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Chapter 6 Resource Masters
StudentWorks PlusTM includes the entire Student Edition text along with the worksheets in
this booklet.
TeacherWorks PlusTM includes all of the materials found in this booklet for viewing, printing,
and editing.
Cover: Jason Reed/Photodisc/Getty Images
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is
granted to reproduce the material contained herein on the condition that such materials
be reproduced only for classroom use; be provided to students, teachers, and families
without charge; and be used solely in conjunction with the Glencoe Precalculus
program. Any other reproduction, for sale or other use, is expressly prohibited.
Send all inquiries to:
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, OH 43240 - 4027
ISBN: 978-0-07-893807-8
MHID: 0-07-893807-4
Printed in the United States of America.
2 3 4 5 6 7 8 9 10 079 18 17 16 15 14 13 12 11 10
Contents
Teacher’s Guide to Using the Chapter 6
Resource Masters ........................................... iv
Lesson 6-4
Partial Fractions
Study Guide and Intervention .......................... 22
Practice............................................................ 24
Word Problem Practice ................................... 25
Enrichment ...................................................... 26
Chapter Resources
Student-Built Glossary ....................................... 1
Anticipation Guide (English) .............................. 3
Anticipation Guide (Spanish) ............................. 4
Lesson 6-5
Lesson 6-1
Linear Optimization
Study Guide and Intervention .......................... 27
Practice............................................................ 29
Word Problem Practice ................................... 30
Enrichment ...................................................... 31
Multivariable Linear Systems and Row
Operations
Study Guide and Intervention ............................ 5
Practice.............................................................. 7
Word Problem Practice ..................................... 8
Enrichment ........................................................ 9
TI-Nspire Activity ............................................. 10
Assessment
Chapter 6 Quizzes 1 and 2 ............................. 33
Chapter 6 Quizzes 3 and 4 ............................. 34
Chapter 6 Mid-Chapter Test ............................ 35
Chapter 6 Vocabulary Test ............................. 36
Chapter 6 Test, Form 1 ................................... 37
Chapter 6 Test, Form 2A................................. 39
Chapter 6 Test, Form 2B................................. 41
Chapter 6 Test, Form 2C ................................ 43
Chapter 6 Test, Form 2D ................................ 45
Chapter 6 Test, Form 3 ................................... 47
Chapter 6 Extended-Response Test ............... 49
Standardized Test Practice ............................. 50
Lesson 6-2
Matrix Multiplication, Inverses, and
Determinants
Study Guide and Intervention .......................... 11
Practice............................................................ 13
Word Problem Practice ................................... 14
Enrichment ...................................................... 15
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 6-3
Solving Linear Systems Using Inverses and
Cramer’s Rule
Study Guide and Intervention .......................... 16
Practice............................................................ 18
Word Problem Practice ................................... 19
Enrichment ...................................................... 20
Spreadsheet Activity ........................................ 21
Chapter 6
Answers ........................................... A1–A23
iii
Glencoe Precalculus
Teacher’s Guide to Using the
Chapter 6 Resource Masters
The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These
materials include worksheets, extensions, and assessment options. The answers for these
pages appear at the back of this booklet.
Practice This master closely follows the
types of problems found in the Exercises
section of the Student Edition and includes
word problems. Use as an additional
practice option or as homework for
second-day teaching of the lesson.
Chapter Resources
Student-Built Glossary (pages 1–2) These
masters are a student study tool that
presents up to twenty of the key vocabulary
terms from the chapter. Students are to
record definitions and/or examples for each
term. You may suggest that students
highlight or star the terms with which they
are not familiar. Give this to students before
beginning Lesson 6-1. Encourage them to
add these pages to their mathematics study
notebooks. Remind them to complete the
appropriate words as they study each lesson.
Word Problem Practice This master
includes additional practice in solving word
problems that apply to the concepts of the
lesson. Use as an additional practice or as
homework for second-day teaching of
the lesson.
Enrichment These activities may extend
the concepts of the lesson, offer an historical
or multicultural look at the concepts, or
widen students’ perspectives on the
mathematics they are learning. They are
written for use with all levels of students.
Graphing Calculator, TI–Nspire, or
Spreadsheet Activities These activities
present ways in which technology can be
used with the concepts in some lessons of
this chapter. Use as an alternative approach
to some concepts or as an integral part of
your lesson presentation.
Lesson Resources
Study Guide and Intervention These
masters provide vocabulary, key concepts,
additional worked-out examples and Guided
Practice exercises to use as a reteaching
activity. It can also be used in conjunction
with the Student Edition as an instructional
tool for students who have been absent.
Chapter 6
iv
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Anticipation Guide (pages 3–4) This
master, presented in both English and
Spanish, is a survey used before beginning
the chapter to pinpoint what students may
or may not know about the concepts in the
chapter. Students will revisit this survey
after they complete the chapter to see if
their perceptions have changed.
Leveled Chapter Tests
Assessment Options
The assessment masters in the Chapter 6
Resource Masters offer a wide range of
assessment tools for formative (monitoring)
assessment and summative (final)
assessment.
• Form 1 contains multiple-choice questions
and is intended for use with below grade
level students.
• Forms 2A and 2B contain multiple-choice
questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Forms 2C and 2D contain free-response
questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Form 3 is a free-response test for use
with above grade level students.
All of the above mentioned tests include a
free-response Bonus question.
Quizzes Four free-response quizzes offer
assessment at appropriate intervals in
the chapter.
Mid-Chapter Test This 1-page test
provides an option to assess the first half of
the chapter. It parallels the timing of the
Mid-Chapter Quiz in the Student Edition
and includes both multiple-choice and
free-response questions.
Vocabulary Test This test is suitable for
all students. It includes a list of vocabulary
words and questions to assess students’
knowledge of those words. This can also be
used in conjunction with one of the leveled
chapter tests.
Extended-Response Test Performance
assessment tasks are suitable for all
students. Sample answers are included for
evaluation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Standardized Test Practice These three
pages are cumulative in nature. It includes
two parts: multiple-choice questions with
bubble-in answer format and short-answer
free-response questions.
Answers
• The answers for the Anticipation Guide
and Lesson Resources are provided as
reduced pages.
• Full-size answer keys are provided for the
assessment masters.
Chapter 6
v
Glencoe Precalculus
NAME
DATE
6
PERIOD
This is an alphabetical list of key vocabulary terms you will learn in Chapter 6.
As you study this chapter, complete each term’s definition or description.
Remember to add the page number where you found the term. Add these pages to
your Precalculus Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term
Found
on Page
Definition/Description/Example
augmented matrix
coefficient matrix
constraints
Cramer’s Rule
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
determinant
feasible solutions
Gaussian elimination
Gauss-Jordan elimination
identity matrix
inverse
(continued on the next page)
Chapter 6
1
Glencoe Precalculus
Chapter Resources
Student-Built Glossary
NAME
DATE
6
PERIOD
Student-Built Glossary
Vocabulary Term
Found
on Page
Definition/Description/Example
inverse matrix
invertible
linear programming
multivariable linear system
objective function
optimization
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
partial fraction
partial fraction
decomposition
reduced row-echelon form
(esh-a-lon)
row-echelon form
singular matrix
square system
Chapter 6
2
Glencoe Precalculus
NAME
6
DATE
PERIOD
Anticipation Guide
Step 1
Before you begin Chapter 6
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or disagree,
write NS (Not Sure).
STEP 1
A, D, or NS
STEP 2
A or D
Statement
1. The augmented matrix of a system is derived from the
coefficients and constant terms of the linear equations.
2. The row-echelon form of a matrix is unique.
3. If a matrix has an inverse, then it is a singular matrix.
4. The product of an m × r matrix and an r × n matrix results
in an m × n matrix.
5. Cramer’s Rule uses inverses to solve systems.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. Cramer’s Rule applies when the determinant of the
coefficient matrix is 0.
7. To find the partial fraction decomposition of an improper
f(x)
d(x) r(x)
f(x)
algorithm − = q(x) + − to rewrite it as the sum of a
d(x)
d(x)
rational expression − , you must use the division
polynomial and a proper rational expression.
8. Graphically, a rational function and its partial fraction
decomposition are different.
9. Linear programming can be used to solve applications
involving systems of equations.
10. In a linear programming problem, one evaluates the objective
function at each vertex of the feasible region to maximize or
minimize the function.
Step 2
After you complete Chapter 6
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an example
of why you disagree.
Chapter 6
3
Glencoe Precalculus
Chapter Resources
Systems of Equations and Matrices
NOMBRE
6
FECHA
PERÍODO
Ejercicios preparatorios
Sistemas de ecuaciones y matrices
Paso 1
Antes de que comiences el Capítulo 6
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a), escribe NS (no estoy
seguro(a)).
PASO 1
A, D o NS
PASO 2
AoD
Enunciado
1. La matriz aumentada de un sistema se deriva de los
coeficientes y los términos constantes de las ecuaciones
lineales.
2. La forma escalón por filas de una matriz es única.
3. Si una matriz tiene inversa, entonces es una matriz singular.
4. El producto de una matriz m × r por una matriz r × n es una
matriz m × n.
5. La regla de Cramer permite resolver sistemas mediante el
uso de inversas.
6. La regla de Cramer se aplica cuando el determinante de la
matriz de coeficientes es cero.
f(x)
d(x)
expresión racional impropia − , se debe usar el algoritmo de
f(x)
d(x)
r(x)
d(x)
división − = q(x) + − para poder reformular dicha
expresión como la suma de un polinomio y una expresión
racional propia.
8. Una función racional y su descomposición en fracciones
parciales son gráficamente distintas.
9. La programación lineal se usa para resolver aplicaciones que
impliquen sistemas de ecuaciones.
10. En problemas de programación lineal, se evalúa la función
objetivo en cada vértice de la región factible para maximizar
o minimizar la función.
Paso 2
Después de que termines el Capítulo 6
• Relee cada enunciado y escribe A o D en la última columna.
• Compara la última columna con la primera. ¿Cambiaste de opinión sobre alguno de los
enunciados?
• En los casos en que hayas estado en desacuerdo con el enunciado, escribe en una hoja
aparte un ejemplo de por qué no estás de acuerdo.
Capítulo 6
4
Precálculo de Glencoe
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. Para calcular la descomposición en fracciones parciales de la
NAME
DATE
6-1
PERIOD
Study Guide and Intervention
Multivariable Linear Systems and Row Operations
Gaussian Elimination
You can solve a system of linear equations using
matrices. Solving a system by transforming it into an equivalent system
is called Gaussian elimination. First, create the augmented matrix.
Then use elementary row operations to transform the matrix so that it is in
row-echelon form. Then write the corresponding system of equations and
use substitution to solve the system.
Lesson 6-1
Example
Solve the system of equations using Gaussian
elimination with matrices.
x - 2y + z = -1
2x + y - 3z = -7
3x - y + 2z = 0
Step 1 Write the augmented matrix.
⎡1 -2 1 -1⎤
2 1 -3 -7
⎣3 -1 2 0⎦
⎢
Step 2 Apply elementary row operations to obtain a row-echelon form of the matrix.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a.
⎡1 -2 1 ⎢ -1⎤
R2 - 2R1→ 0 5 -5 -5
⎣3 -1 2 ⎢ 0⎦
b.
d.
e.
⎢
⎢ ⎡ 1 -2 1 -1 ⎤
0 1 -1 -1
R3 - 5R2→ ⎣ 0 0 4 8 ⎦
⎢
⎡ 1 -2 1 ⎢ -1 ⎤
0 5 -5 -5
⎣
R3 - 3R1→ 0 5 -1 ⎢ 3 ⎦
⎢
⎢ c.
⎡ 1 -2 1 -1 ⎤
1 -1 -1
⎣ 0 5 -1 3 ⎦
1
−
R→ 0
5 2
⎢
⎡ 1 -2 1 -1 ⎤
0 1 -1 -1
1
−
R3→ ⎣ 0 0 1 2 ⎦
4
⎢
Step 3 Write the corresponding system of equations and use substitution to
solve the system.
x - 2y + z = -1
y - z = -1
z=2
The solution of the system is x = -1, y = 1, and z = 2 or (-1, 1, 2).
Exercises
Solve each system of equations using Gaussian elimination with matrices.
1. -2x - y + z = 0
2. 5x - y = -13
3. -4x - y - z = -11
x + 2y - z = -3
-3x + 2y - z = 8
x-z=2
3x + y - 2z = -1
x - 4y + z = -10
2y + 4z = 0
Chapter 6
5
Glencoe Precalculus
NAME
DATE
6-1
Study Guide and Intervention
PERIOD
(continued)
Multivariable Linear Systems and Row Operations
Gauss-Jordan Elimination If you continue to apply
elementary row operations to the row-echelon form of any
augmented matrix, you can obtain a matrix in which every column
has one element equal to 1 and the remaining elements equal to 0.
This is called the reduced row-echelon form of the matrix. Solving
a system by transforming an augmented matrix so that it is in reduced
row-echelon form is called Gauss-Jordan elimination.
Example
⎡1 0 0 a⎤
0 1 0 b
⎣0 0 1 c⎦
⎢
Solve the system of equations.
x - 2y + z = 15
-2x - y + 2z = -1
-x + y = -9
Write the augmented matrix. Apply elementary row operations to obtain a
row-echelon form. Then apply elementary row operations to obtain zeros
above the leading 1s in each row.
Augmented Matrix
⎡ 1 -2 1 15 ⎤
-2 -1 2 -1
⎣ -1 1 0 -9 ⎦
⎢
⎡ 1 -2 1 15 ⎤
2R1 + R2→
0 -5 4 29
⎣ -1 1 0 -9 ⎦
⎢
⎡ 1 -2 1 15 ⎤
0 -5 4 29
R1 + R3→ ⎣ 0 -1 1 6 ⎦
Row-echelon form
⎢
⎡ 1 -2 1 15 ⎤
R2 + 2R3→ 0 1 0 -5
⎣0 0 1 1⎦
⎢
⎡ 1 -2 1 15 ⎤
0 1 -2 -7
⎣
-R3→ 0 0 1 1 ⎦
⎢
R1 + 2R2→ ⎡ 1 0 1 5 ⎤
0 1 0 -5
⎣0 0 1 1⎦
⎢
⎢
⎡ 1 -2 1 15 ⎤
0 1 -2 -7
⎣
R2 + R3→ 0 0 -1 -1 ⎦
Reduced
row-echelon form
R1 - R3→ ⎡ 1 0 0 4 ⎤
0 1 0 -5
⎣0 0 1 1⎦
⎢
The solution of the system is x = 4, y = -5, and z = 1 or (4, -5, 1).
Exercises
Solve each system of equations using Gaussian or Gauss-Jordan elimination.
