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CH 3 Student Companion ANSWERS

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Solving Systems Using
Tables and Graphs
3-1
Vocabulary
Review
1. Cross out the equation that is NOT in slope-intercept form.
y 5 217 x
r5s
a 5 !3b 1 5
3x 1 7y 5 13
Vocabulary Builder
linear system
linear system (noun)
LIN
ee ur SIS tum
5x 7y 1
4x y 9
Related Words: independent system, dependent system
Use Your Vocabulary
The graphs below show the possible types of solutions for a system of two equations
in two variables. Write T for true or F for false.
Intersecting Lines
y
Coinciding Lines
y
x
x
O
one solution
Consistent
Independent
Parallel Lines
y
O
O
infinitely many solutions
Consistent
Dependent
no solution
Inconsistent
F
2. Inconsistent linear systems intersect at two points.
T
3. An independent linear system has one solution.
F
4. A dependent linear system has no solutions.
T
5. Two unique lines with the same slope form an inconsistent system.
Chapter 3
58
x
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Definition: A linear system is a collection of linear equations involving the same set
of variables. The system above is two equations in two variables.
Problem 1 Using a Graph or Table to Solve a System
Got It? What is the solution of the system?
e
3x 2 2y 5 4
3x 1 2y 5 5
6. Circle the graph of the equations.
y
y
4
2
4 2
O
y
2
x
2
4
2
x
O
4 2
2
4
2
4
4 2
2
O
x
2
4
2
4
7. Circle the row in the calculator screen
that contains the solution of the system.
X
Y1
Y2
0
1
2
3
4
–2
–1.5
–1
–.5
0
5
2
–1
–4
–7
8. The solution of the system of equations is Q 2 , 21 R .
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Problem 2 Using a Table to Solve a Problem
Got It? Biology The equation y1 5 1.5x 1 22 models the length, in centimeters, of
a Spiny Dogfish shark x years old. The equation y2 5 0.75x 1 37 models the length,
in centimeters, of a Greenland shark x years old. If the growth rates continue, how
long will each shark be when it is 25 years old?
9. Reasoning Which shark will be longer at age 25? How do you know?
Answers may vary. Sample: The Spiny Dogfish has a faster growth
_______________________________________________________________________
rate, so it will be longer at age 25.
_______________________________________________________________________
10. Underline the correct phrase to complete the sentence.
To solve the problem, I need to find
an x-value / a y-value / two x-values / two y-values .
11. Complete the table for x 5 25.
x
y1 1.5x 22
y1 0.75x 37
25
59.5
55.75
59
Lesson 3-1
12. Spiny Dogfish sharks will be
59.5
cm long and Greenland sharks will be
55.75 cm long when they are 25 years old.
Problem 3
Using Linear Regression
Got It? The table shows the populations of the San Diego and Detroit metropolitan
regions. When were the populations of these regions equal? What was that population?
Populations of San Diego and Detroit (1950–2000)
San Diego
Detroit
1950
1960
1970
1980
1990
2000
334,387
573,224
696,769
875,538
1,110,549
1,223,400
1,849,568
1,670,144
1,511,482
1,203,339
1,027,974
951,270
SOURCE: U.S. Census Bureau
13. Circle the first calculator step in solving the problem.
Calculate the intersection.
Enter the data into lists.
Enter y1 .
14. Write the name of the list you will use (L1 , L2 , or L3 ) next to the data type.
L2
population of San Diego
L3
population of Detroit
L1
years since 1950
y1 5 17816.597 cx 1 356896.238 c
y1 5 217816.597x c 1 356896.238 c
y2 5 219217.551 cx 1 1849401.619 c
y2 5 19217.551x c 1 1849401.619 c
16. Cross out the graph that does NOT show the regression lines.
17. Underline the correct word to complete each sentence.
The x-axis / y-axis corresponds to the number of years since 1950.
The x-axis / y-axis corresponds to the populations of San Diego and Detroit.
18. The scale of the graphs is 260 # x # 60 by 10s and 0 # y # 1,500,000 by 100,000s.
Use the scale to estimate the coordinates of the point of intersection. Accept reasonable estimates.
Exact answers are given.
1,074,919
R.
The coordinates are Q 40.3 ,
The populations of San Diego and Detroit were equal sometime during the year 1990 .
The population was about
Chapter 3
1,075,000
.
