-tiles
2
1
2
Solve a linear system using algebra tiles.
3x 2 y 5 5
x1y53
Equation 1
Equation 2
STEP 1 Model equations
1
Model each equation using algebra
tiles. Arrange the algebra tiles so that
one equation is directly below the other
equation.
1
2
5
1
1
51 1 1
2
1
111
51 1
111
STEP 2 Add equations
Combine the two equations to form one
equation. Notice that the new equation has
one positive y-tile and one negative y-tile.
The y-tiles can be removed because the pair
of y-tiles has a value of 0.
1
1
1
1
1
11
1
11
5
1
11
1
11
STEP 3 Solve for x
Divide the remaining tiles into four equal
groups. Each x-tile is equal to two 1-tiles.
So, x 5 2.
STEP 4 Solve for y
To find the value of y, use the model for Equation 2.
Because x 5 2, you can replace the x-tile with two
1-tiles. Solve the new equation for y. So y 5 1, and
the solution of the system is (2, 1).
DR AW CONCLUSIONS
111
11
1
11
1
51 1 1
Use your observations to complete these exercises
Use algebra tiles to model and solve the linear system.
1. x 1 3y 5 8
4x 2 3y 5 2
2. 2x 1 y 5 5
22x 1 3y 5 7
3. 5x 2 2y 5 22
x 1 2y 5 14
4. x 1 2y 5 3
2x 1 3y 5 2
5. REASONING Is it possible to solve the linear system 3x 2 2y 5 6 and
2x 1 y 5 11 using the steps shown above? Explain your reasoning.
443
7.3
Before
Now
Why?
Key Vocabulary
equations, p. 427
Solve Linear Systems by
Adding or Subtracting
You solved linear systems by graphing and using substitution.
You will solve linear systems using elimination.
So you can solve a problem about arranging flowers, as in Ex. 42.
When solving a linear system, you can sometimes add or subtract the
equations to obtain a new equation in one variable. This method is called
elimination.
For Your Notebook
KEY CONCEPT
Solving a Linear System Using the Elimination Method
STEP 1 Add or subtract the equations to eliminate one variable.
STEP 2 Solve the resulting equation for the other variable.
STEP 3 Substitute in either original equation to find the value of the
eliminated variable.
EXAMPLE 1
Use addition to eliminate a variable
2x 1 3y 5 11
22x 1 5y 5 13
Equation 1
Equation 2
Solution
ADD EQUATIONS
When the coefficients
of one variable are
opposites, add the
equations to eliminate
the variable.
STEP 1 Add the equations to
eliminate one variable.
2x 1 3y 5 11
22x 1 5y 5 13
8y 5 24
STEP 2 Solve for y.
y53
STEP 3 Substitute 3 for y in either equation and solve for x.
2x 1 3y 5 11
Write Equation 1.
2x 1 3(3) 5 11
Substitute 3 for y.
x51
Solve for x.
2x 1 3y 5 11
22x 1 5y 5 13
2(1) 1 3(3) 0 11
22(1) 1 5(3) 0 13
11 5 11 ✓
444
13 5 13 ✓
EXAMPLE 2
Use subtraction to eliminate a variable
4x 1 3y 5 2
5x 1 3y 5 22
Equation 1
Equation 2
Solution
SUBTRACT
EQUATIONS
When the coefficients
of one variable are
the same, subtract the
equations to eliminate
the variable.
STEP 1 Subtract the equations to
4x 1 3y 5 2
5x 1 3y 5 22
2x
STEP 2 Solve for x.
54
x 5 24
STEP 3 Substitute 24 for x in either equation and solve for y.
4x 1 3y 5 2
Write Equation 1.
4(24) 1 3y 5 2
Substitute 24 for x.
y56
Solve for y.
EXAMPLE 3
Arrange like terms
8x 2 4y 5 24
4y 5 3x 1 14
Equation 1
Equation 2
Solution
STEP 1 Rewrite Equation 2 so that the like terms are arranged in columns.
AVOID ERRORS
Make sure that the
equal signs are in the
same column, just as
the like terms are.
8x 2 4y 5 24
4y 5 3x 1 14
8x 2 4y 5 24
23x 1 4y 5 14
5x
STEP 2 Add the equations.
5 10
x52
STEP 3 Solve for x.
STEP 4 Substitute 2 for x in either equation and solve for y.
4y 5 3x 1 14
Write Equation 2.
4y 5 3(2) 1 14
Substitute 2 for x.
y55
Solve for y.
✓
GUIDED PRACTICE
for Examples 1, 2, and 3
Solve the linear system.
1. 4x 2 3y 5 5
2. 25x 2 6y 5 8
3. 6x 2 4y 5 14
4. 7x 2 2y 5 5
5. 3x 1 4y 5 26
6. 2x 1 5y 5 12
22x 1 3y 5 27
7x 2 3y 5 4
5x 1 2y 5 4
2y 5 3x 1 6
23x 1 4y 5 1
5y 5 4x 1 6
445
EXAMPLE 4
Write and solve a linear system
KAYAKING During a kayaking trip, a kayaker travels 12 miles upstream
(against the current) and 12 miles downstream (with the current), as shown.
The speed of the current remained constant during the trip. Find the average
speed of the kayak in still water and the speed of the current.
Upstream: 3 hours
DIRECTION OF CURRENT
Downstream: 2 hours
STEP 1 Write a system of equations. First find the speed of the kayak
going upstream and the speed of the kayak going downstream.
Upstream: d 5 rt
Downstream: d 5 rt
12 5 r p 3
12 5 r p 2
45r
65r
Use the speeds to write a linear system. Let x be the average speed
of the kayak in still water, and let y be the speed of the current.
COMBINE SPEEDS
When you go upstream,
the speed at which
you can travel in still
water is decreased
by the speed of the
current. The opposite
is true when you go
downstream.
Equation 1: Going upstream
Speed of kayak
in still water
2
Speed of
current
5
Speed of kayak
going upstream
x
2
y
5
4
Equation 2: Going downstream
Speed of kayak
in still water
1
Speed of
current
5
Speed of kayak
going downstream
x
1
y
5
6
STEP 2 Solve the system of equations.
x 2y54
x 1y56
2x
5 10
x55
Write Equation 1.
Write Equation 2.
Add equations.
Solve for x.
Substitute 5 for x in Equation 2 and solve for y.
51y56
y51
Substitute 5 for x in Equation 2.
c The average speed of the kayak in still water is 5 miles per hour, and the
speed of the current is 1 mile per hour.
"MHFCSB
446
at classzone.com
✓
GUIDED PRACTICE
for Example 4
7. WHAT IF? In Example 4, suppose it takes the kayaker 5 hours to travel
10 miles upstream and 2 hours to travel 10 miles downstream. The speed
of the current remains constant during the trip. Find the average speed of
the kayak in still water and the speed of the current.
7.3
EXERCISES
HOMEWORK
KEY
Exs. 2, 15, 22, 36, and 44
Ex. 42
SKILL PRACTICE
1. VOCABULARY Give an example of a linear system in two variables that
can be solved by first adding the equations to eliminate one variable.
2.
EXAMPLE 1
on p. 444
for Exs. 3–8
★ WRITING Explain how to solve
the linear system shown using the
elimination method.
on p. 445
for Exs. 9–15
Equation 1
2x 1 3y 5 22
Equation 2
USING ADDITION Solve the linear system using elimination.
3. x 1 2y 5 13
4. 9x 1 y 5 2
5. 23x 2 y 5 8
6. 3x 2 y 5 30
7. 29x 1 4y 5 217
8. 23x 2 5y 5 27
2x 1 y 5 5
23x 1 7y 5 6
EXAMPLE 2
2x 2 y 5 2
24x 2 y 5 217
9x 2 6y 5 3
7x 1 y 5 212
24x 1 5y 5 14
USING SUBTRACTION Solve the linear system using elimination.
9. x 1 y 5 1
22x 1 y 5 4
12. 6x 1 y 5 210
5x 1 y 5 210
15.
10. x 2 y 5 24
11. 2x 2 y 5 7
13. 5x 1 6y 5 50
14. 4x 2 9y 5 221
x 1 3y 5 4
2x 1 6y 5 26
2x 1 7y 5 31
4x 1 3y 5 29
★
MULTIPLE CHOICE Which ordered pair is a solution of the linear
system 4x 1 9y 5 22 and 11x 1 9y 5 26?
A (22, 4)
B (2, 24)
C (4, 22)
D (4, 2)
EXAMPLE 3
ARRANGING LIKE TERMS Solve the linear system using elimination.
on p. 445
for Exs. 16–22
16. 2x 2 y 5 32
17. 28y 1 6x 5 36
18. 2x 2 y 5 211
19. 2x 2 y 5 14
20. 11y 2 3x 5 18
21. 25x 1 y 5 223
y 2 5x 5 13
x 5 5y 2 38
22.
6x 2 y 5 15
23x 5 216y 1 33
y 5 22x 2 13
2y 5 3x 2 9
★
MULTIPLE CHOICE Which ordered pair is a solution of the linear
system 2x 1 y 5 10 and 3y 5 2x 1 6?
A (23, 24)
B (3, 4)
C (24, 3)
D (4, 3)
447
ERROR ANALYSIS Describe and correct the error in finding the value of one
of the variables in the given linear system.
23. 5x 2 7y 5 16
24. 3x 2 2y 5 23
2x 2 7y 5 8
5y 5 60 2 3x
3x 2 2y 5 23
23x 1 5y 5 60
5x 2 7y 5 16
2x 2 7y 5 8
3y 5 57
y 5 19
5 24
x56
4x
SOLVING LINEAR SYSTEMS Solve the linear system using elimination.
1
25. 2x 1 }
y 5 219
2
1
26. }
x2}
y57
1
27. 8x 2 }
y 5 238
28. 5.2x 1 3.5y 5 54
29. 1.3x 2 3y 5 217.6
30. 22.6x 2 3.2y 5 4.8
2
4
31. }
x1}
y 5 14
32. 2.7x 1 1.5y 5 36
33. 4 2 4.8x 5 1.7y
2
3
4
1
2
}x 1 }y 5 24
2
3
x 2 y 5 12
23.6x 1 3.5y 5 10
21.3x 1 4.5y 5 25.1
5
5
2
1
}y 1 }x 5 11
5
5
3.5y 5 2.7x 2 6
2
1
1
}x 2 }y 5 27
4
2
1.9x 2 3.2y 5 24.2
12.8 1 1.7y 5 213.2x
34. WRITING AN EQUATION OF A LINE Use the following steps to write an
equation of the line that passes through the points (1, 2) and (24, 12).
a. Write a system of linear equations by substituting 1 for x and 2 for y
in y 5 mx 1 b and 24 for x and 12 for y in y 5 mx 1 b.
b. Solve the system of linear equations from part (a). What is the slope
of the line? What is the y-intercept?
c. Write an equation of the line that passes through (1, 2) and (24, 12).
35.
36.
GEOMETRY The rectangle has a perimeter
P of 14 feet, and twice its length l is equal to 1
less than 4 times its width w. Write and solve
a system of linear equations to find the length
and the width of the rectangle.