1. 3x - 2y + z = -22
2. x - 4z = 6
3. -2x - y - z = 1
-4x + z = 31
-2y + 3z = -2
-x + 3y - 2z = 24
2x - 5y = -24
2x - 5y = 6
4x + 2y + z = 2
Chapter 6
6
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎡ 1 -2 1 15 ⎤
R2 - 6R3→ 0 1 -2 -7
⎣ 0 -1 1 6 ⎦
⎢
NAME
6-1
DATE
PERIOD
Practice
Multivariable Linear Systems and Row Operations
Write each system of equations in triangular form using Gaussian
elimination. Then solve the system.
⎢
⎡ -5 -3 0 -2 ⎤
3.
0 -2 6 24
⎣ 4 0 -7 2 ⎦
⎢
Write the augmented matrix for each system of linear equations.
4. 5x - 2y = 14
5. 3x + 4y + 7z = -8
-3x + y = -7
6. -4x - 2y - z = 5
-2x - 3y + z = 6
2x - z = 8
5x - 2y + z = 4
y - 2z = -4
Solve each system of equations using Gauss-Jordan elimination.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. -4x - 2y = -6
8. -2x - 5y + z = 6
x + 3y = -11
9. 8x - y + 3z = -38
3x + 2y - 4z = -1
-2x + 5y - 4z = 32
5x - y + 2z = -6
x - y + z = -9
10. FRUIT Three customers bought fruit at Michael’s Groceries. The table
shows the amount of fruit bought by each person. Write and solve a
system of equations to determine the price of each type of fruit.
Chapter 6
Name
Apples
Oranges
Pears
Total Cost ($)
Rosario
5
4
3
13.50
Lindsay
7
2
4
14.20
Edwin
3
8
2
15.30
7
Glencoe Precalculus
Lesson 6-1
⎡ 2 3 1 -23 ⎤
2. -3 -1 4
-5
⎣ -1 5 -1 -19 ⎦
⎡ 1 -1 -12 ⎤
1. ⎢
⎣ -3 2 32 ⎦
NAME
6-1
DATE
PERIOD
Word Problem Practice
Multivariable Linear Systems and Row Operations
1. FOOD Mark bought 5 hamburgers
and 3 bags of chips at a cost of $16.25.
Henry bought 4 hamburgers and 8 bags
of chips at a cost of $20. Write and solve
a system of equations to determine the
cost of a hamburger and a bag of chips.
4. MOVIES The table shows the number of
individuals attending the movies over the
weekend at the Majestic Theater.
Determine the costs for a child, adult,
and senior citizen to attend the movies.
2. MANUFACTURING A company
manufactures tables, chairs, and stools.
Last week, it built a total of 275 items.
The number of chairs built was four
times the total number of tables and
stools built. The total value of these
items is $42,125 with a chair selling for
$150, a table for $200, and a stool for
$75. Write and solve a system of
equations to determine the number of
each item built last week.
Day
Child
Adult
Senior
Citizen
Total
Paid($)
Fri
80
110
25
1755
Sat
100
175
40
2685
Sun
45
85
30
1385
a. Write a system of equations
representing Mr. Wiley’s
investment pattern.
3. COINS Tina has 31 nickels, dimes, and
quarters in her purse. She has 5 more
nickels then the total number of dimes
and quarters. If the total value of the
coins is $3.25, how many of each coin
does Tina have in her purse? Write and
solve a system to determine the number
of coins.
b. Write the augmented matrix for
the system of equations that you
wrote in part a.
c. Solve the system that you wrote in
part b using Gauss-Jordan
elimination.
Chapter 6
8
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. INVESTING Mr. Wiley invested $5000 in
three different accounts at the beginning
of last year, yielding him a total of
$182.50 of interest at the end of the year.
The three accounts were a simple savings
account earning 1%, a certificate of
deposit earning 3.5%, and municipal
bonds earning 4.3%. His municipal
bond investment was 5 times the
amount of money invested in the simple
savings account.
NAME
DATE
6-1
PERIOD
Enrichment
Circles
The general form equation for a circle is Ax2 + Ay2 + Dx + Ey + F = 0,
where A ≠ 0. Suppose you want to find the equation of a circle passing
through the points (-1, 2), (3, 4), and (2, -1). How can you use the general
form equation, systems of equations, and matrices to answer the question?
Because A ≠ 0, divide both sides of the equation by A. The resulting
D
E
F
equation is x2 + y2 + Dx + Ey + F = 0, where D = −
, E = −
, and F = −
.
A
A
Lesson 6-1
A
Because the three points are on the circle, they satisfy this equation.
Use substitution to get the following system.
1 + 4 -1D + 2E + F = 0
9 + 16 + 3D + 4E + F = 0
4 + 1 + 2D -1E + F = 0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The augmented matrix for this system is
-D + 2E + F = -5
3D + 4E + F = -25
2D + -E + F = -5
or
⎡ -1 2 1 -5 ⎤
3 4 1 -25 .
⎣ 2 -1 1 -5 ⎦
⎢
⎡ 1 -2 -1 ⎢
5⎤
Using Gaussian elimination, you can find the equivalent matrix to be 0 1 -5
5 .
⎣ 0 0 18 ⎢ -30 ⎦
⎢
⎢
30
5
10
10
Using substitution, you can find that F = - −
or - −
, E = - −
, and D = - −
.
18
3
3
3
Substituting these values back into x2 + y2 + Dx + Ey + F = 0, you get
10
10
5
x-−
y-−
= 0. Multiplying both sides of the equation by 3 results
x2 + y2 - −
3
3
3
in the equation of the circle in general form: 3x2 + 3y2 - 10x - 10y - 5 = 0.
Exercises
Find an equation of the circle passing through the given points.
1. (1, 0), (-1, 2), (3, 1)
Chapter 6
2. (3, 6), (5, 4), (3, 2)
9
Glencoe Precalculus
NAME
6-1
DATE
PERIOD
TI-Nspire Activity
Reduced Row-Echelon Form
You can solve a system of equations by entering the augmented
matrix into a TI-Nspire and finding the reduced row-echelon form
of the matrix.
Example
Solve the system of equations.
x+y+z=5
2x + 3y - z = 55
-x + 4y + 2z = 4
Step 1: Add a CALCULATOR page. Enter the augmented
matrix by pressing / and the multiplication key.
Select the 3 by 3 matrix. Change the number of
columns to 4. Type in the elements. Press ·.
Step 2: Press menu and choose MATRIX & VECTOR > REDUCED
ROW-ECHELON FORM. Press / v to enter the
matrix above which is considered the previous
answer. Press ·.
Step 3: Use the matrix to solve the system. The solution is (8, 9, -12).
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
To solve another system of three equations in three variables, you can copy
the left side of the previous line onto the current line by highlighting it and
pressing ·. Then simply update the numbers in the cells and press ·.
Exercises
Solve each system of equations.
1. x + 2y + z = 39
2. x - y -z = -11
6x + y + z = 8
2x + 3y + 2z = 24
-x + 5y - 3z = -4
4x - y + 4z = -22
3. 2x - z = 19
4. x + y + z = 10
x + 3y = 29
x - y - 2z = 31
2x - y - z = 14
2x - 2y + 3z = 97
Chapter 6
10
Glencoe Precalculus
NAME
DATE
6-2
PERIOD
Study Guide and Intervention
Matrix Multiplication, Inverses, and Determinants
Multiply Matrices
To multiply matrix A by matrix B, the number of
columns in A must be equal to the number of rows in B. If A has dimensions
m × r and B has dimensions r × n, their product, AB, is an m × n matrix.
If the number of columns in A does not equal the number of rows in B, the
matrices cannot be multiplied.
⎡ a b ⎤ ⎡ e f ⎤ ⎡ae + bg af + bh ⎤
⎢
=⎢
·⎢
⎣ c d ⎦ ⎣ g h ⎦ ⎣ ce + dg cf + dh⎦
Example
if possible.
⎡ 4 -2 ⎤
⎡ -1
Use matrices A = ⎢
and B = ⎢
⎣ -1 3 ⎦
⎣ -2
⎡ 4 -2⎤ ⎡ -1
AB = ⎢
·⎢
⎣ -1 3⎦ ⎣ -2
2 3⎤
to find AB,
4 -1 ⎦
2 3⎤
4 -1⎦
To find the first entry in AB, write the sum of the products of the entries in
row 1 of A and in column 1 of B.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎡ 4 -2⎤ ⎡ -1
⎢
·⎢
⎣ -1 3⎦ ⎣ -2
⎤
⎦
2 3⎤ ⎡ 4(-1) + (-2)(-2)
=⎢
4 -1⎦ ⎣
Follow this same procedure to find the entry for row 1, column 2 of AB.
⎡ 4 -2⎤ ⎡ -1
·⎢
⎢
⎣ -1 3⎦ ⎣ -2
2 3⎤ ⎡ 4(-1) + (-2)(-2)
=⎢
4 -1⎦ ⎣
4(2) + (-2)(4)
⎤
⎦
Continue multiplying each row by each column to find the sum for each entry.
⎡ 4 -2⎤ ⎡ -1
⎢
·⎢
⎣ -1 3⎦ ⎣ -2
2 3⎤ ⎡ 4(-1) + (-2)(-2) 4(2) + (-2)(4) 4(3) + (-2)(-1) ⎤
=⎢
4 -1⎦ ⎣ (-1)(-1) + 3(-2) (-1)(2) + 3(4) (-1)(3) + 3(-1) ⎦
Then simplify each sum.
⎡ 4 -2⎤ ⎡ -1
·⎢
⎢
⎣ -1 3⎦ ⎣ -2
2 3⎤ ⎡ 0 0 14⎤
=⎢
4 -1⎦ ⎣ -5 10 -6⎦
Exercises
Find AB and BA, if possible.
⎡-1 5⎤
⎡ 2 4⎤
1. A = ⎢
, B = ⎢
⎣ 0 -2⎦
⎣-3 -1⎦
Chapter 6
⎡-2 4 0⎤
⎡-1 3⎤
2. A = ⎢
, B = ⎢
⎣ –3 –1 2⎦
⎣-3 2⎦
11
Glencoe Precalculus
Lesson 6-2
A is a 2 × 2 matrix and B is a 2 × 3 matrix. Because the number of columns
for A is equal to the number of rows for B, the product AB exists.
NAME
DATE
6-2
Study Guide and Intervention
PERIOD
(continued)
Matrix Multiplication, Inverses, and Determinants
Inverses and Determinants The identity matrix is an n × n matrix consisting of
all 1s on its main diagonal, from upper left to lower right, and 0s for all other elements. Let
In be the identity matrix of order n and let A be an n × n matrix. If there exists a matrix B
such that AB = BA = In, then B is called the inverse of A and is written as A–1. If a matrix
has an inverse, it is invertible. The determinant of a 2 × 2 matrix can be used to
determine whether or not a matrix is invertible.
⎡a b⎤
⎡ d -b⎤
1
If A = ⎢
, det(A) = ad - cb. If ad - cb ≠ 0, then A-1 = −
⎢
.
ad - cb ⎣-c a⎦
⎣ c d⎦
Example 1
⎡ 7 -4⎤
⎡3 4 ⎤
Determine whether A = ⎢
and B = ⎢
are
⎣-5 3⎦
⎣5 7⎦
inverse matrices.
If A and B are inverse matrices, then AB = BA = I.
⎡ 7 -4⎤ ⎡3 4⎤ ⎡7(3) + (-4)(5)
AB = ⎢
·⎢
=⎢
⎣-5 3⎦ ⎣5 7⎦ ⎣ -5(3) + 3(5)
7(4) + (-4)(7) ⎤
⎡1 0⎤
or ⎢
⎣0 1⎦
-5(4) + 3(7)⎦
⎡3 4⎤ ⎡ 7 -4⎤ ⎡3(7) + 4(-5) 3(-4) + 4(3) ⎤
⎡1 0⎤
BA = ⎢
·⎢
=⎢
or ⎢
⎣5 7⎦ ⎣-5 3⎦ ⎣ 5(7) + 7(-5) 5(-4) + 7(3)⎦
⎣0 1⎦
Because AB = BA = I, B = A-1 and A = B-1.
Example 2
⎢2 -2⎢
det(A) = ⎢
⎢
⎢3 -6⎢
1 ⎡-6 2⎤
A-1 = - −
⎢
6 ⎣-3 2⎦
⎡
1⎤
⎢1 -−
3
=⎢
1
1
⎢−
-−
3⎦
⎣2
= 2(-6) - 3(-2) or -6
Since det(A) ≠ 0, A is invertible.
Exercises
Determine whether A and B are inverse matrices. Explain your reasoning.
⎡11 5⎤
⎡ 1 -5⎤
1. A = ⎢
, B = ⎢
⎣ 2 1⎦
⎣-2 11⎦
⎡3 2⎤
⎡1 2⎤
2. A = ⎢
, B = ⎢
⎣4 1⎦
⎣4 3⎦
⎡ 5 -1⎤
3. Find the determinant of A = ⎢
. Then find A-1, if it exists.
⎣-10 2⎦
⎡3 2⎤
4. Find the determinant of A = ⎢
. Then find A–1, if it exists.
⎣1 -1⎦
Chapter 6
12
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎡ 2 -2⎤
Find the determinant of A = ⎢
. Then find A–1, if it exists.
⎣ 3 -6⎦
NAME
DATE
6-2
PERIOD
Practice
Matrix Multiplication, Inverses, and Determinants
Find AB and BA, if possible.
⎡-1 6 0⎤
⎡ 2 -4⎤
1. A = ⎢
, B = ⎢
⎣ 3 -2 1⎦
⎣ -1 3⎦
⎡ 3 0⎤
⎡ 3 5⎤
2. A = ⎢
, B = ⎢
⎣-1 2⎦
⎣ -2 0⎦
Company
Club Type and Quantity
Club
Club Value ($)
1-Wood
3-Wood
5-Wood
Putter
1-Wood
210
A
600
520
310
300
3-Wood
170
B
210
400
450
400
5-Wood
150
Putter
120
Write each system of equations as a matrix equation, AX = B. Then use
Gauss-Jordan elimination on the augmented matrix to solve for X.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. x1 - 2x2 + 3x3 = 4
5. 2x1 + x2 + 2x3 = 11
5x1 + 3x2 - x3 = 13
-5x1 - x2 + 4x3 = 1
4x1 - x2 + 4x3 = 11
3x1 - 2x2 + 8x3 = 28
Determine whether A and B are inverse matrices.
⎡1 2⎤
⎡ 3 -2⎤
6. A = ⎢
, B = ⎢
⎣1 3⎦
⎣-1 1⎦
⎡5 -2⎤
⎡-1 0⎤
7. A = ⎢
, B = ⎢
⎣4 -3⎦
⎣ 2 -8⎦
Find the determinant of each matrix. Then find its inverse, if it exists.
⎡6 5⎤
8. ⎢
⎣2 2⎦
Evaluate.