60
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
15. Circle the pair of equations you will graph.
Problem 4 Classifying a System Without Graphing
23x 1 y 5 4
Got It? Without graphing, is the system e x 2 1 y 5 1
3
independent, dependent,
or inconsistent?
19. Write each equation in slope-intercept form.
x 2 13 y 5 1
y 5 3x 1 4
23x 1 y 5 4
y 5 3x 2 3
20. The slope of 23x 1 y 5 4 is 3 and the slope of x 2 13 y 5 1 is 3 .
21. The y-intercept of 23x 1 y 5 4 is 4 and the y-intercept of x 2 13 y 5 1 is 23 .
22. Underline the correct words to complete the sentence.
Because the slopes of the lines are equal / not equal and the y-intercepts are
the same / different , the system is inconsistent / independent / dependent .
Lesson Check • Do you UNDERSTAND?
Vocabulary Is it possible for a system of equations to be both independent and
inconsistent? Explain.
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Write T for true or F for false.
T
23. The graphs of an inconsistent system are parallel.
T
24. The graphs of an independent system intersect at one point.
25. Now answer the question.
Answers may vary. Sample: No; inconsistent lines are parallel and
_______________________________________________________________________
never intersect. Independent lines have one point of intersection.
_______________________________________________________________________
Therefore, a system cannot be both independent and inconsistent.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
system of equations
dependent
independent
consistent
inconsistent
Rate how well you can solve a linear system using a graph or table.
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61
Lesson 3-1
3-2
Solving Systems
Algebraically
Vocabulary
Review
1. Circle the equations that are in standard form.
4x 1 3y 5 2
y 5 3x 2 5
4x 2 3 5 2y
2x 1 5y 5 0
Write each equation in standard form.
2. y 5 5x 2 3
3. y 2 4 5 6x
5x 2 y 5 3
4. 2y 1 3x 2 12 5 0
6x 2 y 5 24
3x 2 y 5 12
Vocabulary Builder
LOO
shun
Main Idea: If two numbers are substituted for x and y in a
system of equations and they make both equations true,
then the ordered pair (x, y) is a solution of the system.
Definition: A solution is any ordered pair that makes an equation in two variables true.
Use Your Vocabulary
5. Write three ordered pairs that are solutions of the equation y 5 25x 2 2. Answers may vary.
Check students’ work.
Q 0 , 22 R
Q 25 , 23 R
Q 5 , 227 R Samples are given.
Use the system at the right. Write T for true or F for false.
F
6. The system has a unique solution.
F
7. The system has infinitely many solutions.
T
8. The system has no solution.
F
9. The solution is (0, 0).
T
y
2x 1 2 5 3
e
4x 1 y 5 2
10. The solution of a system can be found by graphing the equations of the system.
Chapter 3
62
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
solution (noun) suh
(2, 1) is a solution of
y 3x 5 because
1 3(2) 5.
Problem 1 Solving by Substitution
Got It? What is the solution of the system of equations? e
x 1 3y 5 5
22x 2 4y 5 25
11. Follow the steps to find the solution.
Solve the п¬Ѓrst equation for x.
x 3y 5
1
2
x
5
3y
Substitute the expression for x in the
2x 4y 5
second equation. Then solve for y.
2+ 3y 5 , 4y 5
4y 5
6y 10
2
3
Substitute the value for y in either
equation. Solve for x.
y
5
y
2.5
x 3y 5
x 3+
,5
2.5
x
5
7.5
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
x 2.5
12. The solution of the system is Q 22.5 ,
R.
2.5
Problem 2 Using Substitution to Solve a Problem
Got It? Music An online music company offers 15 downloads for $19.75 and
40 downloads for $43.50. Each price includes a one-time registration fee. What is
the cost of each download and the registration fee?
13. Complete the model to write a system of equations.
Relate
Define
Write
total cost
is
number of
downloads
times
cost of one
download
plus
registration
fee
Let c the cost of one download and let r the registration fee.
$19.75
15
r
c
r
$ 43.50
40
r
c
r
63
Lesson 3-2
14. Circle the equation that expresses r in terms of c in the first equation.
r 5 215c 1 19.75
r 5 15c 1 19.75
r 5 215c 2 19.75
15. Substitute the equation you chose in Exercise 14 into the second equation of the
system and solve for c.
43.5 5 40c 1 r
Write the original equation.
43.5 5 40c 1
215 c 1 19.75
Substitute for r.
43.5 5
25
c 1 19.75
Simplify.
23.75 5
25
c
Use the Addition Property of Equality.