P 5 14 ft
w
l
★
SHORT RESPONSE Find the solution of the system of linear equations
below. Explain your steps.
x 1 3y 5 8
x 2 6y 5 219
5x 2 3y 5 214
Equation 1
Equation 2
Equation 3
37. CHALLENGE For a Þ 0, what is the solution of the system ax 1 2y 5 4
and ax 2 3y 5 26?
38. CHALLENGE Solve for x, y, and z in the system of equations below.
Explain your steps.
x 1 7y 1 3z 5 29
3z 1 x 2 2y 5 27
5y 5 10 2 2x
448
on p. WS1
Equation 1
Equation 2
Equation 3
★ 5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
PROBLEM SOLVING
EXAMPLE 4
39. ROWING During a practice, a 4 person crew team rows a rowing shell
upstream (against the current) and then rows the same distance
downstream (with the current). The shell moves upstream at a speed
of 4.3 meters per second and downstream at a speed of 4.9 meters per
second. The speed of the current remains constant. Use the models
below to write and solve a system of equations to find the average speed
of the shell in still water and the speed of the current.
on p. 446
for Exs. 39–41
Upstream
Speed of shell
in still water
2
Speed of
current
5
Speed
of shell
1
Speed of
current
5
Speed
of shell
Downstream
Speed of shell
in still water
40. OIL CHANGE Two cars get an oil change at the same service center. Each
customer is charged a fee x (in dollars) for the oil change plus y dollars
per quart of oil used. The oil change for the car that requires 5 quarts of
oil costs $22.45. The oil change for the car that requires 7 quarts of oil
costs $25.45. Find the fee and the cost per quart of oil.
41. PHONES Cellular phone ring tones can be monophonic or polyphonic.
Monophonic ring tones play one tone at a time, and polyphonic ring
tones play multiple tones at a time. The table shows the ring tones
downloaded from a website by two customers. Use the information
to find the cost of a monophonic ring tone and a polyphonic ring
tone, assuming that all monophonic ring tones cost the same and all
polyphonic ring tones cost the same.
Monophonic
ring tones
Polyphonic
ring tones
Julie
3
2
12.85
Tate
1
2
8.95
Customer
42.
Total cost
(dollars)
MULTIPLE REPRESENTATIONS For a floral arrangement class, Alicia
has to create an arrangement of twigs and flowers that has a total of
9 objects. She has to pay for the twigs and flowers that she uses in her
arrangement. Each twig costs $1, and each flower costs $3.
a. Writing a System Alicia spends $15 on the twigs and flowers. Write
and solve a linear system to find the number of twigs and the number
of flowers she used.
b. Making a Table Make a table showing the number of twigs in the
arrangement and the total cost of the arrangement when the number
of flowers purchased is 0, 1, 2, 3, 4, or 5. Use the table to check your
answer to part (a).
449
43. MULTI-STEP PROBLEM On a typical day with light winds, the 1800 mile
flight from Charlotte, North Carolina, to Phoenix, Arizona, takes longer
than the return trip because the plane has to fly into the wind.
MI
#HARLOTTE.#
0HOENIX!:
a. The flight from Charlotte to Phoenix is 4 hours 30 minutes long, and
the flight from Phoenix to Charlotte is 4 hours long. Find the average
speed (in miles per hour) of the airplane on the way to Phoenix and
on the return trip to Charlotte.
b. Let s be the speed (in miles per hour) of the plane with no wind, and
let w be the speed (in miles per hour) of the wind. Use your answer
to part (a) to write and solve a system of equations to find the speed
of the plane with no wind and the speed of the wind.
44.
★
SHORT RESPONSE The students in the graduating classes at the three
high schools in a school district have to pay for their caps and gowns.
A cap-and-gown set costs x dollars, and an extra tassel costs y dollars.
At one high school, students pay $3262 for 215 cap-and-gown sets and
72 extra tassels. At another high school, students pay $3346 for 221 capand-gown sets and 72 extra tassels. How much will students at the third
high school pay for 218 cap-and-gown sets and 56 extra tassels? Explain.
45. CHALLENGE A clothing manufacturer makes men’s dress shirts. For the
production process, an ideal sleeve length x (in centimeters) for each
shirt size and an allowable deviation y (in centimeters) from the ideal
length are established. The deviation is expressed as 6y. For a specific
shirt size, the minimum allowable sleeve length is 62.2 centimeters and
the maximum allowable sleeve length is 64.8 centimeters. Find the ideal
sleeve length and the allowable deviation.
MIXED REVIEW
Graph the equation.
46. x 2 5y 5 212 (p. 225)
47. 22x 1 3y 5 215 (p. 225)
48. y 1 9 5 2(x 1 2) (p. 302)
2
49. y 2 4 5 }
(x 1 1) (p. 302)
3
Solve the linear system by graphing. Check your solution. (p. 427)
50. y 5 x 2 3
PREVIEW
Prepare for
Lesson 7.4 in
Exs. 53–55.
450
y 5 2x 1 1
51. y 5 2
2x 2 3y 5 6
52. 2x 1 y 5 6
6x 2 2y 5 212
Find the least common multiple of the pair of numbers. (p. 910)
53. 9, 12
54. 18, 24
Chapter 7EXTRA
SystemsPRACTICE
of Equationsfor
and
Inequalities
Lesson
7.3, p. 944
55. 15, 20
7.4
Before
by Multiplying First
You solved linear systems by adding or subtracting.
Now
You will solve linear systems by multiplying first.
Why
So you can solve a problem about preparing food, as in Ex. 39.
Key Vocabulary
• least common
multiple, p. 910
In a linear system like the one below, neither variable can be eliminated by
adding or subtracting the equations. For systems like these, you can multiply
one or both of the equations by a constant so that adding or subtracting the
equations will eliminate one variable.
5x 1 2y 5 16
The new system is
equivalent to the
original system.
10x 1 4y 5 32
32
3x 2 4y 5 20
3x 2 4y 5 20
EXAMPLE 1
Multiply one equation, then add
6x 1 5y 5 19
2x 1 3y 5 5
Equation 1
Equation 2
Solution
STEP 1 Multiply Equation 2 by 23 so that the coefficients of x are opposites.
ANOTHER WAY
You can also multiply
Equation 2 by 3 and
subtract the equations.
6x 1 5y 5 19
2x 1 3y 5 5
6x 1 5y 5 19
26x 2 9y 5 215
3 (23)
24y 5 4
y 5 21
STEP 3 Solve for y.
STEP 4 Substitute 21 for y in either of the original equations and solve
for x.
2x 1 3y 5 5
2x 1 3(21) 5 5
2x 1 (23) 5 5
2x 5 8
x54
Write Equation 2.
Substitute 21 for y.
Multiply.
Equation 1
Equation 2
6x 1 5y 5 19
2x 1 3y 5 5
6(4) 1 5(21) 0 19
2(4) 1 3(21) 0 5
19 5 19 ✓
555✓
451
MULTIPLYING BOTH EQUATIONS To eliminate one variable when adding or
subtracting equations in a linear system, you may need to multiply both
equations by constants. Use the least common multiple of the coefficients of
one of the variables to determine the constants.
2x 2 9y 5 1
34
8x 2 36y 5 4
7x 2 12y 5 23
33
21x 2 36y 5 69
The least common multiple of 29 and 212 is 236.
EXAMPLE 2
Multiply both equations, then subtract
4x 1 5y 5 35
2y 5 3x 2 9
Equation 1
Equation 2
Solution
STEP 1 Arrange the equations so that like terms are in columns.
4x 1 5y 5 35
Write Equation 1.
23x 1 2y 5 29
ANOTHER WAY
You can also multiply
Equation 1 by 3 and
Equation 2 by 4.
Then add the revised
equations to eliminate x.
Rewrite Equation 2.
STEP 2 Multiply Equation 1 by 2 and Equation 2 by 5 so that the coeffcient
of y in each equation is the least common multiple of 5 and 2, or 10.
4x 1 5y 5 35
32
8x 1 10y 5 70
23x 1 2y 5 29
35
215x 1 10y 5 245
STEP 3 Subtract the equations.
5 115
23x
x55
STEP 4 Solve for x.
STEP 5 Substitute 5 for x in either of the original equations and solve for y.
4x 1 5y 5 35
Write Equation 1.
4(5) 1 5y 5 35
Substitute 5 for x.
y53
Solve for y.
Equation 1
Equation 2
4x 1 5y 5 35
2y 5 3x 2 9
4(5) 1 5(3) 0 35
2(3) 0 3(5) 2 9
35 5 35 ✓
"MHFCSB
✓
656✓
at classzone.com
GUIDED PRACTICE
Solve the linear system using elimination.
1. 6x 2 2y 5 1
22x 1 3y 5 25
452
2. 2x 1 5y 5 3
3x 1 10y 5 23
3. 3x 2 7y 5 5
9y 5 5x 1 5
★
EXAMPLE 3
Darlene is making a quilt that has alternating stripes of regular quilting
fabric and sateen fabric. She spends $76 on a total of 16 yards of the
two fabrics at a fabric store. Which system of equations can be used to
find the amount x (in yards) of regular quilting fabric and the amount
y (in yards) of sateen fabric she purchased?
Sateen fabric costs $ 6 per yard.
Quilting fabric costs $4 per yard.
ELIMINATE CHOICES
You can eliminate
choice A because x + y
cannot equal both 16
and 76.
A x 1 y 5 16
x 1 y 5 76
B x 1 y 5 16
4x 1 6y 5 76
C x 1 y 5 76
4x 1 6y 5 16
D x 1 y 5 16
6x 1 4y 5 76
Solution
Write a system of equations where x is the number of yards of regular quilting
fabric purchased and y is the number of yards of sateen fabric purchased.
Equation 1: Amount of fabric
Amount
of quilting
fabric
1
Amount
of sateen
fabric
5
Total
yards of
fabric
x
1
y
5
16
Equation 2: Cost of fabric
Quilting
fabric price
(dollars/yd)
4
p
Amount
of quilting
fabric (yd)
1
Sateen
fabric price
p
x
1
6
The system of equations is:
c The correct answer is B.
✓
GUIDED PRACTICE
(dollars/yd)
x 1 y 5 16
4x 1 6y 5 76
p
Amount
of sateen
fabric (yd)
5
p
y
5
Total
cost
(dollars)
76
Equation 1
Equation 2
A B C D
for Example 3
4. SOCCER A sports equipment store is having a sale on soccer balls. A
soccer coach purchases 10 soccer balls and 2 soccer ball bags for $155.
Another soccer coach purchases 12 soccer balls and 3 soccer ball bags
for $189. Find the cost of a soccer ball and the cost of a soccer ball bag.