⎡-1 5⎤
A =⎢
⎣ 3 0⎦
10. AB + C
Chapter 6
⎡-2
4⎤
9. ⎢
⎣ 3 -6⎦
⎡-4 2 -1⎤
B =⎢
⎣ 0 -5 3⎦
⎡-1 0 -4⎤
C =⎢
⎣ 3 -2 1⎦
11. A(B - C)
13
Glencoe Precalculus
Lesson 6-2
3. GOLF The number of golf clubs manufactured daily by two different companies is
shown, as well as the selling price of each type of club. Use this information to
determine which company’s daily production has the highest retail value. How much
greater is the value?
NAME
DATE
6-2
PERIOD
Word Problem Practice
Matrix Multiplication, Inverses, and Determinants
1. INVENTORY A hardware company keeps
three types of lawnmowers in stock at
each of its three stores. The current
inventory and retail price for each mower
is shown. Determine which store’s
inventory has the greatest value. What is
this value?
3. LANDSCAPING Two dump trucks have
capacities of 10 tons and 12 tons. They
make a total of 20 round trips to haul
226 tons of topsoil for a landscaping
project. How many round trips does each
truck make?
Store
Mower Type
4 HP
A
B
C
5
4
3
4.5 HP
3
5
4
5 HP
7
2
3
Mower Type
4 HP
4.5 HP
5 HP
Retail Value ($)
250
300
350
4. CRAFTS A craft store orders beads from
three different vendors, A, B, and C. One
month, the store ordered a total of 150
units of beads from these vendors. The
shipping charges are as shown.
C
I
6
4
2
Joelle
3
5
1
Luisa
2
4
6
System A
System B
SS
20%
40%
C
50%
30%
I
30%
30%
C
40
30
b. Write the system of equations that
you found in part a as a matrix
equation, DX = E.
One of two weighted systems shown
below is used.
Criteria
B
35
c. Solve the system that you found in
part b to determine how many units
of beads were purchased from each of
the vendors.
Use matrices to determine which system
favors each skater.
a. Holly
b. Joelle
c. Luisa
Chapter 6
14
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
SS
Holly
A
Charge per unit ($)
The total delivery cost was $5375. The
store ordered twice the number of units
of beads from vendor C than it ordered
from vendor A.
a. Write a system of equations
representing this situation.
2. ICE SKATING Holly, Joelle, and Luisa
are competitive skaters. Their routines
are judged on skating skill (SS),
choreography (C), and interpretation (I).
In a recent competition, they received the
following scores.
Skater
Vendor
NAME
DATE
6-2
PERIOD
Enrichment
Travel
1
Suppose three state parks P1, P2, and P3 are connected by
roads as shown in the figure. As you can see, there are only
two ways to travel from P1 to P2 without going through P3.
There are only three ways to travel from P1 to P3 without
going through P2. The matrix M represents the number of
ways to go from one park to another without traveling
through the third park.
P1
P1
M=
P2
P3
P2
1
1
P3
⎡ 0 2 3⎤
2 0 1
⎣ 3 1 0⎦
⎢
Lesson 6-2
Confirm the numbers in each cell. For example, the element in row 2,
column 3 indicates that there is only one way to travel from P2 to P3 without
going through P1. The element in row 1, column 3 indicates there are three
ways to travel from P1 to P3 without going through P2.
It can be shown that if we square matrix M, we can determine how many
ways there are to travel from one park to another by traveling through
the third park.
P1
P2
P3
P1
P2
P3
P1
P2
P3
⎡ 0 2 3⎤ P1 ⎡ 0 2 3⎤ P1 ⎡13 3 2⎤
2
M = P2 2 0 1 P2 2 0 1 = P2 3 5 6
P3 ⎣ 3 1 0⎦ P3 ⎣ 3 1 0⎦
P3 ⎣ 2 6 10⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
P1
⎢
⎢
⎢
Consider the product of row 1 and column 2.
0
P1→P1
×
2
P1→P2
+
2
P1→P2
×
0
P2→P2
+
3
P1→P3
×
1
P3→P2
=
3
The result indicates that there are three ways to travel from P1 to P2 by going through P3.
Similarly, consider the product of row 3 and column 2.
3
P3→P1
×
2
P1→P2
+
1
P3→P2
×
0
P2→P2
+
0
P3→P3
×
1
P3→P2
=
6
The result indicates that there are six ways to travel from P3 to P2 by going through P1.
Exercises
1. What does the product of row 1 and column 3 indicate?
2. What does the product of row 2 and column 1 indicate?
Chapter 6
15
Glencoe Precalculus
NAME
DATE
6-3
PERIOD
Study Guide and Intervention
Solving Linear Systems Using Inverses and Cramer’s Rule
Use Inverse Matrices A square system has the same
number of equations as variables. If a square matrix has an
inverse, the system has one unique solution.
Example
Use an inverse matrix to solve each system of equations, if possible.
a. 3x - 7y = -16
-x + 2y = 8
b. -2x + y + z = 0
x + 2z = 9
x - 2y - 9z = -31
Write the system in matrix form.
A
· X = B
Write the system in matrix form.
A · X =
B
⎡-2 1 1⎤ ⎡x⎤ ⎡ 0⎤
9
1 0 2 · y =
⎣ 1 -2 -9⎦ ⎣ z⎦ ⎣-31⎦
Use a graphing calculator to find A-1.
⎡-16⎤
⎡ 3 -7 ⎤ ⎡x⎤
⎢
·⎢ = ⎢
y
⎣ 8⎦
⎣-1 2 ⎦ ⎣ ⎦
⎢
Use the formula for an inverse of a 2 × 2
matrix to find the inverse A-1.
⎢ ⎢
⎡ d -b⎤
1
A-1 = −
⎢
a⎦
ad - cb ⎣-c
⎡2
1
= −−
⎢
(2)(3) - (-7)(-1) ⎣1
7⎤
3⎦
Multiply A-1 by B to solve the system.
X=
A-1
· B
⎡ 4 7 2⎤ ⎡ 0⎤ ⎡ 1⎤
= 11 17 5 · 9 = -2
⎣-2 -3 -1⎦ ⎣-3⎦ ⎣ 4⎦
So, the solution of the system is
(1, -2, 4).
⎢
So, the solution of the system is
(-24, -8).
⎢ ⎢ Exercises
Use an inverse matrix to solve each system of equations, if possible.
1. -2x + 5y = 24
2. x - y + 2z = 5
3x - y = -10
x - z = -4
3x + 2y + z = 0
3. 3x + y = 7
4. x + y - z = -5
-2x - 5y = 43
2x - 3y + 2z = 20
y + 4z = 18
Chapter 6
16
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Multiply A-1 by B to solve the system.
X =
A-1 ·
B
⎡-24⎤
⎡-2 -7 ⎤ ⎡-16⎤
= ⎢
·⎢
= ⎢
⎣ -8⎦
⎣-1 -3 ⎦ ⎣ 8⎦
NAME
DATE
6-3
Study Guide and Intervention
PERIOD
(continued)
Solving Linear Systems Using Inverses and Cramer’s Rule
Use Cramer’s Rule Another method, known as Cramer’s Rule, can be used
to solve a square system of equations.
Let A be the coefficient matrix of a system of n linear equations in n
variables given by AX = B. If det(A) ≠ 0, then the unique solution of the
system is given by
|A |
|A |
|A |
|A |
|A|
|A|
|A|
|A|
3
n
1
2
x1 = −
, x2 = −
, x3 = −
, … , xn = −
,
where Ai is the matrix obtained by replacing the ith column of A with the
column of constants B. If det(A) = 0, then AX = B has either no solution or
infinitely many solutions.
Example
Use Cramer’s Rule to find the solution of the system
of linear equations, if a unique solution exists.
-2x1 + x2 = -7
5x1 - 2x2 = 17
⎡-2 1⎤
The coefficient matrix is A = ⎢
. Calculate the determinant of A.
⎣ 5 -2⎦
⎪-25 -21 ⎥ = (-2)(-2) - 5(1) or -1
Because the determinant of A does not equal zero, you can apply
Cramer’s Rule.
x1
-7 1
⎪
-7(-2) - 17(1)
17 -2⎥
=−=−= −
-3
=−
or 3
x2
-2 -7
⎪
-2(17) - 5(-7)
5 17⎥
=−=−= −
1
=−
or -1
|A1|
|A|
-1
-1
|A2|
|A|
-1
-1
Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
|A| =
-1
-1
Therefore, the solution is x1 = 3 and x2 = -1 or (3, -1).
Exercises
Use Cramer’s Rule to find the solution of each system of linear
equations, if a unique solution exists.
1. x - 2y = -5
2. 3x - 3y = -18
-2x - 5y = -8
3. 3x + y = 21
-x + 4y = 9
4. -2x - 4y = 2
-x + 2y = 14
Chapter 6
x + 3y = -3
17
Glencoe Precalculus
NAME
6-3
DATE
PERIOD
Practice
Solving Linear Systems Using Inverses and Cramer’s Rule
Use an inverse matrix to solve each system of equations, if possible.
1. 4x - 7y = 30
2. -2x - 8y = -36
-6x + 2y = -11
4x + 3y = 7
3. x - 2y + 7z = -33
4. x + y - 2z = 5
-4x + 5y - z = 18
x + 2y + z = 8
5x - 3y = -11
2x + 3y - z = 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. TELEVISION During the summer, Manuel watches television M hours
per day, Monday through Friday. Harry watches television H hours per
day, Friday through Sunday. Ellen watches television E hours per day,
Friday and Saturday. Altogether, they watch television 33 hours each
week. On Fridays, they watch a total of 11 hours of television. If the
number of hours Ellen spends watching television on any given day is
twice the number of hours that Manuel spends watching television on
any given day, how many hours of television does each of them watch
each day?
Use Cramer’s Rule to find the solution of each system of linear
equations, if a unique solution exists.
6. -4x - 5y = 1
7. x + y + z = 8
-2x - 3y = -1
3x - z = -22
y + 2z = 20
8. PAPER ROUTE Payton, Santiago, and Queisha each have a paper
route. Payton delivers 5 times as many papers as Santiago. Santiago
delivers twice as many papers as Queisha. If 20 papers were added to
Payton’s route, he would then deliver four times the total number of
papers that Santiago and Queisha deliver. How many papers does each
person deliver?
Chapter 6
18
Glencoe Precalculus
NAME
DATE
6-3
PERIOD
Word Problem Practice
Solving Linear Systems Using Inverses and Cramer’s Rule
1. PERIMETER The perimeter of rectangle
WXYZ is 92 centimeters. The perimeter
of triangle WXZ is 80 centimeters. If the
−−
length of XZ is two more than twice the
−−−
length of WZ, what are the values
of a, b, and c?
B
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
;
C
9
D
4. NUTS A nut company makes three types
of one-pound gift boxes: X, Y,
and Z. The table shows the amount of
each nut in each box.
:
Gift
Box
Cashews
Almonds
Hazelnuts
X
0.2
0.3
0.5
Y
0.4
0
0.6
Z
0.4
0.4
0.2
The company has 10,000 pounds of
cashews, 7000 pounds of almonds,
and 12,000 pounds of hazelnuts in
its gift boxes.
2. BASEBALL In one season, Marty,
Carlos, and Andrew hit a total of 108
home runs. Marty and Andrew together
hit twice as many home runs as did
Carlos, although Carlos had one more
home run than Andrew. How many home
runs did each player hit?
a. Write a system of equations
representing this situation.
b. Solve the system of equations that you
wrote in part a as a matrix equation,
AX = B.
c. Determine how many of each gift box
the company has.
Chapter 6
19
Glencoe Precalculus
Lesson 6-3
8
3. INVESTING A total of $4000 is invested
in three accounts paying 4%, 5%, and
3.5% simple interest. The combined
annual interest is $173.75. If the interest
earned at 5% is $70 more than the
interest earned at 4%, how much money
is invested in each account?
NAME
6-3
DATE
PERIOD
Enrichment
Pick’s Theorem
Consider the simple polygon drawn on square dot paper, shown at
the right. Pick’s Theorem states that the area of the polygon A is
equal to half the number of boundary points b plus the number of
b
interior points n minus 1, or A = −
+ n - 1. In the figure, b = 10
2
10
+ 2 - 1 or 6 square units.
and n = 2. Therefore, A = −
2
You can use systems of equations and matrices to verify Pick’s Theorem. To verify that the
1
coefficients in the equation for A are −
, 1, and -1, you can write a system of three equations
2
of the form A = bx + ny + z, where the values of A, b, and n vary from polygon to polygon.
Start by drawing three simple polygons on square dot paper similar to the ones shown
below. Be sure that the number of boundary points, interior points, and the area of the
figures are different. The table shown below summarizes the information.
Figure
b
n
A
square
8
1
4
rectangle
18
12
20
triangle
11
6
10.5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use the table to write a system and matrix equation.
8x + y + z = 4
⎡ 8 1 1⎤ ⎡x⎤ ⎡ 4 ⎤
18x + 12y + z = 20
18 12 1 y = 20
11x + 6y + z = 10.5
⎣11 6 1⎦ ⎣ z⎦ ⎣10.5⎦
1
Solving this system, we can see that x = −
, y = 1, and z = -1.
⎢
⎢ ⎢
2
Exercises
1. Write the matrix equation that would be
used to verify Pick’s Theorem using the
polygons at the right.
2. Verify Pick’s Theorem using three simple
polygons of your choice.
Chapter 6
20
Glencoe Precalculus
NAME
DATE
6-3
PERIOD
Spreadsheet Activity
Cramer’s Rule
You can use a spreadsheet to solve systems of equations with Cramer's Rule.
A
1
2
3
4
5
6
7
8
9
10
11
12
6
5
B
3
1
C
To use the spreadsheet to solve a
system of equations, write each
equation in the form below.
ax + by = c
In the spreadsheet, the values of
a, b, and c for the first equation are
entered in cells A1, B1, and C1,
respectively. The values of a, b, and c
for the second equation are entered in
cells A2, B2, and C2, respectively. The
values for the system 6x + 3y = -12
and 5x + y = 8 are shown.
D
-12
8
= A1*B2 - B1*A2
= C1*B2 - B1*C2
= A1*C2 - C1*A2
x=
y=
= (A6/A4)
= (A8/A4)
Exercises
2. Write matrices whose determinants are found using the
formulas in cells A6 and A8.
Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. Study the formula in cell A4. Write a matrix whose
determinant is found using this formula.
3. Explain how the values of x and y are found using
Cramer’s Rule.
Use the spreadsheet to solve each system of equations.
4. 6x + 3y = -12
5x + y = 8
5. 5x - 3y = 19
7x + 2y = 8
6. 0.3x + 1.6y = 0.44
0.4x + 2.5y = 0.66
7. 3y = 4x + 28
5x + 7y = 8
Chapter 6
21
Glencoe Precalculus
NAME
6-4
DATE
PERIOD
Study Guide and Intervention
Partial Fractions
Linear Factors The function g(x) shown below can be written
as the sum of two fractions with denominators that are linear
factors of the original denominator.