0.95 5 c
Divide.
16. Now substitute the value of c into one of the equations of the system and solve for r.
Answers may vary. Sample:
r 5 215c 1 19.75
r 5 215(0.95) 1 19.75
r 5 214.25 1 19.75
r 5 5.5
17. The cost of each download is $
and the registration fee is $ 5.50 .
Solving by Elimination
Got It? What is the solution of the system of equations?
18. Add the equations.
22x
5x
1
2
3x
1
22x 1 8y 5 28
5x 2 8y 5 20
19. Now choose one of the original equations.
Substitute and solve.
5
5
28
20
0y
5
12
x
5
4
8y
8y
e
Answers may vary. Sample:
22(4) 1 8y 5 28
28 1 8y 5 28
8y 5 0
y50
Simplify.
20. Circle the ordered pair that is the solution of the system of equations.
( 24, 22)
Problem 4
(4, 22)
(4, 0)
Solving an Equivalent System
Got It? What is the solution of the system of equations?
21. Underline the correct values to complete the sentence.
e
3x 1 7y 5 15
5x 1 2y 5 24
To get additive inverses for the x-term, multiply the first equation by 2 / 3 / 5
and the second equation by 22 / 23 / 27.
Chapter 3
64
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Problem 3
.95
22. Circle the equivalent system that shows additive inverses for the x-term.
e
3x 1 7y 5 15
5x 1 2y 5 24
e
6x 1 14y 5 3c
235x 2 14y 5 28
23. Solve the system for y.
e
15x 1 35y 5 75
215x 2 6y 5 12
24. Then substitute and solve for x.
15x 1 35y 5 75
215x 2 6y 5 12
29y 5 87
y5 3
5x 1 2y 5 24
5x 1 2(3) 5 24
5x 5 210
x 5 22
25. The solution of the system is Q 22 , 3 R .
Problem 5 Solving Systems Without Unique Solutions
Got It? What is the solution of the system of equations? Explain. e
2x 1 y 5 22
2x 2 2y 5 0
26. Circle the first step in solving the system.
Multiply 2x 1 y 5 22 by 21.
Multiply 2x 1 y 5 22 by 2.
27. Add 22x 1 2y 5 24 and 2x 2 2y 5 0.
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
22x
2x
1
2
2y
2y
5
5
24
0
0
5
24
28. What is the solution of the system?
Place a вњ“ in the box if the response is correct.
Place an вњ— if it is incorrect.
вњ“ The system has no solution.
вњ— The system has infinitely many solutions.
Lesson Check • Do you UNDERSTAND?
Vocabulary Give an example of two equivalent systems.
29. Cross out the system of equations that is NOT equivalent to the others.
e
4y 1 5x 5 13
4y 2 x 5 3
e
y 1 5x 5 12
4y 2 x 5 3
e
8y 1 40x 5 96
8y 2 2x 5 6
Math Success
Check off the vocabulary words that you understand.
substitution
elimination
equivalent equation
unique solutions
Rate how well you can solve linear systems algebraically.
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review
0
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get it!
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65
Lesson 3-2
3-3
Systems of Inequalities
Vocabulary
Review
Complete each statement with the correct word or phrase from the list below.
Use each word or phrase only once.
greater than or equal to
at least
at most
1. The Florida football team needs 9 two more wins to
clinch the division title.
at least
2. The 9 speed a car may travel on the Florida freeway is
40 miles per hour.
minimum
3. If you have $45 and sweaters sell for $20 each, you can buy
9 two sweaters.
at most
4. The height of a rider must be 9 42 inches in order to ride
the Summit Plummet.
greater than or equal to
Vocabulary Builder
inequality symbols
inequality (noun) in ee KWAL uh tee
., /, ≤, ≥, U
Definition: An inequality is a mathematical statement indicating
that one quantity is less than or less than or equal to a second quantity.
Use Your Vocabulary
5. Place a вњ“ next to the math statements that are inequalities. Place an вњ— next to the
math statements that are not.
вњ“
15x . 3y
вњ“
вњ“
y#x
вњ—
r2s
p 5 t 2 17
Complete each inequality with R, S, K, or L.
6. y , 6, so 6 S y
7. b $ 10, so 10 K b
8. r # s 1 t , so s 1 t L r
9. Write an inequality symbol to represent each verbal expression.
p is at most 10
u is greater than m
z is at least 9
p K 10
u S m
z L 9
Chapter 3
66
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
minimum
Problem 1 Solving a System by Using a Table
Got It? Assume that x and y are whole numbers.