453
For Your Notebook
CONCEPT SUMMARY
Methods for Solving Linear Systems
Method
Example
Table (p. 426)
When to Use
x
y 5 2x
y 5 3x 2 1
0
0
21
1
2
2
2
4
5
Graphing (p. 427)
When x-values are integers, so that
equal values can be seen in the table
When you want to see the lines that the
equations represent
y
3x 2 2y 5 2
x1y54
1
1
x
Substitution (p. 435)
y 5 4 2 2x
4x 1 2y 5 8
When one equation is already solved
for x or y
Addition (p. 444)
4x 1 7y = 15
6x 2 7y = 5
When the coefficients of one variable
are opposites
Subtraction (p. 445)
3x 1 5y = 213
3x 1 y = 25
When the coefficients of one variable
are the same
Multiplication (p. 451)
9x 1 2y = 38
3x 2 5y = 7
When no corresponding coefficients are
the same or opposites
7.4
EXERCISES
HOMEWORK
KEY
Exs. 2, 18, 34, 41, and 42
Ex. 40
SKILL PRACTICE
1. VOCABULARY What is the least common multiple of 12 and 18?
2.
EXAMPLE 1
on p. 451
for Exs. 3–8
454
★ WRITING Explain how to solve the linear
2x 2 3y 5 24
Equation 1
system using the elimination method.
7x 1 9y 5 25
Equation 2
3. x 1 y 5 2
2x 1 7y 5 9
4. 3x 2 2y 5 3
2x 1 y 5 1
5. 4x 1 3y 5 8
6. 10x 2 9y 5 46
22x 1 3y 5 10
7. 8x 2 5y 5 11
4x 2 3y 5 5
8. 11x 2 20y 5 28
x 2 2y 5 13
3x 1 4y 5 36
EXAMPLE 2
on p. 452 for
Exs. 9–20
9. 4x 2 3y 5 8
10. 22x 2 5y 5 9
11. 7x 2 6y 5 21
12. 7x 1 3y 5 212
13. 9x 2 8y 5 4
14. 12x 2 7y 5 22
15. 9x 1 2y 5 39
16. 27x 1 10y 5 11
17. 214x 1 15y 5 15
5x 2 2y 5 211
3x 1 11y 5 4
2x 1 5y 5 38
2x 2 3y 5 24
6x 1 13y 5 29
18.
5x 2 4y 5 1
28x 1 11y 5 14
28x 1 15y 5 34
21x 2 20y 5 210
15x 1 8y 5 6 and 25x 1 12y 5 14?
A (23, 22)
B (23, 2)
C (22, 23)
D (2, 23)
ERROR ANALYSIS Describe and correct the error when solving the linear
system.
19.
20.
2x 2 3y 5 29 32 4x 2 6y 5 218
5x 2 6y 5 29
5x 2 6y 5 29
9x
5 227
9x 1 8y 5 11 33 27x 1 24y 5 11
7x 1 6y 5 9
28x 1 24y 5 9
34
2x
52
x 5 23
x 5 22
SOLVING LINEAR SYSTEMS Solve the linear system using any algebraic
method.
21. 3x 1 2y 5 4
22. 4x 2 5y 5 18
23. 8x 2 9y 5 215
24. 0.3x 1 0.1y 5 20.1
25. 4.4x 2 3.6y 5 7.6
26. 3x 2 2y 5 220
27. 0.2x 2 1.5y 5 21
28. 1.5x 2 3.5y 5 25
21.2x 1 2.5y 5 1
29. 4.9x 1 2.4y 5 7.4
30. x 1 y 5 0
1
31. 3x 1 y 5 }
32. }x 2 }y 5 23
2y 5 8 2 5x
2x 1 y 5 3
x 2 4.5y 5 1
1
52
2
1
2
}x 2 }y
33.
3x 5 y 1 11
24x 5 19 1 y
x2y51
x 1 1.2y 5 6.4
0.7x 1 3.6y 5 22.2
3
8
2x 2 3y 5 }
3
5
3
4
2
5
1
3
}x 1 }y 5 8
3
GEOMETRY A rectangle has a perimeter of 18 inches.
A new rectangle is formed by doubling the width w and
tripling the length l, as shown. The new rectangle has a
perimeter P of 46 inches.
a. Write and solve a system of linear equations to find
P 5 46 in.
the length and width of the original rectangle.
2w
3l
b. Find the length and width of the new rectangle.
34.
★ WRITING For which values of a can you solve the linear system
ax 1 3y 5 2 and 4x 1 5y 5 6 without multiplying first? Explain.
CHALLENGE Find the values of a and b so that the
linear system has the given solution.
35. (4, 2)
ax 2 by 5 4
Equation 1
bx 2 ay 5 10
Equation 2
36. (2, 1)
455
PROBLEM SOLVING
EXAMPLE 3
37. BOOK SALE A library is having a book sale to raise money. Hardcover
books cost $4 each and paperback books cost $2 each. A person spends
$26 for 8 books. How many hardcover books did she purchase?
on p. 453
for Exs. 37–39
38. MUSIC A website allows users to download individual songs or an entire
album. All individual songs cost the same to download, and all albums cost
the same to download. Ryan pays $14.94 to download 5 individual songs and
1 album. Seth pays $22.95 to download 3 individual songs and 2 albums.
How much does the website charge to download a song? an entire album?
39. FARM PRODUCTS The table shows the number of apples needed to make
the apple pies and applesauce sold at a farm store. During a recent apple
picking at the farm, 169 Granny Smith apples and 95 Golden Delicious
apples were picked. How many apple pies and batches of applesauce can
be made if every apple is used?
Granny Smith
Golden Delicious
Needed for a pie
5
3
Needed for a batch of applesauce
4
2
Type of apple
40.
MULTIPLE REPRESENTATIONS Tickets for admission to a high school
football game cost $3 for students and $5 for adults. During one game,
$2995 was collected from the sale of 729 tickets.
a. Writing a System Write and solve a system of linear equations to find
the number of tickets sold to students and the number of tickets sold
to adults.
b. Drawing a Graph Graph the system of linear equations. Use the
graph to determine whether your answer to part (a) is reasonable.
41.
★ SHORT RESPONSE A dim sum
restaurant offers two sizes of dishes:
small and large. All small dishes
cost the same and all large dishes
cost the same. The bills show the
cost of the food before the tip is
included. What will 3 small and
2 large dishes cost before the tip
is included? Explain.
42.
456
★ OPEN – ENDED
Describe a
real-world problem that can be solved
using a system of linear equations. Then
solve the problem and explain what the
solution means in this situation.
Chapter 7 Systems
of Equations and Inequalities
on p. WS1
★ 5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
43. INVESTMENTS Matt invested $2000 in stocks and bonds. This year
the bonds paid 8% interest, and the stocks paid 6% in dividends. Matt
received a total of $144 in interest and dividends. How much money did
he invest in stocks? in bonds?
44. CHALLENGE You drive a car 45 miles at an average speed r (in miles per
hour) to reach your destination. Due to traffic, your average speed on the
3
4
return trip is }r. The round trip took a total of 1 hour 45 minutes. Find the
average speed for each leg of your trip.
MIXED REVIEW
Graph the equation. (pp. 215, 225, 244)
PREVIEW
Prepare for
Lesson 7.5 in
Exs. 51–52.
45. x 2 8y 5 210
46. 22x 1 5y 5 215
1
1
47. }
x1}
y58
2
48. y 5 }
x21
3
49. y 5 29x 1 2
50. y 2 1 5 2x 2 7
4
2
Determine which lines are parallel. (p. 244)
51.
a
y
b
52.
c
(2, 5)
c
(0, 3)
(3, 3)
2
(0, 1)
b
a (2, 2)
y
2
(6, 0)
(0, 1)
1
(6, 3)
(22, 21)
x
(0, 22)
(3, 1)
2
(5, 22.5)
x
Solve the linear system using any method. (pp. 427, 435, 444, 451)
53. y 5 4x 2 9
54. y 5 22x 1 20
55. x 1 2y 5 0
56. x 1 2y 5 2
57. 7x 2 8y 5 215
58. 8x 1 y 5 5
y 5 28x 1 15
2x 1 y 5 211
y 5 26x 1 40
2x 5 y 1 3
5x 1 8y 5 3
22x 1 y 5 0
Solve the linear system using elimination. (pp. 444, 451)
1. x 1 y 5 4
23x 1 y 5 28
2. 2x 2 y 5 2
6x 2 y 5 22
3. x 1 y 5 5
4. x 1 3y 5 210
5. x 1 3y 5 10
6. x 1 7y 5 10
7. 4x 2 y 5 22
8. x 1 3y 5 1
9. 3x 1 y 5 21
10. 2x 2 3y 5 25
5x 1 2y 5 16
11. 7x 1 2y 5 13
4x 1 3y 5 13
1
12. }
x 1 5y 5 23
2x 1 5y 5 230
3x 1 2y 5 7
3x 2 y 5 13
5x 1 6y 5 14
2x 1 y 5 23
x 1 2y 5 28
x1y51
3
2
2}
x 1 6y 5 210
3
457
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 7.1–7.4
1. MULTI-STEP PROBLEM Flying into the wind,
a helicopter takes 15 minutes to travel
15 kilometers. The return flight takes
12 minutes. The wind speed remains
constant during the trip.
a. Find the helicopter’s average speed (in
kilometers per hour) for each leg of the trip.
b. Write a system of linear equations that
represents the situation.
c. What is the helicopter’s average speed in
still air? What is the speed of the wind?
4. OPEN-ENDED Describe a real-world problem
that can be modeled by a linear system.
Then solve the system and interpret the
solution in the context of the problem.
5. SHORT RESPONSE A hot air balloon is
launched at Kirby Park, and it ascends at
a rate of 7200 feet per hour. At the same
time, a second hot air balloon is launched
at Newman Park, and it ascends at a rate of
4000 feet per hour. Both of the balloons stop
ascending after 30 minutes. The diagram
shows the altitude of each park. Are the hot
air balloons ever at the same height at the
same time? Explain.
,JSCZ1BSL
/FXNBO1BSL
FT
FT
2. SHORT RESPONSE At a grocery store, a
customer pays a total of $9.70 for 1.8 pounds
of potato salad and 1.4 pounds of coleslaw.
Another customer pays a total of $6.55 for
1 pound of potato salad and 1.2 pounds of
coleslaw. How much do 2 pounds of potato
salad and 2 pounds of coleslaw cost? Explain.
3. GRIDDED ANSWER During one day, two
computers are sold at a computer store. The
two customers each arrange payment plans
with the salesperson. The graph shows the
amount y of money (in dollars) paid for the
computers after x months. After how many
months will each customer have paid the
same amount?
Amount paid
(dollars)
y
458
.OTDRAWNTOSCALE
6. EXTENDED RESPONSE A chemist needs
500 milliliters of a 20% acid and 80% water
mix for a chemistry experiment. The chemist
combines x milliliters of a 10% acid and
90% water mix and y milliliters of a 30% acid
and 70% water mix to make the 20% acid and
80% water mix.
a. Write a linear system that represents the
situation.
b. How many milliliters of the 10% acid
and 90% water mix and the 30% acid and
70% water mix are combined to make the
20% acid and 80% water mix?
c. The chemist also needs 500 milliliters of
400
200
0
3%!,%6%,
0
2
4
6
8
10
Months since purchase
x
a 15% acid and 85% water mix. Does the
chemist need more of the 10% acid and
90% water mix than the 30% acid and
70% water mix to make this new mix?