3x - 1
2
-1
=−
+−
g(x) = −
2
2x - 3x + 1
x-1
2x - 1
Each fraction in the sum is a partial fraction. The sum of these partial
fractions make up the partial fraction decomposition of the original
rational function. If the denominator of a rational expression contains a
repeated linear factor, the partial fraction decomposition must include a
partial fraction with its own constant numerator for each power of this factor.
x + 11
x - 3x - 4
Example
.
Find the partial fraction decomposition of −
2
Rewrite the equation as partial fractions with constant numerators, A and B,
and denominators that are the linear factors of the original denominator.
x + 11
x - 3x - 4
A
B
−
=−
+−
2
x-4
Form a partial fraction decomposition.
x+1
x + 11 = A(x + 1) + B(x - 4)
Multiply each side by the LCD, x2 - 3x - 4.
x + 11 = Ax + A + Bx - 4B
Distributive Property
Group like terms.
1x + 11 = (A + B)x + (A + (-4B))
A+B=1
A + (-4B) = 11
→
C
·
X
=
D
⎡1 1⎤
⎢
⎣1 -4⎦
·
⎡ A⎤
⎢ ⎣B⎦
=
⎡ 1⎤
⎢
⎣ 11 ⎦
x + 11
3
-2
Solving for X yields A = 3 and B = -2. Therefore, −
=−
+−
.
2
x - 3x - 4
x-4
x+1
Exercises
Find the partial fraction decomposition of each rational expression.
5x - 34
1. −
2
x - x - 12
x2 + 1
2x(x - 1)
3. −2
Chapter 6
-7x + 13
x - 5x - 14
2. −
2
2
-x-1
−
4. 5x
2
x (x - 1)
22
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Equate the coefficients on the left and right side of the equation to obtain a
system of two equations. To solve the system, write it in matrix form CX = D
and solve for X.
NAME
DATE
6-4
PERIOD
Study Guide and Intervention
(continued)
Partial Fractions
Irreducible Quadratic Factors
Not all rational expressions
can be written as the sum of partial fractions using only linear
factors in the denominator. If the denominator of a rational expression
contains an irreducible quadratic factor, the partial fraction decomposition
must include a partial fraction with a linear numerator of the form Bx + C
for each power of this factor.
4x4 - 2x3 - 13x2 + 7x + 9
x(x - 3)
Example
Find the partial fraction decomposition of −−
.
2
2
This expression is proper. The denominator has one linear factor and one
irreducible factor of multiplicity 2.
4x4 - 2x3 - 13x2 + 7x + 9
Bx + C
Dx + E
A
−−
=−
+−
+−
x2 - 3
x(x2 - 3)2
(x2 - 3)2
x
4x4 - 2x3 -13x2 + 7x + 9 = A(x2 - 3)2 + (Bx + C)x(x2 - 3) + (Dx + E)x
4x4 - 2x3 -13x2 + 7x + 9 = Ax4 + Bx4 + Cx3- 6Ax2 - 3Bx2 + Dx2 - 3Cx + Ex + 9A
4x4 - 2x3 -13x2 + 7x + 9 = (A + B)x4 + Cx3 + (-6A - 3B + D)x2 + (-3C + E)x + 9A
A+B=4
A
=
1
C = -2
B
=
3
C
= -2
-3C + E = 7
D
=
2
9A = 9
E
=
1
-6A - 3B + D = -13
→
4x4 - 2x3 - 13x2 + 7x + 9
2x + 1
3x - 2
1
Therefore, −−
=−
+−
+−
.
2
2
2
2
2
x(x - 3)
x
x -3
(x - 3)
Exercises
Find the partial fraction decomposition of each rational expression.
5
1. −
3
3x3 - 2x2 - 8x + 5
(x - 3)
x + 5x
2. −−
2
2
2x3 + x + 3
(x + 1)
4. −
2
2
3. −
2
2
Chapter 6
x3 + 2x2 + 2
(x + 1)
23
Glencoe Precalculus
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write and solve the system of equations obtained by equating coefficients.
NAME
6-4
DATE
PERIOD
Practice
Partial Fractions
Find the partial fraction decomposition of each rational expression.
3x - 7
1. −
2
2
- 10x - 2
−
2. 6x
3
2
9x + 15
3. −
2
x
4. −
2
x - 7x + 12
x + 3x + 2
x + x - 2x
2x - 9x + 9
Find the partial fraction decomposition of each improper rational expression.
-5x2 - 11x + 54
x + 2x - 8
3x2 + 5x + 2
x + 2x
6. −
2
6x2 + 17x + 2
x +x
8. −
2
5. −
2
7. −
2
-8x2 + 22x - 10
(2x - 3)
5x4 - 7x3 - 12x2 + 6x + 21
(x - 3)(x - 2)
-2x2 + 29x - 100
x - 10x + 25x
10. −−
2
2
2x2 + 5
x + 6x + 9x
12. −−
2
2
9. −−
3
2
11. −
3
2
4x4 + 8x3 + 6x2 + 6x + 5
(3x + 2)(x + 1)
13. GROWTH When working with exponential growth in calculus, it is often
1
x(50 - x)
necessary to work with functions of the form f(x) = − and to
decompose these functions into the sum of its partial fractions. Find the
partial decomposition of f(x).
Chapter 6
24
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the partial fraction decomposition of each rational expression with
repeated factors.
NAME
6-4
DATE
PERIOD
Word Problem Practice
Partial Fractions
1. CHEMISTRY A chemist uses the
4. AREA Calculus can be used to find
the area of the shaded region shown
below. The shaded region is bounded
50(25 + x)
function f(x) = − to determine
50 + x
how much acid she must mix with a 25%
acid solution to achieve the desired
percentage. Find the partial fraction
decomposition of f(x).
x - 44
by the graphs of f(x) = −
, y = 0,
2
x - 11x
x = 2, and x = 8. To find the area, first
find the partial fraction decomposition
of f(x).
2. INFECTIONS A function of the form
1
, where a > 0 often
g(x) = −
(x + 1)(a - x)
plays a role when studying the spread of
an infection in certain populations. Find
the partial fraction decomposition of
g(x) when a = 350.
[0, 10] scI: 1 by [-2, 10] scI: 1
3. VOLUME Consider the domain
0 ≤ x ≤ 10 for the graph of f(x)
shown below.
10
y
8
6
b. Write the matrix form AX = B for the
system of equations found in part a.
f (x) =
60
x(x + 3)2
c. Find the partial fraction decomposition
of the rational expression.
4
2
0
2
4
6
8
x
10
Suppose you were to revolve the graph
of f(x) around the x-axis, creating a
three-dimensional object. Using calculus,
you could find the volume of the object.
But first, you would need to find the
partial fraction decomposition of f(x).
Find the partial decomposition of f(x).
5. KAYAKING The total time it takes
for a kayaker to travel 10 miles upstream
and 10 miles downstream with a
paddling rate of 4 miles per hour in
still water is given by the function
-80
f(x) = −
. Find the partial fraction
2
x - 16
decomposition of f(x).
Chapter 6
25
Glencoe Precalculus
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Write the system of equations
obtained by equating the coefficients.
NAME
6-4
DATE
PERIOD
Enrichment
Heaviside Method
Another method that can be used to find the partial fraction decomposition
of a rational expression is called the Heaviside Method. Consider the
equation shown below.
2x - 47
A
B
−
+−
=−
2
x+4
x - 3x - 28
x-7
Multiply both sides of the equation by the least common denominator (x + 4)(x - 7).
⎡ A
2x - 47
B ⎤
= (x + 4)(x - 7) ⎢−
(x + 4)(x - 7) −
+−
2
x - 7⎦
x -3x - 28
⎣x + 4
2x - 47 = A(x - 7) + B(x + 4)
To solve for A, let x = -4. This eliminates B.
2(-4) - 47 = A(-4 - 7) + B(-4 + 4)
-55 = -11A
5 = A
To solve for B, let x = 7. This eliminates A.
2(7) - 47 = A(7 - 7) + B(7 + 4)
-33 = 11B
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-3 = B
Substitute A = 5 and B = -3 into the original equation to find the partial
fraction decomposition.
2x - 47
5
-3
−
+−
= −
2
x - 3x - 28
x+4
x-7
Exercises
Use the Heaviside Method to write the partial fraction
decomposition of each rational expression.
x + 39
x - 3x - 18
1. −
2
2
- 8x - 4
−
2. 3x
2
3
13x - 51
3. −
2
x - 8x + 15
4. −
2
18x2 + 39x - 30
x - x - 6x
-x - 114
6. −
2
5. −
3
2
Chapter 6
x + x - 2x
-6x + 18
x - 10x + 24
x + 3x - 54
26
Glencoe Precalculus
NAME
DATE
6-5
PERIOD
Study Guide and Intervention
Linear Optimization
Linear Programming Linear programming is a process for finding a minimum or
maximum value for a specific quantity. The following steps can be used to solve a linear
programming problem.
Step
Step
Step
Step
1
2
3
4
Write an objective function and a list of constraints to model the situation.
Graph the region corresponding to the solution of the system of constraints.
Find the coordinates of the vertices of the region formed.
Evaluate the objective function at each vertex to find the minimum
or maximum.
A leather company wants to add belts and wallets to its product
line. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets
require 3 hours of cutting time and 3 hours of sewing time. The cutting machine
is available 12 hours a week and the sewing machine is available 18 hours per
week. Belts will net $18 in profit and wallets will net $12. How much of each
product should be produced to achieve maximum profit?
Let x represent the number of belts and y represent the number of wallets.
The objective function is then given by f(x, y) = 18x + 12y.
Write the constraints.
x ≥ 0; y ≥ 0
Numbers of items cannot be negative.
2x + 3y ≤ 12
Cutting time
6x + 3y ≤ 18
Sewing time
Graph the system. The solution is the shaded region, including its
y
boundary segments. Find the coordinates of the four vertices by solving
the system of boundary equations for each point of intersection. The
coordinates are (0, 0), (0, 4), (1.5, 3), and (3, 0).
(1.5, 3)
(0, 4)
Evaluate the objective function for each ordered pair.
Point
f(x, y) = 18x + 12y
Result
(0, 0)
f(0, 0) = 18(0) + 12(0)
0
(0, 4)
f(0, 4) = 18(0) + 12(4)
48
(1.5, 3)
f(1.5, 3) = 18(1.5) + 12(3)
63
(3, 0)
f(3, 0) = 18(3) + 12(0)
54
0
(3, 0)
x
← Maximum
Since f is greatest at (1.5, 3), the company will maximize profit if it makes and sells
1.5 belts for every 3 wallets.
Exercises
Find the maximum and minimum values of the objective function f(x, y) and for
what values of x and y they occur, subject to the given constraints.
1. f(x, y) = 3x - 2y
2x + y ≤ 10
x + 2y ≤ 8
x≥0
y≥0
Chapter 6
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example
2. f(x, y) = x + 2y
x+y≤4
x + 3y ≤ 6
x≥0
y≥0
27
Glencoe Precalculus
NAME
6-5
DATE
PERIOD
Study Guide and Intervention
(continued)
Linear Optimization
No or Multiple Optimal Solutions
Linear programming models
can have one, multiple, or no optimal solutions. If the graph of the objective
function f to be optimized is coincident with one side of the region of feasible
solutions, f has multiple optimal solutions. If the region does not form
a polygon, but instead is unbounded, f may have no minimum value or
maximum value.
Example
Find the maximum value of the objective function
f(x, y) = 6x + 3y and for what values of x and y it occurs, subject to
the following constraints.
2x + y ≤ 8
y≤4
x≤3
x≥0
y≥0
Graph the region bounded by the given constraints. Find the value of the
objective function f(x, y) = 6x + 3y at each vertex.
f(0, 0) = 6(0) + 3(0) or 0
y
f(0, 4) = 6(0) + 3(4) or 12
(2, 4)
f(2, 4) = 6(2) + 3(4) or 24
(3, 2)
f(3, 2) = 6(3) + 3(2) or 24
x
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f(3, 0) = 6(3) + 3(0) or 18
Because f(x, y) = 24 at (2, 4) and (3, 2), the problem has multiple optimal
solutions. An equation of the line through these two vertices is y = -2x + 8.
Therefore, f has a maximum value of 24 at every point on y = -2x + 8
for 2 ≤ x ≤ 3.
Exercises
Find the maximum and minimum values of the objective function
f(x, y) and for what values of x and y they occur, subject to the
given constraints.
1. f(x, y) = 2x + y
2. f(x, y) = 3x + 6y
2x + y ≤ 11
x - y ≥ -4
y≤5
x + 2y ≤ 20
y≥0
x≥0
x≤5
x≤8
x≥0
y≥0
Chapter 6
28
Glencoe Precalculus
NAME
DATE
6-5
PERIOD
Practice
Linear Optimization
Find the maximum and minimum values of the objective function
f(x, y) and for what values of x and y they occur, subject to the given
constraints.
1. f(x, y) = 2x + 5y
2. f(x, y) = 4x + 3y
x≥0
x≥0
y≥0
y≥0
x+y≤7
2x + 3y ≥ 6
2x + 3y ≤ 18
x+y≤8
4. f(x, y) = 3x + 3y
x≥0
x≥0
x≤7
y≥0
y≥0
y≤8
y≤5
x + y ≤ 10
x + 2y ≥ 14
3x + 2y ≤ 24
5. SKATES A manufacturer produces roller skates and ice skates.
Manufacturer Information
Roller Skates
Ice Skates
Maximum Time Available
Assembling
5 minutes
4 minutes
200 minutes
Checking and Packaging
1 minute
4 minutes
120 minutes
Profit per Skate
$40
$30
a. Write an objective function and list the constraints that model the
given situation.
c. How many roller skates and ice skates should be
manufactured to maximize profit? What is the maximum
profit?
d. Describe why the company would choose a number of roller
skates and ice skates different from the answer in part c.
Chapter 6
29
60
y
50
40
30
20
10
0
x
10 20 30 40 50 60
Roller Skates
Glencoe Precalculus
Lesson 6-5
b. Sketch a graph of the region determined by the constraints
from part a to find the set of feasible solutions for the
objective function.
Ice Skates
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. f(x, y) = 2x - 3y
NAME
DATE
6-5
PERIOD
Word Problem Practice
Linear Optimization
1. FARMING Mr. Fields owns a 360-acre
farm on which he plants corn and
soybeans. The table shows the cost of
labor and the profit per acre for each
crop. Mr. Fields can spend up to $60,000
for spring planting.
Per Acre
Corn
Soybeans
Labor ($)
300
150
Profit ($)
340
300
2. NUTRITION A certain diet recommends
at least 140 milligrams of Vitamin A and
at least 145 milligrams of Vitamin B
daily. These requirements can be
obtained from two types of food. Type X
contains 10 milligrams of Vitamin A and
20 milligrams of Vitamin B per pound.
Type Y contains 30 milligrams
of Vitamin A and 15 milligrams of
Vitamin B per pound. Type X costs $12
per pound. Type Y costs $8 per pound.
a. Write an objective function and
list the constraints that model
this situation.
a. Write an objective function and
list the constraints that model
this situation.