What is the solution of the system of inequalities?
e
x1yS4
3x 1 7y K 21
10. Circle the inequality that has a finite number of whole-number solutions.
x1y.4
3x 1 7y # 21
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
11. Use the inequality you circled in Exercise 10 and whole numbers to complete the table of values.
x
y
0
0, 1, 2, 3
1
0, 1, 2
2
0, 1 , 2
3
0 , 1
4
0, 1
5
0
6
0
7
0
12. Write the ordered pairs of values from the table that satisfy the first inequality.
(4, 1)
(5, 0)
(6, 0)
(7, 0)
Problem 2 Solving a System by Graphing
Got It? What is the solution of the system of inequalities?
e
x 1 2y K 4
y L 2x 2 1
13. Circle the equivalent system that shows the equations in slope-intercept form.
e
y $ 20.5x 1 2
y $ 2x 2 1
e
2y # 4 1 x
y $ 2x 2 1
e
y # 20.5x 1 2
y $ 2x 2 1
67
e
y $ 22 1 0.5x
y $ 2x 2 1
Lesson 3-3
14. Circle the graph of the solution of the system.
y
y
4
y
4
4
2
x
2
4 2
4
x
O
4
x
4
4
2
2
4
Using a System of Inequalities
Problem 3
Got It? A pizza parlor charges $1 for each vegetable topping and $2 for each meat
topping. You want at least five toppings on your pizza. You have $10 to spend on
toppings. How many of each type of topping can you get on your pizza?
15. Complete the model to write a system of inequalities.
5
cost of
vegetable toppings
plus
number of
meat toppings
is at least
5
is
at most
10
5
cost of
meat toppings
Let v the number of vegetable toppings.
Let m the number of meat toppings.
Define
5
Write
plus
v
m
v
2m
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Relate
number of
vegetable toppings
10
16. Circle the system of inequalities that is equivalent to the system in Exercise 15.
e
v1m$5
2m # 10 1 v
e
m#52v
m$51v
e
m$52v
m # 212 v 1 5
17. Number Sense Circle the types of numbers that can represent the number of toppings
you can get on your pizza.
rational numbers
integers
whole numbers
real numbers
18. Use the graph of the system at the right. Underline the correct
number to complete each sentence.
0 / 5 / 10 meat toppings.
If you order 10 vegetable toppings, you can order at most
0 / 5 / 10 meat toppings.
Meat Toppings
If you order 0 vegetable toppings, you can order at most
10
8
6
4
2
0
0
2
4
6
8
Vegetable Toppings
Chapter 3
68
10
Problem 4
Solving a Linear/Absolute-Value System
Got It? What is the solution of the system of inequalities?
e
y R 2 13 x 1 1
y S 2»x 2 1…
y
19. The graph at the right shows the boundaries
of the system. Shade y , 213 x 1 1 vertically.
Shade y . 2u x 2 1 u horizontally. Darken the
region of overlap. Then label each inequality.
y
1
3 x 1
4
y
2Ux 1U
2
6
4
2
O
x
2
6
Lesson Check • Do you UNDERSTAND?
Reasoning Is the solution of a system of linear inequalities a union or an
intersection of the solutions of the two inequalities? Justify your answer.
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Write T for true or F for false.
T
20. Systems of inequalities are similar to systems of equations.
F
21. The solution of a system of equations is the union of all points on the graphs of the lines.
F
22. The solution of a system of inequalities is the union of all points in the graphs of
the inequalities.
23. Now answer the question.
Answers may vary. Sample: The solution of a system of inequalities is the
_______________________________________________________________________
intersection of the solutions of the inequalities.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
linear inequalities
overlap
absolute-value system
Rate how well you can solve systems of linear inequalities.
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69
Lesson 3-3
3-4
Linear Programming
Vocabulary
Review
1. Draw a line from each polygon in Column A to the number of vertices it has in
Column B.
Column A
Column B
pentagon
5
quadrilateral
8
6
octagon
4
4
y
(4, 21)
(3, 5)
C
B
C
2 A
2. Write the letter of each vertex of the quadrilateral at the right.
(6, 4)
B
(1, 2)
D
O
A
x
2
D
6
constraint (noun) kun
STRAYNT
Related Words: constrain, restrict, limit, feasible region
Main Idea: The constraints in a linear programming situation form a system of
inequalities. The graph of this system is the feasible region and contains all the
points that satisfy the constraints.