Explain.
7.5
Before
Now
Why?
Key Vocabulary
• inconsistent system
• consistent
dependent system
equations, p. 427
• parallel, p. 244
Solve Special Types of
Linear Systems
You found the solution of a linear system.
You will identify the number of solutions of a linear system.
So you can compare distances traveled, as in Ex. 39.
A linear system can have no solution or infinitely many solutions. A linear
system has no solution when the graphs of the equations are parallel.
A linear system with no solution is called an inconsistent system.
A linear system has infinitely many solutions when the graphs of the
equations are the same line. A linear system with infinitely many solutions
is called a consistent dependent system.
EXAMPLE 1
A linear system with no solution
Show that the linear system has no solution.
3x 1 2y 5 10
3x 1 2y 5 2
Equation 1
Equation 2
Solution
METHOD 1 Graphing
REVIEW GRAPHING
For help with graphing
linear equations, see
pp. 215, 225, and 244.
Graph the linear system.
y
3x 1 2y 5 2
23
23
3x 1 2y 5 10
2
2
21
2
x
c The lines are parallel because they have the same slope but different
y-intercepts. Parallel lines do not intersect, so the system has no solution.
METHOD 2 Elimination
Subtract the equations.
IDENTIFY TYPES OF
SYSTEMS
The linear system in
Example 1 is called an
inconsistent system
because the lines do
not intersect (are not
consistent).
3x 1 2y 5 10
3x 1 2y 5 2
058
This is a false statement.
c The variables are eliminated and you are left with a false statement
regardless of the values of x and y. This tells you that the system has
no solution.
"MHFCSB
at classzone.com
459
EXAMPLE 2
A linear system with infinitely many solutions
Show that the linear system has infinitely many solutions.
x 2 2y 5 24
Equation 1
1
y5}
x12
Equation 2
2
Solution
METHOD 1 Graphing
y
x 2 2y 5 24
y 5 12 x 1 2
1
1
x
c The equations represent the same line, so any point on the line is a
solution. So, the linear system has infinitely many solutions.
METHOD 2 Substitution
IDENTIFY TYPES OF
SYSTEMS
The linear system in
Example 2 is called a
consistent dependent
system because the
lines intersect (are
consistent) and the
equations are equivalent
(are dependent).
✓
1
Substitute }
x 1 2 for y in Equation 1 and solve for x.
2
x 2 2y 5 24
1
x 2 21 }
x 1 2 2 5 24
2
24 5 24
Write Equation 1.
1
2
Substitute }x 1 2 for y.
Simplify.
c The variables are eliminated and you are left with a statement that is
true regardless of the values of x and y. This tells you that the system has
infinitely many solutions.
GUIDED PRACTICE
Tell whether the linear system has no solution or infinitely many solutions.
Explain.
1. 5x 1 3y 5 6
25x 2 3y 5 3
2. y 5 2x 2 4
26x 1 3y 5 212
IDENTIFYING THE NUMBER OF SOLUTIONS When the equations of a linear
system are written in slope-intercept form, you can identify the number of
solutions of the system by looking at the slopes and y-intercepts of the lines.
Number of solutions
Slopes and y-intercepts
One solution
Different slopes
No solution
Infinitely many solutions
460
Same slope
Different y-intercepts
Same slope
Same y-intercept
EXAMPLE 3
Identify the number of solutions
Without solving the linear system, tell whether the linear system has one
solution, no solution, or infinitely many solutions.
a. 5x 1 y 5 22
–10x 2 2y 5 4
b. 6x 1 2y 5 3
Equation 1
Equation 2
6x 1 2y 5 25
Equation 1
Equation 2
Solution
a. y 5 25x 2 2
Write Equation 1 in slope-intercept form.
y 5 25x 2 2
c Because the lines have the same slope and the same y-intercept,
the system has infinitely many solutions.
3
2
5
y 5 23x 2 }
2
b. y 5 23x 1 }
c Because the lines have the same slope but different y-intercepts,
the system has no solution.
EXAMPLE 4
Write and solve a system of linear equations
ART An artist wants to sell prints of her
paintings. She orders a set of prints for each of
two of her paintings. Each set contains regular
prints and glossy prints, as shown in the table.
Find the cost of one glossy print.
Regular
Glossy
Cost
45
30
$465
15
10
$155
Solution
STEP 1 Write a linear system. Let x be the cost (in dollars) of a regular print,
and let y be the cost (in dollars) of a glossy print.
45x 1 30y 5 465
15x 1 10y 5 155
Cost of prints for one painting
Cost of prints for other painting
STEP 2 Solve the linear system using elimination.
45x 1 30y 5 465
15x 1 10y 5 155
45x 1 30y 5 465
245x 2 30y 5 2465
3 (23)
050
c There are infinitely many solutions, so you cannot determine the cost of
one glossy print. You need more information.
✓
GUIDED PRACTICE
3. Without solving the linear system, tell
whether it has one solution, no solution,
or infinitely many solutions.
x 2 3y 5 215
2x 2 3y 5 218
Equation 1
Equation 2
4. WHAT IF? In Example 4, suppose a glossy print costs $3 more than a
regular print. Find the cost of a glossy print.
461
For Your Notebook
CONCEPT SUMMARY
Number of Solutions of a Linear System
One solution
No solution
y
y
x
7.5
y
x
x
The lines intersect.
The lines are parallel.
The lines coincide.
The lines have
different slopes.
The lines have the same
slope and different
y-intercepts.
The lines have the same
slope and the same
y-intercept.
EXERCISES
HOMEWORK
KEY
Exs. 3, 4, 24, 25, 32, 33, and 40
SKILL PRACTICE
1. VOCABULARY Copy and complete: A linear system with no solution is
called a(n) ? system.
2. VOCABULARY Copy and complete: A linear system with infinitely many
solutions is called a(n) ? system.
3.
★ WRITING Describe the graph of a linear system that has no solution.
4.
★ WRITING Describe the graph of a linear system that has infinitely
many solutions.
INTERPRETING GRAPHS Match the linear system with its graph. Then use
the graph to tell whether the linear system has one solution, no solution, or
5. x 2 3y 5 29
x 2 y 5 21
A.
6. x 2 y 5 24
7. x 1 3y 5 21
B.
C.
23x 1 3y 5 2
y
22x 2 6y 5 2
y
y
1
2
x
2
2
1
462
x
1
x
EXAMPLES
1 and 2
on pp. 459–460
for Exs. 8–25
INTERPRETING GRAPHS Graph the linear system. Then use the graph to tell
whether the linear system has one solution, no solution, or infinitely many
solutions.
8. x 1 y 5 22
y 5 2x 1 5
11. 22x 1 2y 5 216
3x 2 6y 5 30
9. 3x 2 4y 5 12
10. 3x 2 y 5 29
3x 1 5y 5 215
3
y 5 }x 2 3
4
12. 29x 1 6y 5 18
6x 2 4y 5 212
13. 23x 1 4y 5 12
23x 1 4y 5 24
14. ERROR ANALYSIS Describe and
correct the error in solving the
linear system below.
6x 1 y 5 36
5x 2 y 5 8
The lines do not
intersect, so there
is no solution.
y
1
x
2
SOLVING LINEAR SYSTEMS Solve the linear system using substitution or
elimination.
15. 2x 1 5y 5 14
16. 216x 1 2y 5 22
17. 3x 2 2y 5 25
18. 5x 2 5y 5 23
19. x 2 y 5 0
20. x 2 2y 5 7
21. 218x 1 6y 5 24
22. 4y 1 5x 5 15
23. 6x 1 3y 5 9
6x 1 7y 5 10
y 5 x 1 0.6
3x 2 y 5 22
24.
y 5 8x 2 1
5x 2 2y 5 6
x 5 8y 1 3
4x 1 5y 5 47
2x 1 2y 5 7
2x 1 9y 5 27
★ MULTIPLE CHOICE Which of the linear systems has exactly one
solution?
25.
A 2x 1 y 5 9
x2y59
B 2x 1 y 5 9
x 2 y 5 29
C 2x 1 y 5 9
2x 2 y 5 9
D x 2 y 5 29
2x 1 y 5 29
★ MULTIPLE CHOICE Which of the linear systems has infinitely many
solutions?
EXAMPLE 3
on p. 461
for Exs. 26–31
A 15x 1 5y 5 20
6x 2 2y 5 8
B 15x 2 5y 5 20
6x 2 2y 5 28
C 15x 2 5y 5 220
6x 2 2y 5 8
D 15x 2 5y 5 20
6x 2 2y 5 8
IDENTIFYING THE NUMBER OF SOLUTIONS Without solving the linear
system, tell whether the linear system has one solution, no solution, or
26. y 5 26x 2 2
27. y 5 7x 1 13
28. 4x 1 3y 5 27
29. 9x 2 15y 5 24
30. 0.3x 1 0.4y 5 2.4
31. 0.9x 2 2.1y 5 12.3
12x 1 2y 5 26
6x 2 10y 5 16
221x 1 3y 5 39
0.5x 2 0.6y 5 0.2
4x 2 3y 5 227
1.5x 2 3.5y 5 20.5
463
32.
★ OPEN – ENDED
33.
★ OPEN – ENDED
Write a linear system so that it has infinitely many
solutions, and one of the equations is y 5 3x 1 2.
Write a linear system so that it has no solution and
one of the equations is 7x 2 8y 5 29.
34. REASONING Give a counterexample for the following statement:
If the graphs of the equations of a linear system have the same slope,
then the linear system has no solution.
35. CHALLENGE Find values of p, q, and r that produce the solution(s).
a. No solution
px 1 qy 5 r
2x 2 3y 5 5
b. Infinitely many solutions
Equation 1
Equation 2
c. One solution of (4, 1)
PROBLEM SOLVING
EXAMPLE 4
on p. 461
for Exs. 36–38
36. RECREATION One admission to a roller skating rink costs x dollars and
renting a pair of skates costs y dollars. A group pays $243 for admission
for 36 people and 21 skate rentals. Another group pays $81 for admission
for 12 people and 7 skate rentals. Is there enough information to
determine the cost of one admission to the roller skating rink? Explain.
37. TRANSPORTATION A passenger train travels from New York City to
Washington, D.C., then back to New York City. The table shows the
number of coach tickets and business class tickets purchased for each
leg of the trip. Is there enough information to determine the cost of one
coach ticket? Explain.
Coach
tickets
Business class
tickets
Money collected
(dollars)
Washington, D.C.
150
80
22,860
New York City
170
100
27,280
Destination
38. PHOTOGRAPHY In addition to taking pictures on your digital camera,
you can record 30 second movies. All pictures use the same amount of
memory, and all 30 second movies use the same amount of memory. The
number of pictures and 30 second movies on 2 memory cards is shown.
a. Is there enough information given
to determine the amount of memory
used by a 30 second movie? Explain.
b. Given that a 30 second movie uses
50 times the amount of memory
that a digital picture uses, can you
determine the amount of memory
used by a 30 second movie? Explain.