450
400
b. Sketch a graph of the region
determined by the constraints from
part a to find the feasible solutions
for the objective function.
b
y
12
350
300
8
250
4
200
150
0
100
50
0
12
x
c
50
100
150
200
c. How many pounds of each type of food
should be purchased to satisfy the
requirements at the minimum cost?
What is the minimum cost?
250
Acres of Corn
c. How can Mr. Fields maximize his
profit? What is his maximum profit?
Chapter 6
4
30
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Acres of Soybeans
b. Sketch a graph of the region
determined by the constraints from
part a to find the feasible solutions for
the objective function.
NAME
6-5
DATE
PERIOD
Enrichment
Convex Polygons
You have already learned that over a closed convex polygonal region, the
maximum and minimum values of any linear function occur at the vertices
of the polygon. To see why the values of the function at any point on the
boundary of the region must be between the values at the vertices, consider
the convex polygon with vertices P and Q.
−−−
Let W be a point on PQ.
y
PW
If W lies between P and Q, let −
= w.
PQ
8
x
0
Example
If f(x, y) = 3x + 2y, find the maximum value of the
function over the shaded region at the right.
y
The maximum value occurs at the vertex (6, 3). The minimum value
occurs at (0, 0). The values of f(x, y) at W 1 and W 2 are between the
maximum and minimum values.
f(Q) = f(6, 3) = 3(6) + 2(3) or 24
f(W 1) = f(2, 1) = 3(2) + 2(1) or 8
1
f(W 2) = f(5, 2.5) = 3(5) + 2(2.5) or 20
0
f(P) = f(0, 0) = 3(0) + 2(0) or 0
3
2
8
8
x
Exercises
−−
Let P and Q be vertices of a closed convex polygon, and let W lie on PQ.
Let f(x, y) = ax + by.
1. If f(Q) = f(P), what is true of f ? of f(W)?
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Then 0 < w < 1 and the coordinates of W are
((1 - w)x 1 + wx 2, (1 - w)y 1 + wy2). Now consider
the function f(x, y) = 3x - 5y.
f(W) = 3[(1 - w)x 1 + wx 2] - 5[(1 - w)y 1 + wy 2]
= (1 - w)(3x 1) + 3wx 2 + (1 - w)(-5y1) - 5wy 2
= (1 - w)(3x1 - 5y1) + w(3x 2 - 5y2)
= (1 - w)f(P) + wf(Q)
This means that f(W) is between f(P) and f(Q), or that the
greatest and least values of f(x, y) must occur at P or Q.
1
2
2. If f(Q) = f(P), find an equation of the line containing P and Q.
Chapter 6
31
Glencoe Precalculus
NAME
6
DATE
PERIOD
Chapter 6 Quiz 1
SCORE
(Lessons 6-1 and 6-2)
1. 3x - 2y = 1
2. 2x - 5y + 7z = -9
-5x + y = -11
Assessment
Write the augmented matrix for each system of
linear equations.
1.
-x - y + 2z = 1
-3x + 4y - z = 10
Solve each system of equations.
2.
3. -4x + 2y = 22
3.
4. -3x + y - 2z = 2
x - 3y = -8
2x - y = -5
-5x - 2z = 1
4.
⎡ 2 -4 ⎤
.
5. MULTIPLE CHOICE Find the determinant of ⎢
⎣ -3 5 ⎦
A -22
B -2
C 2
D 22
6. Find A-1, if it exists.
⎡ 3 -7 ⎤
A=⎢
⎣-2 5⎦
6.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NAME
6
5.
DATE
PERIOD
Chapter 6 Quiz 2
SCORE
(Lesson 6-3)
Use an inverse matrix to solve each system of equations,
if possible.
1. -4x + y = 18
2. -3x - y = 8
x - 5y = -14
1.
5x + 3z = -16
-4y - z = 10
2.
Use Cramer’s Rule to find the solution of each system of
linear equations, if a unique solution exists.
3. 2x + 5y = 3
4. 3x - y + 2z = -3
-x - 3y = -5
3.
-x + 2y - z = 2
2x - 3y + z = - 1
4.
5. MULTIPLE CHOICE Find the inverse matrix required
⎡ -4 7 ⎤ ⎡ x ⎤ ⎡-19⎤
·⎢ =⎢
to solve ⎢
.
⎣ -1 2 ⎦ ⎣ y ⎦ ⎣ -5⎦
⎡2 7⎤
A ⎢
⎣ 1 -4 ⎦
Chapter 6
⎡ 2 -7 ⎤
B ⎢
⎣ 1 -4 ⎦
⎡ -4 7 ⎤
D ⎢
⎣ -1 2 ⎦
⎡-2 7 ⎤
C ⎢
⎣-1 4 ⎦
33
5.
Glencoe Precalculus
NAME
DATE
6
PERIOD
Chapter 6 Quiz 3
SCORE
(Lesson 6-4)
Find the partial fraction decomposition of each
rational expression.
x-3
1. −
2
2x + 24
x - x -6
1.
6x2 - x + 16
x + 4x
2.
x -x
2. −
2
-x2 + x + 32
x - 8x + 16x
4. −
3
3. −
3
2
5. MULTIPLE CHOICE Which rational expression
is improper?
A
3x2 + 5x -4
−
(x - 2)(x + 1)
5x -1
B −
x2 - x -2
23x - 26
C −
2
x - 32x + 12
4.
x+2
D −2
(x + 1)
5.
NAME
6
3.
DATE
PERIOD
Chapter 6 Quiz 4
SCORE
(Lesson 6-5)
A vertices
C interior points
B exterior points
D constraints
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. MULTIPLE CHOICE If a linear programming problem can be
optimized, it occurs at one of the ___________ of the region
representing the solutions.
1.
Find the maximum and minimum values of the objective
function f(x, y) and for what values of x and y they occur,
subject to the given constraints.
2. f(x, y) = 3x + 4y
x≥0
y≥0
x+y≤7
x + 3y ≤ 15
3. f(x, y) = -2x + y
x≥0
y≤7
y≤x
7x + 2y ≤ 84
2.
3.
4. CLOCKS It costs a clockmaker $30 to make a small clock and
$50 to make a large clock. He has a budget of $1500 to build
them. He will not make more than 40 small clocks.
a. If he makes a profit of $60 on the small clocks and $45 on
the large clocks, how many of each size clock must he make
4a.
and sell to maximize his profit?
4b.
b. What is his maximum profit?
Chapter 6
34
Glencoe Precalculus
NAME
DATE
6
PERIOD
Chapter 6 Mid-Chapter Test
SCORE
(Lessons 6-1 through 6-3)
Assessment
Part I Write the letter for the correct answer in the blank
at the right of each question.
1. Solve the system of equations.
4x - 3y = -22
-2x - 5y = -28
B (-1, 6)
A (-1, -6)
C (1, -6)
D (1, 6)
1.
H (-2, 4, -1)
J (2, 4, -1)
2.
⎡ 2 4⎤
D ⎢
⎣ -3 -6 ⎦
3.
2. Solve the system of equations.
-4x + 2y - z = 17
x - 3y + 2z = -16
2x + y - 4z = 4
F (2, -4, 1)
G (-2, -4, 1)
3. Which of the following matrices is singular?
⎡ 0 1⎤
A ⎢
⎣ -1 2 ⎦
⎡ 7 -1 ⎤
B ⎢
⎣4 4⎦
⎡ 4 -3 ⎤
C ⎢
⎣5 2⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. Use an inverse matrix to solve the system of equations, if possible.
-4x - 2y = -14
-8x - 4y = 4
F no solution
G (-1, 1)
H (1, -2)
J (2, 3)
4.
Part II
5. FRUIT Katie bought 4 apples and 6 pears for $9.80.
Sylvia bought 3 apples and 9 pears for $10.95.
a. Write a set of linear equations for this situation.
5a.
b. Determine the cost of one apple and one pear.
5b.
⎡ -1 1
6. Find the determinant of
1 -1
⎣ 1 -1
⎢
0⎤
1 .
1⎦
6.
7. Find AB if possible.
⎡ -1 ⎤
⎡ -4 3 2 ⎤
A=
0 -1 1 , B =
1
⎣ -2 ⎦
⎣ -2 1 0 ⎦
⎢
Chapter 6
⎢ 7.
35
Glencoe Precalculus
NAME
DATE
PERIOD
6
Chapter 6 Vocabulary Test
SCORE
augmented matrix
coefficient matrix
constraint
Cramer’s Rule
determinant
feasible solutions
Gaussian elimination
Gauss-Jordan elimination
identity matrix
inverse
inverse matrix
invertible
linear programming
multiple optimal solutions
multivariable linear system
nonsingular matrix
objective function
optimization
partial fraction
partial fraction decomposition
reduced row-echelon form
row-echelon form
singular matrix
square system
unbounded
Choose a term from the vocabulary list above to complete each sentence.
1.
2. Transferring a system into an equivalent system is called
_______________.
2.
3. The multiplicative inverse of a square matrix is called its
_______________.
3.
4. The process for finding a minimum or maximum value for a
specific quantity is known as _______________.
4.
5. If a matrix has an inverse, then the matrix is said to be a(n)
_______________ matrix.
5.
6. A method for solving square systems using determinants
instead of row reduction or inverses is known as
_______________.
6.
7. When a rational function is written as the sum of its partial
fractions, this sum is called the _______________ of the
rational function.
7.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. A system of linear equations that has the same number of
equations as variables is called a(n) _______________.
Define each term in your own words.
8. identity matrix
9. objective function
Chapter 6
36
Glencoe Precalculus
DATE
6
PERIOD
Chapter 6 Test, Form 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. What is the augmented matrix for the given system?
2x - 3y = -16
x + 5y = 18
⎡2 −3⎤
⎡ 5 3⎤
⎡2 -3
⎡ 5 3
-16⎤
A ⎢
B ⎢
C ⎢
D ⎢
⎣1 5⎦
⎣−1 2⎦
⎣1 5
⎣-1 2
18⎦
-16⎤
18⎦
1.
2⎤
-1
5⎦
2.
2. Which matrix is not in row-echelon form?
⎡ 1 -3
F
⎢0
⎣0
1
0
2⎤
-6
0⎦
G
⎡1 2
⎢
⎣0 1
4⎤
-2 ⎦
⎡1 7
H ⎢
⎣0 1
⎡ 1 -2
J 1 -4
⎣0 1
-2 ⎤
5⎦
⎢
3. Solve the system of equations using Gaussian elimination.
-3x - 5y = 2
2x + 3y = -2
A (4, 2)
B (-4, -2)
C (4, -2)
D (-4, 2)
3.
4. FOOD The table shows several boxes of assorted candy available at a
candy shop. What is the price per pound for each candy?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Box
Chocolate
Taffy
Nougat
Price ($)
Grand Edition
10
5
0
12.25
Special Edition
10
5
5
16.25
Deluxe Edition
15
10
5
24.25
F ($0.85, $0.75, $0.80)
H ($0.80, $0.75, $0.85)
G ($0.75, $0.80, $0.85)
J ($0.75, $0.85, $0.80)
4.
C 12
D 20
5.
⎡ -2 0 18⎤
J ⎢
⎣-10 28 4⎦
6.
⎡–1 1⎤
D ⎢
⎣ 4 –3⎦
7.
⎡3 -2⎤
5. What is the determinant of ⎢
?
⎣4 0⎦
A -8
B 8
⎡-2 4 6 ⎤
6. Find DE if D = ⎢
and E =
⎣ 5 -7 -1⎦
⎡-20 44⎤
F ⎢
⎣ 8 -42⎦
⎡ -2 -10⎤
G
0 -28
⎣-18 -4⎦
⎢
⎡ 1 -2⎤
⎢
0
⎣-3
4.
4⎦
⎡-20
8⎤
H ⎢
⎣ 44 -42⎦
⎡ 3 -1 ⎤
7. Find the inverse of ⎢
, if it exists.
⎣-4 1⎦
A does not exist
Chapter 6
⎡-1 -1⎤
B ⎢
⎣-4 -3⎦
⎡1 1⎤
C ⎢
⎣4 3⎦
37
Glencoe Precalculus
Assessment
NAME
NAME
DATE
6
Chapter 6 Test, Form 1
PERIOD
(continued)
⎡–4 –6 ⎤
⎡ 3 –2 ⎤
8. What is B if A = ⎢
and AB = ⎢
?
⎣ 1 –4 ⎦
⎣ 2 –12 ⎦
⎡ 2 0⎤
⎡-2 0⎤
⎡-2 0⎤
⎡-2 0⎤
H ⎢
J ⎢
F ⎢
G ⎢
⎣ 1 3⎦
⎣-1 -3⎦
⎣-1 3⎦
⎣-1 3 ⎦
9. Solve the following system of equations using an inverse matrix.
-4x - 2y + z = 6
A (1, 0, -2)
-x - y - 2z = -3
B (-1, 0, -2)
8.
2x + 3y - z = -4
C (-1, 0, 2)
D (1, 0, 2)
9.
10. Use Cramer’s Rule to solve the system of equations.
-3x + 7y = 78
-2x + 5y = 55
G (5, 9)
F (5, -9)
H (-5, 9)
J (-5, -9) 10.
11. FUNDRAISING The cheerleading squad is raising money for new
uniforms by selling popcorn balls and calendars. Tanya raised $70 by
selling 25 popcorn balls and 30 calendars. Nichole raised $53 by selling
20 popcorn balls and 22 calendars. What is the cost of one calendar?
A $1
B $1.25
C $1.50
D $1.75
11.
–2x + 10
(x – 1) (x + 3)
12. Find the partial fraction decomposition of − .
4
–2
H −
+−
2
–4
+−
G −
2
-4
J −
+−
x -1
x+1
x+3
x+1
x–3
x-1
x–3
12.
x+3
13. Find the maximum value of the objective function f(x, y) = 2x + 4y,
subject to the constraints x ≥ 0, x ≤ 8, y ≥ 0, and x + y ≤ 10.
A 40
B 24
C 20
D 16
13.
14. Find the minimum value of the objective function f(x, y) = -2x - 3y,
and for what values of x and y, subject to the constraints x ≥ 0, x ≤ 5,
y ≥ 0, y ≤ 5, and 5y - 2x ≥ 0.
F -40, (12.5, 5)
G -25, (5, 5)
H -15, (0, 5)
J -10, (5, 0) 14.
15. COLLECTIONS A scouting troop is collecting aluminum cans and paper
to recycle. The total weight collected cannot exceed 30 pounds. The troop
cannot collect more than 20 pounds of paper or 15 pounds of aluminum.
If the troop earns $2.50 per pound for aluminum and $0.50 for paper,
what is the maximum profit?
A $35
Bonus
Chapter 6
B $45
C $55
⎡ 2 -3⎤
If A = ⎢
, find (A-1)-1.
⎣-5 8⎦
D $75
15.