Definition: A constraint is a restriction or limitation.
Use Your Vocabulary
Complete each statement with a word from the list. Use each word only once.
constraint
constrain
constrained
3. Weight is one 9 on vehicles allowed on the bridge.
constrained
4. An injury can cause 9 motion.
constrain
5. The rules of a game 9 how you play.
Chapter 3
constraint
70
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Vocabulary Builder
Linear programming is a method for finding the minimum or maximum value of some
quantity, given a set of constraints. The constraints form a system of linear inequalities.
The graph of the solutions is the feasible region.
Key Concept
Vertex Principle of Linear Programming
If there is a maximum or a minimum value of a linear objective function,
it occurs at one or more vertices of the feasible region.
6. The graph at the right shows a feasible region. Write the coordinates
at which a maximum or minimum value of a linear objective
function could occur.
Q 0 , 0 R
Q 3 , 0 R
4
y
2
4 2
O
x
2
4
4
Q 3 , 3 R
4
Problem 1 Testing Vertices
Got It? Use the graph and the constraints below. What values of x and y in the
feasible region maximize P for the objective function P 5 x 1 3y?
7. Label the vertices of the feasible region with their coordinates.
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
6
y
Constraints:
x 1 2y # 5
•x 2 y # 2
x $ 0, y $ 0
4
(0, 2.5)
2
(0, 0)
2 O
(3, 1)
(2, 0) 4
x
8
4
8. Evaluate P 5 x 1 3y at each vertex.
At (0, 0),
P 5 x 1 3y
5 0 1 3(0)
50
At (0, 2.5),
P 5 x 1 3y
5 0 1 3(2.5)
5 7.5
At (3, 1),
P 5 x 1 3y
5 3 1 3(1)
56
At (2, 0),
P 5 x 1 3y
5 2 1 3(0)
52
9. P has a maximum value of 7.5 when x 5 0 and y 5 2.5 .
71
Lesson 3-4
Problem 2
Using Linear Programming to Maximize Profit
Got It? Business You are screen-printing T-shirts and sweatshirts to sell at the Polk
County Blues Festival and are working with the following constraints.
• It takes 10 min to make a 1-color T-shirt.
• It takes 20 min to make a 3-color sweatshirt.
• You have 20 hours at most to make shirts.
• Supplies for a T-shirt cost $4.
• Supplies for a sweatshirt cost $20.
• You want to spend no more than $600 on supplies.
• You want to have at least 50 items to sell.
The profit on a T-shirt is $6. The profit on a sweatshirt is $20. How many of each type of
shirt should you make to maximize your profit?
10. Complete the inequalities that describe the constraints.
x
constraints
1200
x1y$
50
4 x 1 20y #
600
x$
0
y$
0
11. Circle the objective function that models the situation.
P 5 10x 1 y
P 5 10x 1 20y
P 5 4x 1 20y
12. Shade the feasible region on the graph.
y
48
36
24
12
x
O
30
60
90
120
13. The vertices of the feasible region are
Q
50
Q 100,
Chapter 3
, 0R, Q
10
120
, 0R,
R , and Q 25,
25
R.
72
P 5 6x 1 20y
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
10x 1 20y #
14. Evaluate P at each vertex.
At (50, 0),
P 5 6x 1 20y
5 6(50) 1 20(0)
5 300
At (120, 0),
P 5 6x 1 20y
5 6(120) 1 20(0)
5 720
At (100, 10),
P 5 6x 1 20y
5 6(100) 1 20(10)
5 800
At (25, 25),
P 5 6x 1 20y
5 6(25) 1 20(25)
5 650
15. How many of each type of shirt should you make to maximize your profit?
T-shirts 5
sweatshirts 5
100
10
Lesson Check • Do you UNDERSTAND?
Write a system of constraints whose graphs determine a trapezoid.
Write an objective function and evaluate it at each vertex.
16. Write the constraints that produce the feasible region below.
4
y$0
y
yR2
2
4
2
2x 1 y R 3
x
O
2
4
x1yR3
2
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
4
17. Use the objective function P 5 23x 1 2y . Evaluate the function at each vertex of
the feasible region.