464
on p. WS1
Size of card
(megabytes)
64
256
Pictures
450
1800
Movies
7
28
★ 5 STANDARDIZED
TEST PRACTICE
39. MULTI-STEP PROBLEM Two people are
training for a speed ice-climbing event.
During a practice climb, one climber starts
15 seconds after the first climber. The rates
that the climbers ascend are shown.
a. Let d be the distance (in feet) traveled by
a climber t seconds after the first person
starts climbing. Write a linear system that
models the situation.
b. Graph the linear system from part (a).
Does the second climber catch up to the
first climber? Explain.
40.
Climbs 10 feet
every 30 seconds
Climbs 5 feet
every 15 seconds
★ EXTENDED RESPONSE Two employees at a banquet facility are given
the task of folding napkins. One person starts folding napkins at a rate of
5 napkins per minute. The second person starts 10 minutes after the first
person and folds napkins at a rate of 4 napkins per minute.
a. Model Let y be the number of napkins folded x minutes after the first
person starts folding. Write a linear system that models the situation.
b. Solve Solve the linear system.
c. Interpret Does the solution of the linear system make sense in the
context of the problem? Explain.
41. CHALLENGE An airplane has an average air speed of 160 miles per
hour. The airplane takes 3 hours to travel with the wind from Salem to
Lancaster. The airplane has to travel against the wind on the return trip.
After 3 hours into the return trip, the airplane is 120 miles from Salem.
Find the distance from Salem to Lancaster. If the problem cannot be
solved with the information given, explain why.
MIXED REVIEW
Solve the equation, if possible. (p. 154)
42. 61 1 5c 5 7 2 4c
43. 3m 2 2 5 7m 2 50 1 8m
44. 11z 1 3 5 10(2z 1 3)
45. 26(1 2 w) 5 14(w 2 5)
PREVIEW
Solve the inequality. Then graph your solution.
Prepare for
Lesson 7.6 in
Exs. 46–61.
46. x 1 15 < 23 (p. 356)
47. x 2 1 ≥ 10 (p. 356)
x
48. }
> 22.5 (p. 363)
49. 27x ≤ 84 (p. 363)
50. 2 2 5x ≥ 27 (p. 369)
51. 3x 2 9 > 3(x 2 3) (p. 369)
52. 22 < x 1 3 ≤ 11 (p. 380)
53. 27 ≤ 5 2 2x ≤ 7 (p. 380)
54. ⏐x 2 3⏐ ≥ 5 (p. 398)
55. 2⏐2x 2 1⏐ 2 9 ≤ 1 (p. 398)
4
Graph the inequality. (p. 405)
56. x 1 y < 23
57. x 2 y ≥ 1
58. 2x 2 y < 5
59. 22x 2 3y ≤ 9
60. y ≤ 24
61. x > 6.5
465
7.6
Before
Solve Systems of
Linear Inequalities
You graphed linear inequalities in two variables.
Now
You will solve systems of linear inequalities in two variables.
Why
So you can find a marching band’s competition score, as in Ex. 36.
Key Vocabulary
inequalities
• solution of a
system of linear
inequalities
• graph of a system
of linear
inequalities
A system of linear inequalities in two variables, or simply a system of
inequalities, consists of two or more linear inequalities in the same
variables. An example is shown.
x2y>7
2x 1 y < 8
Inequality 1
Inequality 2
A solution of a system of linear inequalities is an ordered pair that is a
solution of each inequality in the system. For example, (6, 25) is a solution
of the system above. The graph of a system of linear inequalities is the
graph of all solutions of the system.
For Your Notebook
KEY CONCEPT
Graphing a System of Linear Inequalities
STEP 1 Graph each inequality (as you learned to do in Lesson 6.7).
STEP 2 Find the intersection of the half-planes. The graph of the system
is this intersection.
EXAMPLE 1
Graph a system of two linear inequalities
Graph the system of inequalities.
REVIEW GRAPHING
INEQUALITIES
For help with graphing
a linear inequality in
two variables, see
p. 405.
y > 2x 2 2
y ≤ 3x 1 6
Inequality 1
Inequality 2
Solution
Graph both inequalities in the same coordinate
plane. The graph of the system is the intersection
of the two half-planes, which is shown as the
darker shade of blue.
y
CHECK Choose a point in the dark blue region,
such as (0, 1). To check this solution, substitute
0 for x and 1 for y into each inequality.
1?
> 022
1?
≤ 016
1 > 22 ✓
"MHFCSB
466
1≤6✓
at classzone.com
1
(0, 1)
1
x
THE SOLUTION REGION In Example 1, the half-plane for each inequality is
shaded, and the solution region is the intersection of the half-planes. From
this point on, only the solution region will be shaded.
EXAMPLE 2
Graph a system of three linear inequalities
y ≥ 21
x > 22
x 1 2y ≤ 4
Graph the system of inequalities.
Inequality 1
Inequality 2
Inequality 3
Solution
Graph all three inequalities in the same coordinate plane. The graph of the
system is the triangular region shown.
y
The region is to
the right of the
line x 5 22.
The region is on
and below the
line x 1 2y 5 4.
1
1
The region is on
and above the
line y 5 21.
✓
GUIDED PRACTICE
x
Graph the system of linear inequalities.
1. y < x 2 4
2. y > 2x 1 2
y > 2x 1 3
EXAMPLE 3
y<4
x<3
3. y > 2x
y>x24
y<5
Write a system of linear inequalities
Write a system of inequalities for the shaded region.
REVIEW EQUATIONS
OF LINES
For help with writing an
equation of a line, see
pp. 283, 302, and 311.
y
Solution
INEQUALITY 1: One boundary line for the shaded
region is y 5 3. Because the shaded region is
above the solid line, the inequality is y ≥ 3.
2
1
x
INEQUALITY 2: Another boundary line for the
shaded region has a slope of 2 and a y-intercept
of 1. So, its equation is y 5 2x 1 1. Because the
shaded region is above the dashed line, the
inequality is y > 2x 1 1.
c The system of inequalities for the shaded region is:
y≥3
y > 2x 1 1
Inequality 1
Inequality 2
467
EXAMPLE 4
Write and solve a system of linear inequalities
BASEBALL The National Collegiate Athletic Association (NCAA)
regulates the lengths of aluminum baseball bats used by college
baseball teams. The NCAA states that the length (in inches) of
the bat minus the weight (in ounces) of the bat cannot exceed 3.
Bats can be purchased at lengths from 26 to 34 inches.
x
a. Write and graph a system of linear inequalities that
describes the information given above.
b. A sporting goods store sells an aluminum bat that is 31 inches
long and weighs 25 ounces. Use the graph to determine if
this bat can be used by a player on an NCAA team.
Solution
a. Let x be the length (in inches) of the bat, and let y be the weight
(in ounces) of the bat. From the given information, you can write
the following inequalities:
WRITING SYSTEMS
OF INEQUALITIES
Consider the values
of the variables when
writing a system of
inequalities. In many
real-world problems,
the values cannot be
negative.
x2y≤3
The difference of the bat’s length and weight can be at most 3.
x ≥ 26
The length of the bat must be at least 26 inches.
x ≤ 34
The length of the bat can be at most 34 inches.
y≥0
The weight of the bat cannot be a negative number.
Graph each inequality in the system. Then
y
identify the region that is common to all of
the graphs of the inequalities. This region
is shaded in the graph shown.
(31, 25)
b. Graph the point that represents a bat that
is 31 inches long and weighs 25 ounces.
c Because the point falls outside the
solution region, the bat cannot be used
by a player on an NCAA team.
✓
GUIDED PRACTICE
5
10
x
Write a system of inequalities that defines the shaded region.
4.
5.
y
y
1
2
x
1
1
x
6. WHAT IF? In Example 4, suppose a Senior League (ages 10–14) player
wants to buy the bat described in part (b). In Senior League, the length
(in inches) of the bat minus the weight (in ounces) of the bat cannot
exceed 8. Write and graph a system of inequalities to determine whether
the described bat can be used by the Senior League player.
468
7.6
EXERCISES
HOMEWORK
KEY
Exs. 2, 21, 22, 33, and 40
SKILL PRACTICE
1. VOCABULARY Copy and complete: A(n) ? of a system of linear
inequalities is an ordered pair that is a solution of each inequality in
the system.
2.
★ WRITING Describe the steps you would take to
x2y<7
y≥3
graph the system of inequalities shown.
Inequality 1
Inequality 2
CHECKING A SOLUTION Tell whether the ordered pair is a solution of the
system of inequalities.
3. (1, 1)
5. (3, 21)
4. (0, 6)
y
y
y
1
1
1
1
1
x
1
x
EXAMPLE 1
MATCHING SYSTEMS AND GRAPHS Match the system of inequalities with
on p. 466
for Exs. 6217
its graph.
6. x 2 4y > 28
x≥2
7. x 2 4y ≥ 28
x<2
8. x 2 4y > 28
A.
B.
C.
y
y≥2
y
1
y
1
1
1
x
x
1
1
x
x
GRAPHING A SYSTEM Graph the system of inequalities.
9. x > 25
10. y ≤ 10
y>x
12. y < 22x 1 3
13. y ≥ 0
14. y ≥ 2x 1 1
15. x < 8
16. y ≥ 22
17. y 2 2x < 7
19. x ≥ 0
20. x 1 y ≤ 10
y≥4
y < 2.5x 2 1
x 2 4y ≤ 28
EXAMPLE 2
11. x > 3
y≥6
x<2
2x 1 3y > 26
18. x < 4
21.
y 1 2x > 21
y≥0
6x 2 y < 12
y>1
y ≥ 2x 1 1
on p. 467
for Exs. 18221
y < 2x 1 4
x2y≥2
y≥2
★ MULTIPLE CHOICE Which ordered pair is a solution of the system
2x 2 y ≤ 5 and x 1 2y > 2?
A (1, 21)
B (4, 1)
C (2, 0)
D (3, 2)
469
EXAMPLE 2
22.
on p. 467
for Exs. 22–23
★ MULTIPLE CHOICE The graph of which
system of inequalities is shown?
A y < 2x
2x 1 3y < 6
B y < 2x
2x 1 3y > 6
C y > 2x
2x 1 3y < 6
D y > 2x
2x 1 3y > 6
y
1
1
x
23. ERROR ANALYSIS Describe and correct
y
the error in graphing this system of
inequalities:
x1y<3
x > 21
x≤3
Inequality 1
Inequality 2
Inequality 3
1
x
1
EXAMPLE 3
WRITING A SYSTEM Write a system of inequalities for the shaded region.
on p. 467
for Exs. 24–29
24.
25.
y
26.
y
y
1
1
2
1
x
1
27.
28.
y
29.
y
3
1
x
GRAPHING A SYSTEM Graph the system of inequalities.