B:
38
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-2
4
+−
F −
DATE
6
PERIOD
Chapter 6 Test, Form 2A
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. What is the augmented matrix for the system?
4x + 2y = 6
-3x - 2y = -4
⎡ 4 2
A ⎢
⎣ -3 -2
6⎤
-4⎦
⎡ 4 -3
B ⎢
⎣ 2 -2
6⎤
-4⎦
⎡ 4 2⎤
C ⎢
⎣ -3 -2⎦
2. Which matrix is not in row-echelon form?
⎡ 1 4 0 -2 ⎤
⎡ 1 -2 3 5 ⎤
⎡ 1 -2
G 0 0 0 0
H ⎢
F 0 1 3
1
⎣0 1
5 ⎦
⎣0 0 1
⎣0 0 1 2⎦
⎢
⎢
-7 ⎤
2⎦
⎡-2 2 ⎤
D ⎢
⎣ 3 4⎦
1.
⎡1 5
J ⎢
⎣0 1
2.
3⎤
-4 ⎦
3. Solve the system of equations using Gaussian elimination.
-2x - 3y + z = 4
4x + y - 2z = -13
-x + 2y - 4z = -8
A (2, 1, 3)
B (-2, 1, 3)
C (2, -1, 3)
D (-2, -1, -3)
3.
4. SEWING The table shows several packages of assorted spools of
thread available at a store. What is the price per spool of each
kind of thread?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Package
Red
White
Blue
Price ($)
Bargain Spools
5
5
0
5.50
Simply Spools
5
2
5
7.50
Mega Spools
10
10
10
18.00
F ($0.50, $0.60, $0.70)
H ($0.70, $0.60, $0.50)
G ($0.70, $0.50, $0.60)
J ($0.60, $0.50, $0.70)
4.
⎡ 1 -2 4 ⎤
5. What is the determinant of 0 5 3 ?
⎣-5 -2 1 ⎦
⎢
A -151
B -141
C 141
D 151
5.
⎡ -1
⎡-0.4 1.2⎤
3⎤
6. Find AB if A = ⎢
and B = ⎢
.
⎣0.5 -0.2⎦
⎣
5 -0.1⎦
⎡ 0.62 -1.2⎤
F ⎢
⎣-1.5 15.4⎦
⎡-0.62
⎡ 15.4 -1.5⎤
1.2⎤
G ⎢
H ⎢
⎣ 1.5 -15.4⎦
⎣-1.2 0.62⎦
⎡ 6 8⎤
7. Find the inverse of ⎢
, if it exists.
⎣-3 -4⎦
⎡-4 -8 ⎤
⎡-6 -3 ⎤
⎡ 6 3⎤
A ⎢
B ⎢
C ⎢
⎣ 3 6⎦
⎣ 8 4⎦
⎣ -8 -4 ⎦
Chapter 6
39
⎡-15.4
1.5⎤
J ⎢
6.
⎣ 1.2 -0.62⎦
D does not exist
7.
Glencoe Precalculus
Assessment
NAME
NAME
DATE
6
Chapter 6 Test, Form 2A
PERIOD
(continued)
⎡-4 -1 ⎤
⎡-8 13 ⎤
and DE = ⎢
, find E.
8. Given D = ⎢
⎣ 2 3⎦
⎣ 4 -9 ⎦
⎡ 2 -3 ⎤
⎡-2 3 ⎤
⎡-1 0 ⎤
G ⎢
H ⎢
F ⎢
⎣ 0 -1 ⎦
⎣ 0 1⎦
⎣-3 2 ⎦
⎡1 0⎤
J ⎢
⎣ 3 -2 ⎦
8.
9. Solve the following system of equations using an inverse matrix.
-x + 2y - 3z = 11
A (-3, -4, 2)
2x + z = 4
B (3, 4, -2)
x - y + 2z = -5
C (3, -4, 2)
D (-3, 4, -2)
9.
10. Use Cramer’s Rule to solve the system of equations.
-4x + 2y = 30
F (4, 7)
-x - y = -3
G (4, -7)
J (-4, 7)
H (-4, -7)
10.
11. FOOD On Friday, Lila raised $147.50 by selling 20 hamburgers and
35 hot dogs. On Saturday, she raised $107.50 by selling 15 hamburgers
and 25 hot dogs. What was the selling price of one hot dog?
A $2.50
B $2.75
C $3.00
D $3.25
11.
19x -1
12. Find the partial fraction decomposition of −
.
2
3x -10x + 3
7
-2
+−
H −
-7
2
+−
G −
7
-2
+−
J −
x -3
x+3
3x - 1
x+3
x -3
3x + 1
3x + 1
12.
3x - 1
13. Find the maximum value of the objective function f(x, y) = 3x - y subject to
the constraints x ≥ 0, y ≥ 0, x + y ≤ 8, and x + 6y ≤ 24.
A -4
B 15
C 24
D 48
13.
14. Find the minimum value of the objective function f(x, y) = 2x + 3y and for
what values of x and y, subject to the constraints y ≥ 0, y ≤ 6, and y - x ≤ 3.
F 0, (0, 0)
G 9, (0, 3)
H 18, (0, 6)
J 24, (3, 6)
14.
15. SALES Tony can sell a maximum of 20 boxes of birthday cards and holiday
cards. He cannot sell more than 12 boxes of birthday cards or 15 boxes of
holiday cards. If he earns $3.50 per box of birthday cards and $2.50 per box
of holiday cards, what is his maximum profit?
A $79.50
Bonus
Chapter 6
B $62
C $55
D $37.50
If A and B are inverse 2 × 2 matrices, what matrix represents the
product of A and B?
40
15.
B:
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-7
2
+−
F −
DATE
6
PERIOD
Chapter 6 Test, Form 2B
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. What is the augmented matrix for the given system?
-5x + 2y = -21
3x - 4y = 21
⎡-5 2⎤
⎡-4 -2⎤
⎡-5 2 -21⎤
⎡-5 3
A ⎢
B ⎢
C ⎢
D ⎢
⎣ 3 -4⎦
⎣-3 -5⎦
⎣ 3 -4
⎣ 2 -4
21⎦
2. Which matrix is not in row-echelon form?
⎡ 1 -2 0
⎡ 1 -1 0
5⎤
3⎤
⎡ 1 -5
F 0 1 -3
2 G 0 1 -2
4 H ⎢
⎣0 1
⎣ 1 0 0 -1 ⎦
⎣ 0 0 1 -2 ⎦
⎢
⎢
⎡1 0
3⎤
J ⎢
⎣0 1
6⎦
-21⎤
1.
21⎦
-7 ⎤
2⎦
2.
3. Solve the system of equations using Gaussian elimination.
2x - y + 3z = -4
-x + 2y - z = 0
x + 2y + z = -4
A (2, 2, 2)
B (1, 1, 1)
C (-2, -2, -2)
D (-1, -1, -1)
3.
4. CRAFTS The table shows several packages of assorted shapes
available at a craft store. What is the price per shape?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Package
Hearts
Stars
Diamonds
Price ($)
Mini Shapes
20
0
20
3.00
More Shapes
40
20
10
6.00
Mighty Shapes
50
50
50
15.00
F ($0.05, $0.10, $0.15)
H ($0.05, $0.15, $0.10)
G ($0.10, $0.15, $0.05)
J ($0.15, $0.10, $0.05)
⎡-2
4.
0⎤
5. What is the determinant of 3 -4 -1 ?
⎣ 2 -1 0⎦
⎢
A -12
4
B -6
D 12
5.
⎡ 0.3 -13⎤
H ⎢
⎣-0.1 -4⎦
⎡-0.3 -13⎤
J ⎢
⎣-0.1 -4⎦
6.
⎡ 5 4⎤
C ⎢
⎣-1 -1⎦
D does not exist
7.
C 6
⎡-3 2⎤
⎡-0.1 3⎤
6. Find EF if E = ⎢
and F = ⎢
.
⎣-1 0.5⎦
⎣
0 -2⎦
⎡-0.3 -13⎤
F ⎢
⎣ 0.1 -4⎦
⎡ 0.3 -13⎤
G ⎢
⎣ 0.1 -4⎦
⎡-5 4⎤
7. Find the inverse of ⎢
, if it exists.
⎣ 1 -1⎦
⎡-1 -4⎤
A ⎢
⎣-1 -5⎦
Chapter 6
⎡ 5 -4⎤
B ⎢
⎣-1 1⎦
41
Glencoe Precalculus
Assessment
NAME
NAME
DATE
6
Chapter 6 Test, Form 2B
PERIOD
(continued)
⎡ 24 -4 ⎤
⎡-4 -1 ⎤
and AC = ⎢
, find C.
8. Given A = ⎢
⎣ -18 -14 ⎦
⎣ 3 5⎦
⎡ 6 -2 ⎤
⎡-4 -2 ⎤
⎡ 6 0⎤
G ⎢
H ⎢
F ⎢
⎣0 4⎦
⎣ 0 -6 ⎦
⎣ -2 4 ⎦
⎡-6 2 ⎤
J ⎢
⎣ 0 -4 ⎦
8.
9. Solve the following system of equations using an inverse matrix.
-3x + 2y - z = 19
A (4, -2, 3)
x - 4y + 2z = -18
B (4, 2, 3)
–x + 3z = -5
C (-4, 2, -3)
D (-4, -2, -3)
9.
10. Use Cramer’s Rule to solve the system of equations.
6x - 2y = 28
-x + y = -8
G (-3, 5)
F (3, -5)
H (-3, -5)
J (3, 5)
10.
11. PETS Willy’s Pet Store sells puppies and kittens. Last week, there
were 5 puppies and 4 kittens sold, earning $388. This week, there
were 8 puppies and 7 kittens sold, earning $643. What was the
selling price of one kitten?
A $51
B $48
C $42
D $37
11.
9x -16
.
12. Find the partial fraction decomposition of −
3x2 + 11x - 4
3
-4
H −
+−
3
-4
G −
+−
-3
4
J −
+−
x+4
x -4
3x - 1
x+4
x -4
3x + 1
3x - 1
3x + 1
12.
13. Find the maximum value of the objective function f(x, y) = 2x + 3y subject to
the constraints x ≥ 0, y ≥ 0, x + y ≤ 6, and y ≤ 3.
A 9
B 12
C 15
D 24
13.
14. Find the minimum value of the objective function f(x, y) = -x + 2y and for
what values of x and y, subject to the constraints x ≥ 0, x ≤ 3, y ≥ 0, y ≤ 2,
and x + y ≥ 1.
F -7, (3, -2)
G -3, (3, 0)
H 2, (0, 1)
J 4, (0, 2)
14.
15. DRINKS Homer can buy at most 14 bottles of water and juice, but he
cannot buy more than 8 bottles of water or 10 bottles of juice. If a bottle
of water cost $2 and a bottle of juice costs $3.50, what is the most money
he can spend?
A $47
B $43
C $35
D $28
15.
⎡ 1 0⎤
Bonus What is the inverse of matrix A, if A = ⎢
?
⎣ 0 1⎦
Chapter 6
42
B:
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-3
4
F −
+−
NAME
6
DATE
PERIOD
Chapter 6 Test, Form 2C
SCORE
-4x + y = 12
3x - 2y = -14
Assessment
1. Write the augmented matrix for the given system of equations.
1.
2. Give an example of a matrix that is not in row-echelon form.
2.
3. Solve the system of equations using Gaussian elimination.
-4x + 2y - 3z = 8
-x + y + 2z = 3
x - 3y - z = -7
3.
4. FISHING Ned, Reed, and Mel bought hooks, bobbers, and
sinkers at the same fishing store. The table shows how
many of each they bought and what they each paid.
Determine the price per item.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Fishermen
Hooks
Bobbers
Sinkers
Paid ($)
Ned
4
3
0
8.15
Reed
5
2
1
8.90
Mel
1
3
4
8.45
5. Find the determinant of
⎡-1 5 3 ⎤
0 2 -4 .
⎣ 3 -2 1 ⎦
⎢
4.
5.
⎡-1 9 ⎤
⎡-4 3 ⎤
and B = ⎢
.
6. Find AB if A = ⎢
⎣ 0 -4 ⎦
⎣ 5 -2 ⎦
6.
⎡3 -6 ⎤
7. Find the inverse of ⎢
, if it exists.
⎣1 -2⎦
7.
8. Given M and MN, find N.
⎡4 -3 ⎤
⎡-13 36 ⎤
MN = ⎢
M=⎢
⎣5 2 ⎦
⎣ 1 22 ⎦
8.
9. Solve the system of equations using an inverse matrix.
-3x + y + z = 2
5x + 2y - 4z = 21
x - 3y - 7z = -10
9.
10. Use Cramer’s Rule to solve the system of equations.
2x + y = 7
x+y=5
Chapter 6
10.
43
Glencoe Precalculus
NAME
6
DATE
Chapter 6 Test, Form 2C
PERIOD
(continued)
11. DINER Last week, the owner of a diner spent $91.25 on
15 gallons of milk and 11 pounds of butter. This week, he
spent $70.40 on 12 gallons of milk and 8 pounds of butter.
Find the cost of one pound of butter.
11.
12. Find all values of n such that the system represented by
the given augmented matrix cannot be solved using an
inverse matrix.
⎡-4 -n -5 ⎤
⎢
⎣ 4n 9
2⎦
12.
Find the partial decomposition of each rational expression.
5x - 41
13. −
2
13.
x -2x -15
x2 - 13x + 61
(x + 1)(x - 4)
14.
8x2 - 4x + 18
x + 3x
15.
14. −2
15. −
3
10
16. GROWTH The function f(x) = −
is a logistic growth
2
100x - x
function. Find the partial decomposition of f(x).
16.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the maximum value of the objective function f(x, y)
subject to the given constraints.
17. f(x, y) = 2x - y
x≥0
y≥0
y≤5
x+y≤8
y≥0
-x + y ≤ 6
17.
18. f(x, y) = 3x - 5y
x≥0
x≤4
2x + y ≤ 12
18.
19. Find the minimum value of the objective function f(x, y)
and for what values of x and y it occurs, subject to the
given constraints.
f(x, y) = 3x + y
x≥0
x≤5
y≥0
y≤7
y-x≤4
19.
20. VOLUNTEERING Tanya packs baskets at a local food pantry.
She can put at most 20 canned and bottled items in a basket,
but she cannot put more than 12 canned items or more than
14 bottled items in a basket. If a canned item costs the pantry
$0.40 and a bottled item costs the pantry $0.60, what is the
20.
most expensive basket Tanya can pack?
Bonus Matrix D is the product of invertible matrices A, B, and C.
In terms of A, B, C, and/or D, what does A-1D equal?
B:
Chapter 6
44
Glencoe Precalculus
NAME
DATE
6
PERIOD
Chapter 6 Test, Form 2D
SCORE
Assessment
1. Write the augmented matrix for the given system of equations.
5x - 2y = -3
1.
4x + 7y = -11
2. Give an example of a matrix that is in row-echelon form.
2.