(–3, 0)
(–1, 2)
P 5 23x 1 2y
5 23(23) 1 2(0)
59
(1, 2)
P 5 23x 1 2y
5 23(21) 1 2(2)
57
P 5 23x 1 2y
5 23(1) 1 2(2)
51
(3, 0)
P 5 23x 1 2y
5 23(3) 1 2(0)
5 29
Math Success
Check off the vocabulary words that you understand.
constraint
linear programming
objective function
feasible region
Rate how well you can solve linear programming problems.
Need to
review
0
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73
Lesson 3-4
Systems With Three
Variables
3-5
Vocabulary
Review
1. Circle the number of points that determine a plane.
1
2
3
4
Vocabulary Builder
ordered triple (noun)
AWR
durd TRIP ul
Definition: An ordered triple (x, y, z) has three coordinates and describes a point in
three-dimensional space.
Use Your Vocabulary
2. Circle the ordered triple that has x-coordinate 0.
(1, 0, 1)
Problem 1
(1, 1, 0)
(0, 1, 1)
(1, 0, 0)
Solving a System Using Elimination
Got It? What is the solution of the system at the right? Use elimination.
Check your answer in all three original equations.
3. How can you combine A and B to eliminate y?
add B to A
4. How can you combine B and C to eliminate y?
subtract B from A
add B to C
5. Combine A and B to eliminate y.
x2
x1
y1
y1
2 x1
0 y1
Chapter 3
A x 2 y 1 z 5 21
B • x 1 y 1 3z 5 23
C 2x 2 y 1 2z 5 0
subtract B from C
6. Combine B and C to eliminate y.
z 5 21
3z 5 23
4 z 5 24
74
x1
2x 2
y1
y1
3 x1
0 y1
3z 5 23
2z 5 20
5 z 5 23
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Main Idea: The first coordinate of an ordered triple represents the point’s location
along the x-axis, the second coordinate its location along the y-axis, and the third
coordinate its location along the z-axis.
7. Use the equations from Exercises 5 and 6 to write and solve a system of two
equations in two variables.
Answers may vary. Sample:
2x 1 4z 5 24
2x 5 24z 2 4
x 5 22z 2 2
3x 1 5z 5 23
3(22z 2 2) 1 5z 5 23
26z 2 6 1 5z 5 23
2z 5 3
z 5 23
8. Use the solutions from Exercise 7 and
вћЂ to substitute and solve for y.
2x 1 4z 5 24
2x 1 4(23) 5 24
2x 2 12 5 24
2x 5 8
x54
9. The solution of the original system is
Q 4 , 2 , 23 R .
x 2 y 1 z 5 21
4 2 y 1 (23) 5 21
1 2 y 5 21
y52
10. Now check your answer in all three original equations.
x 2 y 1 z 5 21
x 1 y 1 3z 5 23
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Check:
x 2 y 1 z 5 21
4 2 2 1 (23) 0 21
21 5 21
Problem 2
2x 2 y 1 2z 5 0
Check:
x 1 y 1 3z 5 23
4 1 2 1 3(23) 0 23
23 5 23
Check:
2x 2 y 1 2z 5 0
2(4) 2 2 1 2(23) 0 0
050
Solving an Equivalent System
Got It? What is the solution of the system at the right? Use elimination.
11. First, add a multiple of A to B to eliminate x. Circle the number you will
use to multiply A.
21
22
1
A x 2 2y 1 3z 5 12
B • 2x 2 y 2 2z 5 5
C 2x 1 2y 2 z 5 4
2
12. Next, add a multiple of B to C to eliminate x. Circle the number you will use to
multiply B.
21
22
1
2
13. Write the two equations you get by eliminating x.
3 y 1 28 z 5 219
3 y1
14. Now find the solution of the two-variable
system of equations.
3y 2 8z 5 219
2 (3y 1 z 5 21)
29z 5 218
z52
1 z 5 21
15. Substitute the values of y and z into
A to solve for x.
3y 1 z 5 21
3y 1 1(2) 5 21
3y 5 23
y 5 21
x 2 2y 1 3z 5 12
x 2 2(21) 1 3(2) 5 12
x 1 8 5 12
x54
75
Lesson 3-5
Solving a System Using Substitution
Problem 3
Got It? What is the solution of the system? Use substitution.
A x 2 2y 1 z 5 24
B • 24x 1 y 2 2z 5 1
C 2x 1 2y 2 z 5 10
16. Reasoning Why should you choose A to solve for x? Place a вњ“ in the
box if the response is correct. Place an вњ— if it is incorrect.
вњ“ The coefficient of x is 1.
вњ—
The coefficient of x is negative.