30. x > 4
31. x 1 y < 4
x<9
y≤2
y > 22
33.
x 1 y > 22
x2y≤3
x 2 y ≥ 24
32. x ≤ 10
3x 1 2y ≥ 9
x 2 2y ≤ 6
x1y≤5
★ SHORT RESPONSE Does the system of inequalities have any solutions?
Explain.
x2y>5
x2y<1
Inequality 1
Inequality 2
CHALLENGE Write a system of inequalities for the shaded region described.
34. The shaded region is a rectangle with vertices at (2, 1), (2, 4), (6, 4), and
(6, 1).
35. The shaded region is a triangle with vertices at (23, 0), (3, 2), and (0, 22).
470
on p. WS1
1
x
y
21
x
x
x
1
3
1
★ 5 STANDARDIZED
TEST PRACTICE
PROBLEM SOLVING
EXAMPLE 4
36. COMPETITION SCORES In a marching band competition, scoring is based
on a musical evaluation and a visual evaluation. The musical evaluation
score cannot exceed 60 points, the visual evaluation score cannot exceed
40 points. Write and graph a system of inequalities for the scores that a
marching band can receive.
on p. 468
for Exs. 36–38
37. NUTRITION For a hiking trip, you are making a mix of x ounces of
peanuts and y ounces of chocolate pieces. You want the mix to have less
than 70 grams of fat and weigh less than 8 ounces. An ounce of peanuts
has 14 grams of fat, and an ounce of chocolate pieces has 7 grams of fat.
Write and graph a system of inequalities that models the situation.
38. FISHING LIMITS You are fishing in a marina for surfperch and rockfish,
which are two species of bottomfish. Gaming laws in the marina allow
you to catch no more than 15 surfperch per day, no more than 10 rockfish
per day, and no more than 15 total bottomfish per day.
a. Write and graph a system
of inequalitites that models
the situation.
b. Use the graph to determine
whether you can catch
11 surfperch and 9 rockfish
in one day.
Surfperch
Rockfish
39. HEALTH A person’s maximum heart rate (in beats per minute) is given
by 220 2 x where x is the person’s age in years (20 ≤ x ≤ 65). When
exercising, a person should aim for a heart rate that is at least 70% of
the maximum heart rate and at most 85% of the maximum heart rate.
a. Write and graph a system of inequalitites that models the situation.
b. A 40-year-old person’s heart rate varies from 104 to 120 beats per
minute while exercising. Does his heart rate stay in the suggested
target range for his age? Explain.
40.
★ SHORT RESPONSE A photography shop has a self-service photo center
that allows you to make prints of pictures. Each sheet of printed pictures
costs $8. The number of pictures that fit on each sheet is shown.
a. You want at least 16 pictures
of any size, and you are willing
to spend up to $48. Write and
graph a system of inequalities
that models the situation.
Four 3 inch by 5 inch
pictures fit on one sheet.
b. Will you be able to purchase
12 pictures that are 3 inches
by 5 inches and 6 pictures
that are 4 inches by 6 inches?
Explain.
Two 4 inch by 6 inch
pictures fit on one sheet.
471
41. CHALLENGE You make necklaces and keychains to sell at a craft fair. The
table shows the time that it takes to make each necklace and keychain,
the cost of materials for each necklace and keychain, and the time and
money that you can devote to making necklaces and keychains.
Necklace
Keychain
Available
Time to make (hours)
0.5
0.25
20
Cost to make (dollars)
2
3
120
a. Write and graph a system of inequalities for the number x of
necklaces and the number y of keychains that you can make under
the given constraints.
b. Find the vertices (corner points) of the graph.
c. You sell each necklace for $10 and each keychain for $8. The revenue
R is given by the equation R 5 10x 1 8y. Find the revenue for each
ordered pair in part (b). Which vertex results in the maximum revenue?
MIXED REVIEW
PREVIEW
Evaluate the expression.
Prepare for
Lesson 8.1 in
Exs. 42–47.
42. 13x2 when x 5 2 (p. 8)
43. 64 4 z3 when z 5 2 (p. 8)
44. 2| 2c | 2 32 when c 5 8 (p. 64)
45. 28 1 3y 2 16 when y 5 26 (p. 88)
2 8w
46. 7}
when w 5 5 (p. 103)
47. 21 1 Ï x when x 5 144 (p. 110)
}
11w
Use any method to solve the linear system. (pp. 427, 435, 444, 451)
48. y 5 4x 2 1
49. y 5 22x 2 6
50. x 1 2y 5 21
51. 4x 1 y 5 0
52. 2x 2 y 5 25
53. 3x 1 2y 5 2
y 5 28x 1 23
2x 1 y 5 5
y 5 25x 2 12
y 5 2x 2 5
2x 5 y 2 2
23x 1 y 5 211
Graph the linear system. Then use the graph to tell whether the linear
system has one solution, no solution, or infinitely many solutions. (p. 459)
1. x 2 y 5 1
x2y56
2. 6x 1 2y 5 16
2x 2 y 5 2
3. 3x 2 3y 5 22
26x 1 6y 5 4
Graph the system of linear inequalities. (p. 466)
4. x > 23
x<7
7. x 1 y < 2
2x 1 y > 23
y≥0
472
5. y ≤ 2
6. 4x ≥ y
8. y ≥ 3x 2 4
9. x > 25
y < 6x 1 2
y≤x
y ≥ 25x 2 15
2x 1 4y < 4
x<0
y ≤ 2x 1 7
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 7.5–7.6
600 bricks and 12 bags of sand are needed
for a construction job. Each brick weighs
2 pounds, and each bag of sand weighs
50 pounds. The maximum weight that a
delivery truck can carry is 3000 pounds.
a. Let x be the number of bricks, and let
y be the number of bags of sand. Write a
system of linear inequalities that models
the situation.
b. Graph the system of inequalities.
c. Use the graph to determine whether
700 bricks and 20 bags of sand can be
delivered in one trip.
2. MULTI-STEP PROBLEM Dana decides to
paint the ceiling and the walls of a room.
She spends $120 on 2 gallons of paint for the
ceiling and 4 gallons of paint for the walls.
Then she decides to paint the ceiling and the
walls of another room using the same kinds of
paint. She spends $60 for 1 gallon of paint for
the ceiling and 2 gallons of paint for the walls.
a. Write a system of
linear equations
that models the
situation.
b. Is there enough
information given
to determine the
cost of one gallon of
each type of paint?
Explain.
c. A gallon of ceiling paint costs $3 more
than a gallon of wall paint. What is the
cost of one gallon of each type of paint?
3. SHORT RESPONSE During a sale at a music
and video store, all CDs are priced the same
and all DVDs are priced the same. Karen buys
4 CDs and 2 DVDs for $78. The next day, while
the sale is still in progress, Karen goes back
and buys 2 CDs and 1 DVD for $39. Is there
enough information to determine the cost of
1 CD? Explain.
4. SHORT RESPONSE Two airport shuttles,
bus A and bus B, take passengers to the
airport from the same bus stop. The graph
shows the distance d (in miles) traveled
by each bus t hours after bus A leaves the
station. The distance from the bus stop to
the airport is 25 miles. If bus A and bus B
continue at the same rates, will bus B ever
catch up to bus A? Explain.
Distance (miles)
1. MULTI-STEP PROBLEM A minimum of
d
30
Bus A
Bus B
20
10
0
0
1
Time (hours)
2 t
5. EXTENDED RESPONSE During the summer,
you want to earn at least $200 per week. You
earn $10 per hour working as a lifeguard, and
you earn $8 per hour working at a retail store.
You can work at most 30 hours per week.
a. Write and graph a system of linear
inequalities that models the situation.
b. If you work 5 hours per week as a lifeguard
and 15 hours per week at the retail store,
will you earn at least $200 per week?
Explain.
c. You are scheduled to work 20 hours per
week at the retail store. What is the range
of hours you can work as a lifeguard to
earn at least $200 per week?
6. OPEN-ENDED Describe a real-world situation
that can be modeled by a system of linear
inequalities. Then write and graph the
system of inequalities.
7. GRIDDED ANSWER What is the area (in
square feet) of the triangular garden defined
by the system of inequalities below?
y≥0
x≥0
4x 1 5y ≤ 60
Mixed Review of Problem Solving
473
7
CHAPTER SUMMARY
Big Idea 1
For Your Notebook
BIG IDEAS
Solving Linear Systems by Graphing
The graph of a system of two linear equations tells you how many solutions
the system has.
One solution
No solution
y
y
y
The lines intersect.
Big Idea 2
x
x
x
The lines are parallel.
The lines coincide.
Solving Linear Systems Using Algebra
You can use any of the following algebraic methods to solve a system of linear
equations. Sometimes it is easier to use one method instead of another.
Big Idea 3
Method
Procedure
When to use
Substitution
Solve one equation for x or y.
Substitute the expression for x
or y into the other equation.
When one equation is
already solved for x or y
Addition
Add the equations to eliminate
x or y.
When the coefficients of
one variable are opposites
Subtraction
Subtract the equations to
eliminate x or y.
When the coefficients of
one variable are the same
Multiplication
Multiply one or both equations
by a constant so that adding or
subtracting the equations will
eliminate x or y.
When no corresponding
coefficients are the same
or opposites
Solving Systems of Linear Inequalities
The graph of a system of linear inequalities is the intersection of the
half-planes of each inequality in the system. For example, the graph of
the system of inequalities below is the shaded region.
x≤6
y<2
2x 1 3y ≥ 6
Inequality 1
Inequality 2
Inequality 3
y
1
1
474
x
7
CHAPTER REVIEW
classzone.com
• Multi-Language Glossary
• Vocabulary practice
REVIEW KEY VOCABULARY
• system of linear equations, p. 427
• inconsistent system, p. 459
• solution of a system of linear
equations, p. 427
• consistent dependent
system, p. 459
• consistent independent system,
• system of linear inequalities,
• solution of a system of linear
• graph of a system of linear
p. 466
p. 427
VOCABULARY EXERCISES
1. Copy and complete: A(n) ? consists of two or more linear inequalities
in the same variables.
2. Copy and complete: A(n) ? consists of two or more linear equations in
the same variables.
3. Describe how you would graph a system of two linear inequalities.
4. Give an example of a consistent dependent system. Explain why the
system is a consistent dependent system.
REVIEW EXAMPLES AND EXERCISES
Use the review examples and exercises below to check your understanding of
the concepts you have learned in each lesson of Chapter 7.
7.1
Solve Linear Systems by Graphing
pp. 427–433
EXAMPLE
y5x22
y 5 23x 1 2
Equation 1
Equation 2
y
y5x22
Graph both equations. The lines appear to intersect at
(1, 21). Check the solution by substituting 1 for x and
21 for y in each equation.
y5x22
EXAMPLES
1 and 2
on pp. 427–428
for Exs. 5–7
y 5 23x 1 2
21 0 1 2 2
21 0 23(1) 1 2
21 5 21 ✓
21 5 21 ✓
1
3
x
y 5 23x 1 2
EXERCISES
5. y 5 23x 1 1
y5x27
6. y 5 3x 1 4
y 5 22x 2 1
7. x 1 y 5 3
x2y55
Chapter Review
475
7
CHAPTER REVIEW
7.2
Solve Linear Systems by Substitution
pp. 435–441
EXAMPLE
Solve the linear system: 3x 1 y 5 29
y 5 5x 1 7
Equation 1
Equation 2
STEP 1 Substitute 5x 1 7 for y in Equation 1 and solve for x.