3. Solve the system of equations using Gaussian elimination.
2x - 3y + z = 14
-2x - z = -5
3.
x + 4y = -10
4. SALES The table shows how many of each type of magazine
Romeo sold each day and the money he received from the
sales. Determine the price per magazine.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Day
Sports
News
Eduational
Money Earned ($)
Saturday
3
0
2
19.00
Sunday
5
1
3
34.50
Monday
1
1
4
27.50
⎡ -2
5. Find the determinant of
⎢
⎣
5
3 -4
1 -1
0⎤
1 .
0⎦
⎡ 5
⎡2 -4⎤
6. Find CD if C = ⎢
and D = ⎢
⎣5 -6⎦
⎣-4
4.
5.
0⎤
.
1⎦
⎡ 12 -2⎤
7. Find the inverse of ⎢
, if it exists.
⎣-11 2⎦
6.
7.
8. Given U and UV, find V.
⎡-6 3⎤
U=⎢
⎣ 1 -2⎦
⎡ 15 -9⎤
UV = ⎢
⎣-1 -6⎦
8.
9. Solve the system of equations using an inverse matrix.
x - 2y + z = -5
3x - 2y + z = 3
9.
2x - y + 2z = -7
10. Use Cramer’s Rule to solve the system of equations.
-3x + 3y = -15
6x - 9y = 39
Chapter 6
45
10.
Glencoe Precalculus
NAME
DATE
6
Chapter 6 Test, Form 2D
PERIOD
(continued)
11. FOOD Last week, Gina spent $24.75 on 8 cups of coffee and
5 bagels. This week, she spent $28.50 on 9 cups of coffee and
6 bagels. What is the cost of one bagel?
11.
12. Find all values of n such that the system represented by
the given augmented matrix cannot be solved using an
inverse matrix.
⎡ n2 -6
3⎤
⎢
⎣ -6 4 -1⎦
12.
Find the partial decomposition of each rational expression.
2x + 6
x - 3x
13. −
2
13.
2
- 2x - 5
−
14. 5x
2
3
14.
x +x
-7x2 + x - 10
(x - 2)(2x + 1)
15. −
2
15.
11
is a logistic
16. GROWTH The function f(x) = −
2
121x - x
growth function. Find the partial decomposition of f(x).
16.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the maximum value of the objective function f(x, y)
subject to the given constraints.
17. f(x, y) = -2x + 3y
x≥0
x≤7
y≥0
x + 7y ≤ 28
17.
18. f(x, y) = x + 2y
x≥0
y≥0
y ≤ 10
2x + y ≤ 18
y - 2x ≤ -6
18.
19. Find the minimum value of the objective function f(x, y) and
for what values of x and y it occurs, subject to the given constraints.
f(x, y) = -x + 2y
x≥0
x ≤ 15
y≥0
y ≤ 13
2x + y ≥ 4
19.
20. BAKERY A baker can make at most 100 cupcakes and pies
in one hour, but he cannot make more than 70 cupcakes or
more than 60 pies in an hour. A cupcake costs $1.50 and a pie
costs $6. What is the maximum value of baked goods that he
can produce in one hour?
20.
Bonus Matrix D is the product of invertible matrices A, B,
and C. In terms of A, B, C, and/or D, what does
(AB)-1D equal?
Chapter 6
46
B:
Glencoe Precalculus
NAME
6
DATE
PERIOD
Chapter 6 Test, Form 3
SCORE
3x = 2y + 5
-y = -4x - 7
Assessment
1. Write the augmented matrix for the given system of equations.
1.
2. Give an example of a matrix that is in row-echelon form but
contains one row with all zeros.
2.
3. Solve the system of equations using Gaussian elimination.
4x - 6z = 18
3y = 7x + 33
-8y + 12 = 4z
3.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. RENTALS Julie rents bicycles in a tourist town. The table
shows how many of each type of bicycle Julie rented over the
weekend and the money she received from the rentals.
Determine the cost to rent each kind of bicycle.
Day
Tandem
Child
Adult
Rental ($)
Friday
2
1
7
269
Saturday
4
0
9
370
Sunday
3
2
6
296
⎡ -1
⎢
5. Find the determinant of -4
⎣ 2
3 0⎤
5 -6 .
1 0⎦
4.
5.
⎡ x 4⎤ ⎡1 -3⎤ ⎡2 22⎤
6. Solve ⎢
⎢
=⎢
for x and y.
⎣3 -1⎦ ⎣1 y⎦ ⎣2 -13⎦
6.
⎡ 2 -1 2 ⎤
7. Find the inverse of 1 0 -1 , if it exists.
⎣ 3 2 -1 ⎦
7.
⎡4 -1⎤
⎡-12 6⎤
8. Given A = ⎢
and AB = ⎢
, find B.
⎣3 5⎦
⎣ -9 -7⎦
8.
⎢
9. Solve the system of equations using an inverse matrix.
-3x - y + z - 2w = -1
x + y - w = -1
2x - z + w = 0
3y + 2z = 1
10. Use Cramer’s Rule to solve the system of equations.
11x - 5y = 59
3x - y = 15
Chapter 6
47
9.
10.
Glencoe Precalculus
NAME
6
DATE
Chapter 6 Test, Form 3
PERIOD
(continued)
11. GARDENING Tayshia spent $13.60 on 4 tomato plants,
2 pepper plants, and 1 squash plant. Kiaya spent $16.80 on
6 tomato plants and 3 squash plants. LaTeesha spent $16.45
on 4 tomato plants, 3 pepper plants and 2 squash plants.
What is the cost of one squash plant?
12. Find all values of n such that the system represented by
⎡ n 3
-1⎤
cannot be solved using an inverse matrix.
⎢
4⎦
⎣ -2 n - 5
11.
12.
Find the partial decomposition of each rational expression.
15x2 - 2x - 59
13. −
2
13.
-2x2 + 8x + 12
x + 4x + 4x
14.
4x2 + 7x - 1
x +x
15.
(x - 6)(2x - 5)
14. −
3
2
15. −
2
13
is a logistic
16. GROWTH The function f(x) = −
2
169x - x
growth function. Find the partial decomposition of f(x).
16.
Find the maximum value of the objective function f(x, y)
subject to the given constraints.
x≥0
y-x≤2
y≥0
2x + y ≤ 20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
17. f(x, y) = 3x - y
17.
18. f(x, y) = -x + 2y
x≥0
y≤8
y≥0
x + y ≤ 12
y - 2x ≤ 4
18.
19. Find the minimum and maximum value of the objective
function f(x, y) and for what values of x and y it occurs,
subject to the given constraints.
f(x, y) = 8x + 2y
x≥0
x≤8
y≥0
y ≤ 10
4x + y ≤ 34
19.
20. CLOTHING Ginny sews dresses and gowns. Each dress
sells for $200 and each gown sells for $650. It takes her
2 weeks to sew a dress and 5 weeks to sew a gown. She
accepts orders for at least three times as many dresses
as gowns. In the next 22 weeks, what is the maximum
amount of money she can expect to earn?
20.
Bonus Use a matrix equation to find the value of x for the
given system of equations.
ax + by = c
dx + ey = f
B:
Chapter 6
48
Glencoe Precalculus
6
DATE
PERIOD
Extended-Response Test
SCORE
Demonstrate your knowledge by giving a clear, concise solution
to each problem. Be sure to include all relevant drawings and
justify your answers. You may show your solutions in more than
one way or investigate beyond the requirements of the problem.
1. AMUSEMENT PARKS Gala Amusement Corporation owns three
amusement parks. The fraction of visitors at each park by quarter is shown
in the first table. The total number of visitors for all three parks combined is
shown in the second table, differentiated by quarter and time of day.
Amusement Park
1
2
3
4
Ocean Side
0.40 0.25 0.15 0.35
Big Mountain
0.25 0.35 0.40 0.35
Lazy River
0.35 0.40 0.45 0.30
Quarter Morning Afternoon Evening
1
2000
2500
1500
2
3500
4000
2000
3
3000
4200
1800
4
2500
2000
1000
a. Write a 3 × 4 matrix representing the data found in the first table.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Write a 4 × 3 matrix representing the data found in the second table.
c. Use these two matrices to determine if there were more visitors to Lazy
River Park in the morning or to Big Mountain Park in the afternoon.
2. Consider the following system of equations.
3x - y = -5
-x + 2z = 3
y-z=1
a. Write the system in matrix form.
b. Find the inverse of the coefficient matrix.
c. Use the matrix found in part b to solve the system.
3. CONSTRUCTION A builder constructs two styles of houses: the Executive,
on which he makes a $30,000 profit, and the Suburban, on which he makes
a $25,000 profit. He can complete up to 10 houses each year, with no more
than 6 of them in the Executive style.
a. How many houses of each style should he build to maximize his profit?
Explain.
b. If he keeps the original limit on Executives but raises the limit on the
total number of houses he builds, what would be the effect on the
number of each style built to maximize profit? Why?
Chapter 6
49
Glencoe Precalculus
Assessment
NAME
NAME
DATE
PERIOD
6
Standardized Test Practice
SCORE
(Chapters 1–6)
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
-7
1. State the domain of f(x) = −
.
2
x - x - 12
A
B
C
D
{x|x
{x|x
{x|x
{x|x
≠
≠
≠
≠
-4, 3 , x ∈ }
-4, -3 , x ∈ }
-3, 4 , x ∈ }
3, 4 , x ∈ }
1.
A
B
C
D
1
2. Evaluate −
log64 x.
3
F 262,144
G 256
64
H −
J 4
2.
F
G
H
J
π
C −
D π
3.
A
B
C
D
H sec x
J tan x
4.
F
G
H
J
D (1, -2, 1)
5.
A
B
C
D
6.
F
G
H
J
3
√2
2
3. Find the exact value of sin-1 −.
π
A −
π
B −
6
4
3
4. Simplify sin x tan x + cos x.
F csc x
G cot x
5. What is the solution of the system of equations shown?
-4y + z = -7
2x - 3y + z = -7
A (-1, 2, 1)
B (1, -2, -1)
C (-1, 2, -1)
6. What is [g º f ](x) if f(x) = 3x2 - 1 and g(x) = x - 4?
F [g º f ](x) = 3x2 - 5
H [g º f ](x) = 3x2 - 24x + 48
G [g º f ](x) = x2 - 8x + 16
J
[g º f ](x) = 3x2 - 24x + 47
7. Solve 5x + 2 = 25x - 4.
A 5
B 6
⎡3 -5⎤
8. What is the determinant of ⎢
?
⎣1 3⎦
F -14
G -4
C 8
D 10
7.
A
B
C
D
H 4
J 14
8.
F
G
H
J
9.
A
B
C
D
9. Which of the following is the inverse of f(x) = 2x + 3?
A f -1(x) = -2x - 3
B f
1
(x) = −
x-3
2
-1
Chapter 6
x-3
C f -1(x) = −
D f
50
2
x+3
(x) = −
2
-1
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-x + 3y + 2z = 9
NAME
DATE
6
Standardized Test Practice
PERIOD
(continued)
10. Choose the radian measure that is equal to 405°.
9π
G −
11π
F −
4
4
7π
H −
5π
J −
10.
F
G
H
J
⎡ 1 -4 -3⎤
C ⎢
⎣-2 0 5⎦
⎡18 26⎤
D ⎢
⎣-2 0⎦
11.
A
B
C
D
12.
F
G
H
J
13.
A
B
C
D
J 6
14.
F
G
H
J
3
D −
15.
A
B
C
D
16.
F
G
H
J
17.
A
B
C
D
4
4
11. Find AB, if possible.
⎡-1 4 3 ⎤
⎡-1 0⎤
A=⎢
, B = ⎢
⎣ 2 0 -5⎦
⎣ 4 7⎦
A not possible
⎡7 -4 ⎤
B ⎢
⎣0 -1⎦
12. What is the effect on the graph of f(x) = log x when the equation
is changed to g(x) = log (x + 4)?
F
G
H
J
The
The
The
The
graph
graph
graph
graph
is
is
is
is
translated
translated
translated
translated
4
4
4
4
units
units
units
units
up.
to the left.
to the right.
down.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
13. Find the value of sin 2θ in the interval (0, 90°) given that cos θ = −
.
15
A −
64
4
√
15
D −
4
√
15
8
15
B −
C −
32
5
1
14. What is the solution of −
x +x-4=−
x?
F 2
G 3
H 5
5π
15. Find the exact value of tan −
.
4
1
A −
2
4
B −
5
C 1
2
16. Which is closest to the value of x?
F 35.2
H 45.2
G 44.8
J 54.8
12
x°
17
17. The graph of an odd function is symmetric with respect to which
of the following?
A the x-axis
B the y-axis
Chapter 6
C the origin
D none of these
51
Glencoe Precalculus
Assessment
(Chapters 1–6)
NAME
DATE
6
Standardized Test Practice
PERIOD
(continued)
(Chapters 1–6)
Part 2: Short Response
Instructions: Write your answers in the space provided.
18. Find the vertical asymptotes in the interval [-2π, 2π] for
x
the graph of y = 3 tan −
.
18.
2x - 4
19. Find the partial fraction decomposition of −
.
2
19.
3
3x - 2x
20. List all of the possible rational zeros of f(x) = 2x4 - 3x2 + 5x - 3. 20.
21. Condense 3 ln (x - 2) - 4 ln x.
21.
22. Find all solutions to the given equation in the interval [0, 2π].
2 cos2 θ = cos θ
22.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
23. State the amplitude, period, frequency, phase shift, and vertical
π
+ 2.
shift of y = 3 cos x - −
(
2
)
23.
3x
.
24. Consider g(x) = −
x-1
a. Find the x- and y-intercepts.
24a.
b. Find the vertical asymptote.
24b.
c. Find the horizontal asymptote.
24c.
25. Consider f(x) = x 5 + 3x 4 - x 3 - 6x 2 + 8.
a. Determine the consecutive integer values of x between
which each real zero is located.
25a.
b. Estimate the x-coordinates at which the relative
maxima and relative minima occur.
25b.
Chapter 6
52
Glencoe Precalculus
Chapter 6
A1
Glencoe Precalculus
DATE
Before you begin Chapter 6
Systems of Equations and Matrices
Anticipation Guide
PERIOD
D
D
2. The row-echelon form of a matrix is unique.
After you complete Chapter 6
Chapter 6
3
Chapter Resources
3/23/09 3:48:07 PM
Glencoe Precalculus
• For those statements that you mark with a D, use a piece of paper to write an example
of why you disagree.
• Did any of your opinions about the statements change from the first column?
A
A
9. Linear programming can be used to solve applications
involving systems of equations.
10. In a linear programming problem, one evaluates the objective
function at each vertex of the feasible region to maximize or
minimize the function.
D
A
8. Graphically, a rational function and its partial fraction
decomposition are different.
polynomial and a proper rational expression.
f(x)
rational expression − , you must use the division
d(x) r(x)
f(x)
algorithm − = q(x) + − to rewrite it as the sum of a
d(x)
d(x)
7. To find the partial fraction decomposition of an improper
D
D
5. Cramer’s Rule uses inverses to solve systems.
6. Cramer’s Rule applies when the determinant of the
coefficient matrix is 0.