вњ—
The coefficients of x and y are equal.
17. Now solve the system.
Solve A for x. Then substitute for x in B and C. Next, solve for y, for z, and for x.
x 5 2y 2 z 2 4 , so
24(2y 2 z 2 4) 1 y 2 2z 5
1
28y 1 4z 1 16 1 y 2 2z 5
1
27y 1 2z 5 215
and
2(2y 2 z 2 4) 1 2y 2 z 5 10
4y 2 2z 2 8 1 2y 2 z 5 10
6y 2 3z 5 18
27y 1 2z 5 215
6y 2 3z 5 18
3(27y 1 2z 5 215)
2(6y 2 3z 5 18)
29y
5 29
y51
6y 2 3z 5 18
6(1) 2 3z 5 18
23z 5 12
z 5 24
x 2 2y 1 z 5 24
x 2 2(1) 1 (24) 5 24
x52
Solving a Real-World Problem
Problem 4
Got It? Business You manage a clothing store and budget $5400 to restock
200 shirts. You can buy T-shirts for $12 each, polo shirts for $24 each, and rugby
shirts for $36 each. If you want to have the same number of T-shirts as polo shirts,
how many of each type of shirt should you buy?
19. Use the information in the problem to complete the model below.
Relate
T-shirts polo shirts
T-shirts polo shirts
$12 r T-shirts $
24
r polo shirts
Let
y the number of polo shirts
1
Write
z the number of rugby shirts
Chapter 3
$
2
3
76
36
5
5
x the number of T-shirts
Define
rugby shirts
r rugby shirts
12x 24
200
x y $ 5400
z
x y
36z
200
y
5400
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
18. The solution of the system is Q 2 , 1 , 24 R .
20. Circle the method(s) you will use to solve the system.
elimination
equivalent system
substitution
21. Now solve the system.
Substitute for
y 5 x in A and C.
A x 1 y 1 z 5 200
x 1 x 1 z 5 200
2x 1 z 5 200
Solve the system.
2x 1 y 5 200
e
x 1 z 5 150
Solve for y and z.
x5y
50 5 y
2x 1 z 5 200
2 (x 1 z 5 150)
x 5 50
2x 1 z 5 200
2(50) 1 z 5 200
100 1 z 5 200
z 5 100
C 12x 1 24y 1 36z 5 5400
12x 1 24x 1 36z 5 5400
36x 1 36z 5 5400
x 1 z 5 В 150
22. You should buy
50
T-shirts,
50
100
polo shirts, and
rugby shirts.
Lesson Check • Do you UNDERSTAND?
Writing How many solutions does the system at the right have?
Explain your answer in terms of intersecting planes.
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
23. Solve the system by adding A and B and then adding C.
A 2x 2 3y 1 z 5 5
B • 2x 2 3y 1 z 5 22
C 24x 1 6y 2 2z 5 10
4x 2 6y 1 2z 5 3
24x 1 6y 2 2z 5 10
0 5 13
2x 2 3y 1 z 5 5
2x 2 3y 1 z 5 22
4x 2 6y 1 2z 5 3
24. The system has zero / one / infinitely many solution(s). The graphs of the three
equations are intersecting / parallel / perpendicular planes.
Math Success
Check off the vocabulary words that you understand.
linear system
elimination
substitution
Rate how well you can solve systems with three variables.
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Lesson 3-5
3-6
Solving Systems Using
Matrices
Vocabulary
Review
1. Underline the correct word to complete the sentence.
The partial solution of the system of equations at the right uses
substitution / elimination / equivalent systems.
2x 1 3y 2 z 5 4
2x 1 2y 1 z 5 23
x 1 5y
5 1
Vocabulary Builder
rref (noun)
1 0 0
5
Example: C 0 1 0 † 28 S is a matrix in reduced row echelon form representing
0 0 1
4
the solution (5, 28, 4).
Use Your Vocabulary
2. Draw a line from each rref matrix in Column A to the solution it represents in
Column B.
Column A
Column B
1 0 0 2
C 0 1 0 † 0 S
0 0 1 3
(0, 2, 3)
1 0 0 0
C 0 1 0 † 2 S
0 0 1 3
(2, 0, 3)
1 0 0 2
C 0 1 0 † 3 S
0 0 1 0
(2, 3, 0)
Chapter 3
78
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Definition: The rref (reduced row echelon form) function on a calculator generates
the matrix that represents the solution of a system of equations.