3x 1 y 5 29
Write Equation 1.
3x 1 5x 1 7 5 29
Substitute 5x 1 7 for y.
x 5 22
Solve for x.
STEP 2 Substitute 22 for x in Equation 2 to find the value of y.
y 5 5x 1 7 5 5(22) 1 7 5 210 1 7 5 23
c The solution is (22, 23). Check the solution by substituting 22 for x
and 23 for y in each of the original equations.
EXERCISES
EXAMPLES
1, 2, and 3
on pp. 435–437
for Exs. 8–11
Solve the linear system using substitution.
8. y 5 2x 2 7
9. x 1 4y 5 9
x 1 2y 5 1
10. 2x 1 y 5 215
x2y54
y 2 5x 5 6
11. ART Kara spends $16 on tubes of paint and disposable brushes for an
art project. Each tube of paint costs $3, and each disposable brush costs
$.50. Kara purchases twice as many brushes as tubes of paint. Find the
number of brushes and the number of tubes of paint that she purchases.
7.3
Solve Linear Systems by Adding or Subtracting
EXAMPLE
Solve the linear system: 5x 2 y 5 8
25x 1 4y 5 217
STEP 1 Add the equations to
Equation 1
Equation 2
5x 2 y 5 8
25x 1 4y 5 217
3y 5 29
y 5 23
STEP 2 Solve for y.
STEP 3 Substitute 23 for y in either equation and solve for x.
5x 2 y 5 8
5x 2 (23) 5 8
x51
Write Equation 1.
Substitute 23 for y.
Solve for x.
c The solution is (1, 23). Check the solution by substituting 1 for x
and 23 for y in each of the original equations.
476
pp. 444–450
classzone.com
Chapter Review Practice
EXERCISES
EXAMPLES
1, 2, and 3
on pp. 444–445
for Exs. 12–17
12. x 1 2y 5 13
13. 4x 2 5y 5 14
14. x 1 7y 5 12
15. 9x 2 2y 5 34
16. 3x 5 y 1 1
17. 4y 5 11 2 3x
x 2 2y 5 27
24x 1 y 5 26
5x 2 2y 5 10
7.4
2x 2 y 5 9
22x 1 7y 5 18
3x 1 2y 5 25
Solve Linear Systems by Multiplying First
pp. 451–457
EXAMPLE
Solve the linear system: x 2 2y 5 27
3x 2 y 5 4
Equation 1
Equation 2
STEP 1 Multiply the first equation by 23.
x 2 2y 5 27
3x 2 y 5 4
3 (23)
23x 1 6y 5 21
3x 2 y 5 4
5y 5 25
y55
STEP 3 Solve for y.
STEP 4 Substitute 5 for y in either of the original equations and solve for x.
x 2 2y 5 27
Write Equation 1.
x 2 2(5) 5 27
Substitute 5 for y.
x53
Solve for x.
Equation 1
Equation 2
x 2 2y 5 27
3x – y 5 4
3 2 2(5) 0 27
3(3) 2 5 0 4
27 5 27 ✓
454✓
EXERCISES
EXAMPLES
1 and 2
on pp. 451–452
for Exs. 18–24
18. 2x 1 y 5 24
19. x 1 6y 5 28
20. 3x 2 5y 5 27
21. 8x 2 7y 5 23
22. 5x 5 3y 2 2
23. 11x 5 2y 2 1
2x 2 3y 5 5
6x 2 5y 5 21
2x 2 3y 5 219
3x 1 2y 5 14
24x 1 7y 5 8
3y 5 10 1 8x
24. CAR MAINTENANCE You pay $24.50 for 10 gallons of gasoline and 1 quart
of oil at a gas station. Your friend pays $22 for 8 gallons of the same
gasoline and 2 quarts of the same oil. Find the cost of 1 quart of oil.
Chapter Review
477
7
CHAPTER REVIEW
7.5
Solve Special Types of Linear Systems
pp. 459–465
EXAMPLE
22x 1 y 5 23
y 5 2x 1 1
Show that the linear system has no solution.
Equation 1
Equation 2
y
The lines are parallel because they have
the same slope but different y-intercepts.
Parallel lines do not intersect, so the system
has no solution.
1
2
1
2
2
y 5 2x 1 1
22
x
y 5 2x 2 3
EXERCISES
EXAMPLES
1, 2, and 3
Tell whether the linear system has one solution, no solution, or infinitely
many solutions. Explain.
on pp. 459–461
for Exs. 25–27
25. x 5 2y 2 3
7.6
1.5x 2 3y 5 0
26. 2x 1 y 5 8
27. 4x 5 2y 1 6
x185y
4x 1 2y 5 10
Solve Systems of Linear Inequalities
pp. 466–472
EXAMPLE
y < 22x 1 3
y≥x23
The graph of y < 22x 1 3 is the half-plane below
the dashed line y 5 22x 1 3.
The graph of y ≥ x 2 3 is the half-plane on and
above the solid line y 5 x 2 3.
The graph of the system is the intersection of
the two half-planes shown as the darker shade
of blue.
Inequality 1
Inequality 2
y
y≥x23
1
1
x
y < 22x 1 3
EXERCISES
EXAMPLES
1, 2, 3, and 4
on pp. 466–468
for Exs. 28–31
28. y < x 1 3
y > 23x 2 2
29. y ≤ 2x 2 2
y > 4x 1 1
30. y ≥ 0
x≤2
y<x14
31. MOVIE COSTS You receive a $40 gift card to a movie theater. A ticket to a
matinee movie costs $5, and a ticket to an evening movie costs $8. Write
and graph a system of inequalities for the number of tickets you can
purchase using the gift card.
478
7
CHAPTER TEST
1. 3x 2 y 5 26
2. 22x 1 y 5 5
3. y 5 4x 1 4
4. 5x 2 4y 5 20
5. x 1 3y 5 9
6. 2x 1 7y 5 14
x1y52
x 1 2y 5 4
x 1 y 5 21
2x 2 y 5 4
3x 1 2y 5 12
5x 1 7y 5 27
Solve the linear system using substitution.
7. y 5 5x 2 7
8. x 5 y 2 11
10. 15x 1 y 5 70
11. 3y 1 x 5 17
24x 1 y 5 21
3x 2 2y 5 28
x 2 3y 5 1
x1y58
9. 3x 1 y 5 219
x2y57
12. 0.5x 1 y 5 9
1.6x 1 0.2y 5 13
13. 8x 1 3y 5 29
14. x 2 5y 5 23
15. 4x 1 y 5 17
16. 3x 1 2y 5 25
17. 3y 5 x 1 5
18. 6x 2 5y 5 9
28x 1 y 5 29
x 2 y 5 10
3x 2 5y 5 11
23x 1 8y 5 8
7y 5 4x 2 9
9x 2 7y 5 15
Tell whether the linear system has one solution, no solution, or infinitely
many solutions.
19. 15x 2 3y 5 12
20. 4x 2 y 5 24
21. 212x 1 3y 5 18
22. 6x 2 7y 5 5
23. 3x 2 4y 5 24
24. 10x 2 2y 5 14
y 5 5x 2 4
212x 1 14y 5 10
28x 1 2y 5 2
3x 1 4y 5 24
4x 1 y 5 26
15x 2 3y 5 21
25. y < 2x 1 2
y ≥ 2x 2 1
26. y ≤ 3x 2 2
y>x14
27. y ≤ 3
x > 21
y > 3x 2 3
28. TRUCK RENTALS Carrie and Dave each rent the same size moving truck
for one day. They pay a fee of x dollars for the truck and y dollars per mile
they drive. Carrie drives 150 miles and pays $215. Dave drives 120 miles
and pays $176. Find the amount of the fee and the cost per mile.
29.
GEOMETRY The rectangle has a perimeter P of
58 inches. The length l is one more than 3 times the
width w. Write and solve a system of linear equations
to find the length and width of the rectangle.
P 5 58 in.
w
l
30. COMMUNITY SERVICE A town committee has a budget of $75 to spend on
snacks for the volunteers participating in a clean-up day. The committee
chairperson decides to purchase granola bars and at least 50 bottles of
water. Granola bars cost $.50 each, and bottles of water cost $.75 each.
Write and graph a system of linear inequalities for the number of bottles
of water and the number of granola bars that can be purchased.
Chapter Test
479
7
★ Standardized TEST PREPARATION
MULTIPLE CHOICE QUESTIONS
If you have difficulty solving a multiple choice problem directly, you may be
able to use another approach to eliminate incorrect answer choices and obtain
the correct answer.
PROBLEM 1
1
Which ordered pair is the solution of the linear system y 5 }
x
2
and 2x 1 3y 5 27?
A (2, 1)
B (1, 23)
C (22, 21)
D (4, 2)
Method 1
Method 2
SOLVE DIRECTLY Use substitution to solve the
ELIMINATE CHOICES Substitute the values
given in each answer choice for x and y in both
equations.
linear system.
STEP 1 Substitute }1 x for y in the equation
2
2x 1 3y 5 27 and solve for x.
2x 1 3y 5 27
1
2x 1 3 }
x 5 27
2
1 2
3
2x 1 }
x 5 27
2
7
} x 5 27
2
x 5 22
STEP 2
1
Substitute 22 for x in y 5 }
x to find the
2
value of y.
1
y5}
x
2
1
5}
(22)
2
5 21
The solution of the system is (22, 21).
The correct answer is C. A B C D
Choice A: (2, 1)
Substitute 2 for x and 1 for y.
1
y5}
x
2
1
10}
(2)
2
151✓
2x 1 3y 5 27
2(2) 1 3(1) 0 27
7 5 27 ✗
Choice B: (1, 23)
1
y5}
x
2
1
23 0 }
(1)
2
1
23 5 }
✗
2
Choice C: (22, 21)
1
y5}
x
2
1
21 0 }
(22)
2
21 5 21 ✓
2x 1 3y 5 27
2(22) 1 3(21) 0 27
27 5 27 ✓
480
PROBLEM 2
The sum of two numbers is 21, and the difference of the two numbers
is 5. What are the numbers?
A 25 and 4
B 1 and 6
C 2 and 23
D 22 and 3
Method 1
Method 2
SOLVE DIRECTLY Write and solve a system of
ELIMINATE CHOICES Find the sum and difference
of each pair of numbers. Because the difference
is positive, be sure to subtract the lesser number
from the greater number.
equations for the numbers.
STEP 1 Write a system of equations. Let x and y
be the numbers.
x 1 y 5 21
x2y55
Choice A: 25 and 4
Equation 1
Equation 2
Sum: 25 1 4 5 21 ✓
STEP 2 Add the equations to eliminate one
Difference: 25 2 4 5 29 ✗
variable. Then find the value of the other variable.