Problem 1 Identifying a Matrix Element
Got It? What is element a13 in matrix A?
4 29 17В В 1
`
A5 C 0
5 8В ` В 6 S
`
23 22 10В В 0
3. Underline the correct words to complete the sentence.
The matrix has 3 rows / columns and 4 rows / columns .
4. The element a13 is in row 1 and column 3 .
5. Use the matrix below. Circle element a13 .
4 29 17В В 1
`
A5 C 0
5 8В ` В 6 S
`
23 22 10В В 0
Problem 2 Representing Systems With Matrices
Got It? How can you represent the system of equations at the right
with a matrix?
b
24x 2 2y 5 7
3x 1 y 5 25
6. How many rows and columns will the matrix have?
number of rows 5
2
number of columns 5
3
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
7. Write the matrix.
24
22
7
3
1
25
Problem 3 Writing a System from a Matrix
Got It? What linear system does B
2
0В В 6
`` R represent?
5 22В В 1
8. Underline the correct numbers to complete the sentence.
The matrix represents a system of 2 / 3 / 4 equations in 2 / 3 / 4 variables.
9. Complete the system of equations.
2 x1
0 y5
6
5 x 1 22 y 5
1
79
Lesson 3-6
Key Concept
Row Operations
10. Use the row operation indicated to complete each matrix.
Switch any two rows.
Switch Rows 1 and 2. B
4 5 3
R becomes
3 2 6
3
2
6
4
5
3
Multiply a row by a constant.
Multiply Row 2 by 3. B
4 5 3
R becomes
3 2 6
4
3
5
4
5
3
9
6
18
5
3?
3
3?
3?
2
6
Add one row to another.
Add Row 2 to Row 1. B
4 5 3
R becomes
3 2 6
413 51 2
3
3 1
2
6
7
7
9
3
2
6
5
6
You can combine any of these steps to solve a system using a matrix.
Problem 4
Solving a System Using a Matrix
9x 2 2y 5 5
3x 1 7y 5 17
11. The system is solved below. Write a justification for each step.
B
B
e
Write the matrix for the system.
9 22
5
В ` В R
3
7 17
Multiply Row 2 by 23. Add to Row 1.
9В 022В 205)
23(3В 207В 217)
0В 223В 246)
5
9 22
В ` В R
0 223 246
Replace Row 2 by the sum.
1
(0В 223В 246)
223
1
.
Multiply Row 2 by 223
0В 201В 202)
B
9 22 5
В ` В R
0
1 2
Replace Row 1 by the sum.
Multiply Row 1 by 19 .
1
9 (9В 0В 9)
9 0 9
B
В ` В R
0 1 2
B
Multiply Row 2 by 2. Add to Row 1.
9В 22В 5)
2(0В 21В 2)
9В 20В 9)
1В 0В 1)
Replace Row 1.
1 0 1
В ` В R
0 1 2
12. Circle the solution of the system.
(1, 0)
Chapter 3
(0, 1)
(1, 2)
80
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Got It? What is the solution of the system of equations?
Problem 5 Using a Calculator to Solve a Linear System
Got It? What is the solution of the system of equations?
13. Circle the matrix that models the system.
1 4 6 21
C 2 2 1 † 4 S
0 8 1 21
1
4 6 21
C 2 22 1 † 4 S
0 28 1 21
a 1 4b 1 6c 5 21
• 2a 2 2b 1 c 5 4
28b 1 c 5 21
1
2
0 21
C 4 22 28 † 4 S
6
1
1 21
14. Use the rref() function on your calculator to find the solution.
a5 1
b5
1
2
c5 3
15. Check the solution.
a 1 4b 1 6c 5 21
2a 2 2b 1 c 5 4
Check:
1 1 4(12) 1 6(3) 0 21
1 1 2 1 18 5 21
21 5 21
28b 1 c 5 21
Check:
2(1) 2 2(12) 1 3 0 4
2211354
454
Check:
28(12) 1 3 0 21
24 1 3 5 21
21 5 21
Copyright В© by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Lesson Check • Do you UNDERSTAND?
How many elements are in a 4 3 4 matrix?
16. There are 4 rows, and each row has 4 elements.
17. A 4 3 4 matrix has 4
3 4 , or 16 elements.
Math Success
Check off the vocabulary words that you understand.
matrix
matrix element
row operation
Rate how well you can use matrices to solve systems of equations.
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review
0
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4
6
8
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get it!
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Lesson 3-6
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