Choice B: 1 and 6
x 1 y 5 21
x2y55
Sum: 1 1 6 5 7 ✗
2x 5 4, so x 5 2
Choice C: 2 and 23
STEP 3 Substitute 2 for x in Equation 1 and solve
for y.
Sum: 2 1 (23) 5 21 ✓
Difference: 2 2 (23) 5 5 ✓
2 1 y 5 21, so y 5 23
PRACTICE
Explain why you can eliminate the highlighted answer choice.
1. The sum of two numbers is –27. One number is twice the other. What are
the numbers?
A
9 and 18
B 23 and 24
C 218 and 29 D 214 and 213
2. Which ordered pair is a solution of the linear system 5x 1 2y 5 211
1
and x 5 2}
y 2 4?
2
A
17
1 2}76, 2}
32
B
23 1
,}
1 2}
3 32
C (23, 2)
D
(23, 213)
3. Long-sleeve and short-sleeve T-shirts can be purchased at a concert. A
long-sleeve T-shirt costs $25 and a short-sleeve T-shirt costs $15. During
a concert, the T-shirt vendor collects $8415 from the sale of 441 T-shirts.
How many short-sleeve T-shirts were sold?
A 100
B
180
C 261
D 441
Standardized Test Preparation
481
7
★ Standardized TEST PRACTICE
MULTIPLE CHOICE
1. Which ordered pair is the solution of the
7. How many solutions does the system of
linear equations whose graph is shown have?
3
1
linear system y 5 }
x 1 1 and y 5 }
x 1 4?
2
2
A
1 3, }52 2
B
C
1 }32, }74 2
D (0, 1)
y
1 23, 2 }12 2
1
1
2. How many solutions does the linear system
x
3x 1 5y 5 8 and 3x 1 5y 5 1 have?
A 0
B 1
C 2
D Infinitely many
3. The sum of two numbers is 23, and the
difference of the two numbers is 11. What are
the numbers?
A 4 and 7
B 23 and 8
C 3 and 14
D 27 and 4
A 0
B 1
C 2
D Infinitely many
8. At a bakery, one customer pays $5.67 for
3 bagels and 4 muffins. Another customer
pays $6.70 for 5 bagels and 3 muffins.
Which system of equations can be used to
determine the cost x (in dollars) of one bagel
and the cost y (in dollars) of one muffin at
the bakery?
system of linear equations whose graph
is shown?
y
1
4 x
A x1y57
x1y58
B y 5 3x 1 5.67
y 5 5x 1 6.7
C 3x 1 4y 5 6.7
5x 1 3y 5 5.67
D 3x 1 4y 5 5.67
5x 1 3y 5 6.7
9. The perimeter P (in feet) of each of the two
rectangles below is given. What are the
values of l and w?
A (3, 21)
B (0, 0)
C (0, 2)
D (23, 1)
P 5 42 ft
P 5 24 ft
w ft
1
2
7
2
linear system 3x 1 y 5 21 and y 5 2} x 2 } ?
17
1 }52, 2}
2 2
B (1, 24)
C (21, 23)
D (0, 21)
A
6. Which ordered pair is a solution of the
system x 1 2y ≤ 22 and y ≤ 23x 1 4?
482
A (0, 0)
B (2, 22)
C (22, 2)
D (5, 24)
l ft
A l 5 7 and w 5 5
B l 5 8 and w 5 4
C l 5 11 and w 5 10
D l 5 12 and w 5 9
(l 1 4) ft
2w ft
STATE TEST PRACTICE
classzone.com
GRIDDED ANSWER
SHORT RESPONSE
10. What is the x-coordinate of the solution of
13. Is it possible to find a value for c so that the
the system whose graph is shown?
y
1
x
22
linear system below has exactly one solution?
Explain.
5x 1 3y 5 21
Equation 1
5
y 5 2}
x1c
3
Equation 2
14. A rental car agency charges x dollars per day
plus y dollars per mile to rent any of the midsized cars at the agency. The total costs for
two customers are shown below.
11. What is the x-coordinate of the solution of
1
the system y 5 }
x 1 8 and 2x 2 y 5 2?
3
12. A science museum charges one amount for
admission for an adult and a lesser amount
for admission for a student. Admission to
the museum for 28 students and 5 adults
costs $284. Admission for 40 students and
10 adults costs $440. What is the admission
cost (in dollars) for one student?
Time
(days)
Distance
(miles)
Cost
(dollars)
Jackson
3
150
217.50
Bree
2
112
148.00
Customer
How much will it cost to rent a mid-sized car
for 5 days and drive 250 miles? Explain.
EXTENDED RESPONSE
15. A baseball player’s batting average is the number of hits the player has
divided by the number of at-bats. At the beginning of a game, a player has
a batting average of .360. During the game, the player gets 3 hits during
5 at-bats, and his batting average changes to .375.
a. Write a system of linear equations that represents the situation.
b. How many at-bats has the player had so far this season?
c. Another player on the team has a batting average of .240 at the
beginning of the same game. During the game, he gets 3 hits during
5 at-bats, and his batting average changes to .300. Has this player had
more at-bats so far this season than the other player? Explain.
16. A gardener combines x fluid ounces of a 20% liquid fertilizer and
80% water mix with y fluid ounces of a 5% liquid fertilizer and
95% water mix to make 30 fluid ounces of a 10% liquid fertilizer
and 90% water mix.
a. Write a system of linear equations that represents the situation.
b. Solve the system from part (a).
c. Suppose the gardener combines pure (100%) water and the 20% liquid
fertilizer and 80% water mix to make the 30 fluid ounces of the
10% liquid fertilizer and 90% water mix. Is more of the 20% liquid
fertilizer and 80% water mix used in this mix than in the original mix?
Explain.
483
CUMULATIVE REVIEW
Chapters
1–7
Evaluate the expression.
1. 25 p 2 2 4 4 2 (p. 8)
}
4. Ï144 (p. 110)
2. 24 4 6 1 (9 2 6) (p. 8)
}
5. 2Ï2500 (p. 110)
3. 5[(6 2 2)2 2 5] (p. 8)
}
6. 6Ï400 (p. 110)
Check whether the given number is a solution of the equation or
inequality. (p. 21)
7. 7 1 3x 5 16; 3
10. g 2 3 > 2; 5
8. 21y 1 1 5 1; 0
9. 20 2 12h 5 12; 1
11. 10 ≥ 4 2 x; 0
12. 30 2 4p ≥ 5; 6
13. 5(y 2 1) 1 4 (p. 96)
14. 12w 1 (w 2 2)3 (p. 96)
15. (g 2 1)(24) 1 3g (p. 96)
2 25
16. 10h
} (p. 103)
2 4x
17. 21
} (p. 103)
2 20m
18. 32
} (p. 103)
19. x 2 8 5 21 (p. 134)
20. 21 5 x 1 3 (p. 134)
21. 6x 5 242 (p. 134)
x 5 8 (p. 134)
22. }
23. 5 2 2x 5 11 (p. 141)
2
24. }
x 2 3 5 17 (p. 141)
25. 3(x 2 2) 5 215 (p. 148)
26. 3(5x 2 7) 5 5x 2 1 (p. 154)
27. 27(2x 2 10) 5 4x 2 10
Simplify the expression.
5
27
2
Solve the equation.
3
3
(p. 154)
Graph the equation.
28. x 1 2y 5 28 (p. 225)
29. 22x 1 5y 5 210 (p. 225)
30. 3x 2 4y 5 12 (p. 225)
31. y 5 3x 2 7 (p. 244)
32. y 5 x 1 6 (p. 244)
1 x (p. 253)
33. y 5 2}
3
Write an equation of the line in slope-intercept form with the given slope
and y-intercept. (p. 283)
34. slope: 5
y-intercept: 21
35. slope: 21
y-intercept: 3
36. slope: 27
y-intercept: 0
Write an equation in point-slope form of the line that passes through the
given points. (p. 302)
37. (1, 210), (25, 2)
38. (4, 7), (24, 3)
39. (29, 22), (26, 8)
40. (21, 1), (1, 23)
41. (2, 4), (8, 2)
42. (26, 1), (3, 25)
Solve the inequality. Then graph your solution.
43. x 2 9 < 213 (p. 356)
44. 8 ≤ x 1 7 (p. 356)
45. 8x ≥ 56 (p. 363)
x > 7 (p. 363)
46. }
47. 1 2 2x < 11 (p. 369)
48. 8 > 23x 2 1 (p. 369)
49. 4x 2 10 ≤ 7x 1 8 (p. 369)
50. 7x 2 5 < 6x 2 4 (p. 369)
51. 24 < 3x 2 1 < 5 (p. 380)
52. 3 ≤ 9 2 2x ≤ 15 (p. 380)
53. ⏐3x⏐ < 15 (p. 398)
54. ⏐4x 2 2⏐ ≥ 18 (p. 398)
24
484
Cumulative Review: Chapters 1–7
Solve the linear system using elimination. (p. 451)
55. 4x 1 y 5 8
56. 3x 2 5y 5 5
5x 2 2y 5 23
57. 12x 1 7y 5 3
x 2 5y 5 24
8x 1 5y 5 1
58. ART PROJECT You are making a tile mosaic on the
rectangular tabletop shown. A bag of porcelain tiles
costs $3.95 and covers 36 square inches. How much will
it cost to buy enough tiles to cover the tabletop? (p. 28)
24 in.
30 in.
59. FOOD The table shows the changes in the price for a dozen grade A,
large eggs over 4 years. Find the average yearly change to the nearest
cent in the price for a dozen grade A, large eggs during the period
1999–2002. (p. 103)
Year
1999
2000
2001
2002
Change in price for a dozen
grade A, large eggs (dollars)
20.17
0.04
20.03
0.25
60. HONEY PRODUCTION Honeybees visit about 2,000,000 flowers to make
16 ounces of honey. About how many flowers do honeybees visit to
make 6 ounces of honey? (p. 168)
61. MUSIC The table shows the price p (in dollars) for various lengths of
speaker cable. (p. 253)
Length, l (feet)
Price, p (dollars)
3
5
12
15
7.50
12.50
30.00
37.50
a. Explain why p varies directly with l.
b. Write a direct variation equation that relates l and p.
62. CURRENCY The table shows the exchange rate between the currency of
Bolivia (bolivianos) and U.S. dollars from 1998 to 2003. (p. 335)
Year
1998
1999
2000
2001
2002
2003
Bolivianos per U.S. dollar
5.51
5.81
6.18
6.61
7.17
7.66
a. Find an equation that models the bolivianos per U.S. dollar as a
function of the number of years since 1998.
b. If the trend continues, predict the number of bolivianos per
U.S. dollar in 2010.
63. BATTERIES A manufacturer of nickel-cadmium batteries recommends
storing the batteries at temperatures ranging from 220°C to 45°C. Use
an inequality to describe the temperatures (in degrees Fahrenheit) at
which the batteries can be stored. (p. 380)
Cumulative Review: Chapters 1–7
485

Systems of Equations and Inequalities

Transcription

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