A
4. The product of an m × r matrix and an r × n matrix results
in an m × n matrix.
3. If a matrix has an inverse, then it is a singular matrix.
A
STEP 2
A or D
1. The augmented matrix of a system is derived from the
coefficients and constant terms of the linear equations.
Statement
• Reread each statement and complete the last column by entering an A or a D.
Step 2
STEP 1
A, D, or NS
• Write A or D in the first column OR if you are not sure whether you agree or disagree,
write NS (Not Sure).
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Read each statement.
Step 1
6
NAME
0ii_004_PCCRMC06_893807.indd Sec1:3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
⎢
⎢ ⎢
⎡ 1 -2 1 -1 ⎤
0 1 -1 -1
1
−
R → ⎣0 0 1 2⎦
4 3
⎢
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 5
Chapter 6
Answers
5
2y + 4z = 0
(-2, 3, 4)
(2, -1, 3)
Lesson 6-1
11/17/09 4:38:23 PM
Glencoe Precalculus
(3, -2, 1)
x-z=2
x - 4y + z = -10
3x + y - 2z = -1
3. -4x - y - z = -11
-3x + 2y - z = 8
2. 5x - y = -13
x + 2y - z = -3
1. -2x - y + z = 0
Solve each system of equations using Gaussian elimination with matrices.
Exercises
⎡ 1 -2 1 -1 ⎤
1
−
R → 0 1 -1 -1
5 2
⎣ 0 5 -1 3 ⎦
c.
Step 3 Write the corresponding system of equations and use substitution to
solve the system.
x - 2y + z = -1
y - z = -1
z=2
The solution of the system is x = -1, y = 1, and z = 2 or (-1, 1, 2).
⎡ 1 -2 1 -1 ⎤
0 1 -1 -1
R3 - 5R2→ ⎣ 0 0 4 8 ⎦
⎢
e.
b.
d.
⎢ ⎡ 1 -2 1 ⎢ -1 ⎤
0 5 -5 -5
R3 - 3R1→ ⎣ 0 5 -1 ⎢ 3 ⎦
⎢
⎡1 -2 1 ⎢ -1⎤
R2 - 2R1→ 0 5 -5 -5
⎣3 -1 2 ⎢ 0⎦
a.
Step 2 Apply elementary row operations to obtain a row-echelon form of the matrix.
⎢
⎡1 -2 1 -1⎤
2 1 -3 -7
⎣3 -1 2 0⎦
Step 1 Write the augmented matrix.
Example
Solve the system of equations using Gaussian
elimination with matrices.
x - 2y + z = -1
2x + y - 3z = -7
3x - y + 2z = 0
You can solve a system of linear equations using
matrices. Solving a system by transforming it into an equivalent system
is called Gaussian elimination. First, create the augmented matrix.
Then use elementary row operations to transform the matrix so that it is in
row-echelon form. Then write the corresponding system of equations and
use substitution to solve the system.
Multivariable Linear Systems and Row Operations
Study Guide and Intervention
Gaussian Elimination
6-1
NAME
Answers (Anticipation Guide and Lesson 6-1)
(continued)
PERIOD
A2
⎢
⎢
⎢
R1 - R3→ ⎡ 1 0 0 4 ⎤
0 1 0 -5
⎣0 0 1 1⎦
Reduced
row-echelon form
⎢
⎡ 1 -2 1 15 ⎤
0 1 -2 -7
-R3→ ⎣ 0 0 1 1 ⎦
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 6
6
(-2, -2, -2)
(-7, 2, 3)
Chapter 6
4x + 2y + z = 2
2x - 5y = 6
2x - 5y = -24
Glencoe Precalculus
(-1, 5, -4)
-x + 3y - 2z = 24
3. -2x - y - z = 1
-2y + 3z = -2
2. x - 4z = 6
-4x + z = 31
1. 3x - 2y + z = -22
Solve each system of equations using Gaussian or Gauss-Jordan elimination.
Exercises
⎢
⎢
⎡ 1 -2 1 15 ⎤
0 -5 4 29
R1 + R3→ ⎣ 0 -1 1 6 ⎦
The solution of the system is x = 4, y = -5, and z = 1 or (4, -5, 1).
⎢
R1 + 2R2→ ⎡ 1 0 1 5 ⎤
0 1 0 -5
⎣0 0 1 1⎦
⎡ 1 -2 1 15 ⎤
R2 + 2R3→ 0 1 0 -5
⎣0 0 1 1⎦
⎢
⎢
Row-echelon form
⎡ 1 -2 1 15 ⎤
0 1 -2 -7
R2 + R3→ ⎣ 0 0 -1 -1 ⎦
⎡ 1 -2 1 15 ⎤
R2 - 6R3→ 0 1 -2 -7
⎣ 0 -1 1 6 ⎦
⎢
⎡ 1 -2 1 15 ⎤
⎡ 1 -2 1 15 ⎤
0 -5 4 29
-2 -1 2 -1 2R1 + R2→
⎣ -1 1 0 -9 ⎦
⎣ -1 1 0 -9 ⎦
Augmented Matrix
Example
Solve the system of equations.
x - 2y + z = 15
-2x - y + 2z = -1
-x + y = -9
Write the augmented matrix. Apply elementary row operations to obtain a
row-echelon form. Then apply elementary row operations to obtain zeros
above the leading 1s in each row.
If you continue to apply
elementary row operations to the row-echelon form of any
augmented matrix, you can obtain a matrix in which every column
has one element equal to 1 and the remaining elements equal to 0.
This is called the reduced row-echelon form of the matrix. Solving
a system by transforming an augmented matrix so that it is in reduced
row-echelon form is called Gauss-Jordan elimination.
⎡1 0 0 a⎤
0 1 0 b
⎣0 0 1 c⎦
Multivariable Linear Systems and Row Operations
Study Guide and Intervention
DATE
11/17/09 4:38:36 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 6
Gauss-Jordan Elimination
6-1
NAME
PERIOD
(-2, -5, -4)
⎢
⎡ 2 3 1 -23 ⎤
2. -3 -1 4 -5
⎣ -1 5 -1 -19 ⎦
3.
(4, -6, 2)
⎢
⎡ -5 -3 0 -2 ⎤
0 -2 6 24
⎣ 4 0 -7 2 ⎦
⎣
⎢
⎢ 3 4 7 ⎢ -8 ⎤
6
-2 -3 1
5 -2 1 ⎢ 4 ⎦
⎣
⎢
⎡ -4
(-1, -1, -1)
5x - y + 2z = -6
3x + 2y - 4z = -1
8. -2x - 5y + z = 6
(-3, 2, -4)
x - y + z = -9
-2x + 5y - 4z = 32
9. 8x - y + 3z = -38
⎢
⎢
7
3
Lindsay
Edwin
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 7
Chapter 6
5
8
2
4
Oranges
2
15.30
13.50
14.20
4
Total Cost ($)
3
Pears
7
Lesson 6-1
3/23/09 3:45:49 PM
Glencoe Precalculus
4 3 ⎢ 13.50 ⎤
7 2 4 14.20 ; (1.1, 1.25, 1); apples: $1.10, oranges: $1.25, pears: $1
⎣ 3 8 2 ⎢ 15.30 ⎦
⎡5
Apples
Name
Rosario
10. FRUIT Three customers bought fruit at Michael’s Groceries. The table
shows the amount of fruit bought by each person. Write and solve a
system of equations to determine the price of each type of fruit.
(4, -5)
x + 3y = -11
7. -4x - 2y = -6
-2 -1 5 ⎤
2 0 -1 8
0 1 -2 -4 ⎦
y - 2z = -4
⎡
2x - z = 8
5x - 2y + z = 4
6. -4x - 2y - z = 5
-2x - 3y + z = 6
5. 3x + 4y + 7z = -8
Solve each system of equations using Gauss-Jordan elimination.
⎡ 5 -2 14 ⎤
⎢
⎣ -3 1 -7 ⎦
-3x + y = -7
4. 5x - 2y = 14
Write the augmented matrix for each system of linear equations.
(-8, 4)
⎡ 1 -1 -12 ⎤
1. ⎢
⎣ -3 2 32 ⎦
Multivariable Linear Systems and Row Operations
Practice
DATE
Write each system of equations in triangular form using Gaussian
elimination. Then solve the system.
6-1
NAME
Answers (Lesson 6-1)
Chapter 6
A3
1
1
⎢ 31
⎤
PERIOD
8
85
30
40
25
Senior
Citizen
1385
2685
1755
Total
Paid($)
Glencoe Precalculus
$500: simple savings,
$2000: certificate of deposit,
$2500: municipal bonds
c. Solve the system that you wrote in
part b using Gauss-Jordan
elimination.
⎢
⎤
1
1
5000
0.035
0.043
0.01
182.50
0 ⎦
0
-1
⎣5
⎡1
b. Write the augmented matrix for
the system of equations that you
wrote in part a.
s + c + m = 5000
0.01s + 0.035c + 0.043m
= 182.50
5s - m = 0
a. Write a system of equations
representing Mr. Wiley’s
investment pattern.
5. INVESTING Mr. Wiley invested $5000 in
three different accounts at the beginning
of last year, yielding him a total of
$182.50 of interest at the end of the year.
The three accounts were a simple savings
account earning 1%, a certificate of
deposit earning 3.5%, and municipal
bonds earning 4.3%. His municipal
bond investment was 5 times the
amount of money invested in the simple
savings account.
$4: child, $11: adult,
$9: senior citizen
45
Sun
175
110
80
100
Fri
Adult
Child
Sat
Day
4. MOVIES The table shows the number of
individuals attending the movies over the
weekend at the Majestic Theater.
Determine the costs for a child, adult,
and senior citizen to attend the movies.
10/23/09 11:55:32 AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 8
Chapter 6
⎢ 3.25 ⎦
⎣
18 nickels, 6 dimes, 7 quarters
⎢ 10.05 -10.10 -10.25 ⎢ 5 ;
⎡1
3. COINS Tina has 31 nickels, dimes, and
quarters in her purse. She has 5 more
nickels then the total number of dimes
and quarters. If the total value of the
coins is $3.25, how many of each coin
does Tina have in her purse? Write and
solve a system to determine the number
of coins.
⎢
⎢
1 1 1 ⎢ 275 ⎤
-4 1 -4
0 ; 40 tables,
⎣ 200 150 75 ⎢ 42,125 ⎦
220 chairs, 15 stools
2. MANUFACTURING A company
manufactures tables, chairs, and stools.
Last week, it built a total of 275 items.
The number of chairs built was four
times the total number of tables and
stools built. The total value of these
items is $42,125 with a chair selling for
$150, a table for $200, and a stool for
$75. Write and solve a system of
equations to determine the number of
each item built last week.
hamburger: $2.50, chips: $1.25
⎡ 5 3 16.25 ⎤
; (2.5, 1.25);
⎢
⎣ 4 8 20
⎦
⎡
DATE
Multivariable Linear Systems and Row Operations
Word Problem Practice
1. FOOD Mark bought 5 hamburgers
and 3 bags of chips at a cost of $16.25.
Henry bought 4 hamburgers and 8 bags
of chips at a cost of $20. Write and solve
a system of equations to determine the
cost of a hamburger and a bag of chips.
6-1
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enrichment
DATE
PERIOD
A
A
⎢
⎢
-D + 2E + F = -5
3D + 4E + F = -25
2D + -E + F = -5
⎡ -1 2 1 -5 ⎤
3 4 1 -25 .
⎣ 2 -1 1 -5 ⎦
or
⎢
18
3
3
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 9
Chapter 6
9
Lesson 6-1
10/23/09 12:00:51 PM
Glencoe Precalculus
x2 + y2 - 6x - 8y + 21 = 0
2. (3, 6), (5, 4), (3, 2)
Answers
3x2 + 3y2 - 7x - 13y + 4 = 0
1. (1, 0), (-1, 2), (3, 1)
Find an equation of the circle passing through the given points.
Exercises
in the equation of the circle in general form: 3x2 + 3y2 - 10x - 10y - 5 = 0.
10
10
5
x-−
y-−
= 0. Multiplying both sides of the equation by 3 results
x2 + y2 - −
3
3
3
Substituting these values back into x2 + y2 + Dx + Ey + F = 0, you get
30
5
10
10
Using substitution, you can find that F = - −
or - −
, E = - −
, and D = - −
.
⎡ 1 -2 -1 ⎢ 5 ⎤
Using Gaussian elimination, you can find the equivalent matrix to be 0 1 -5
5 .
⎣ 0 0 18 ⎢ -30 ⎦
The augmented matrix for this system is
1 + 4 -1D + 2E + F = 0
9 + 16 + 3D + 4E + F = 0
4 + 1 + 2D -1E + F = 0
Because the three points are on the circle, they satisfy this equation.
Use substitution to get the following system.
A
D
E
F
equation is x2 + y2 + Dx + Ey + F = 0, where D = −
, E = −
, and F = −
.
Because A ≠ 0, divide both sides of the equation by A. The resulting
The general form equation for a circle is Ax2 + Ay2 + Dx + Ey + F = 0,
where A ≠ 0. Suppose you want to find the equation of a circle passing
through the points (-1, 2), (3, 4), and (2, -1). How can you use the general
form equation, systems of equations, and matrices to answer the question?
Circles
6-1
NAME
Answers (Lesson 6-1)
TI-Nspire Activity
Solve the system of equations.
x+y+z=5
2x + 3y - z = 55
-x + 4y + 2z = 4
A4
10
(23, -18, 5)
(14, 5, 9)
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 10
Chapter 6
2x - 2y + 3z = 97
x - y - 2z = 31
2x - y - z = 14
x + 3y = 29
4. x + y + z = 10
(-2, 10, -1)
(-4, 11, 21)
3. 2x - z = 19
2x + 3y + 2z = 24
4x - y + 4z = -22
6x + y + z = 8
2. x - y -z = -11
-x + 5y - 3z = -4
1. x + 2y + z = 39
Solve each system of equations.
Exercises
To solve another system of three equations in three variables, you can copy
the left side of the previous line onto the current line by highlighting it and
pressing ·. Then simply update the numbers in the cells and press ·.
Step 3: Use the matrix to solve the system. The solution is (8, 9, -12).
Step 2: Press menu and choose MATRIX & VECTOR > REDUCED
ROW-ECHELON FORM. Press / v to enter the
matrix above which is considered the previous
answer. Press ·.
Step 1: Add a CALCULATOR page. Enter the augmented
matrix by pressing / and the multiplication key.
Select the 3 by 3 matrix. Change the number of
columns to 4. Type in the elements.
Example
Glencoe Precalculus
PERIOD
You can solve a system of equations by entering the augmented
matrix into a TI-Nspire and finding the reduced row-echelon form
of the matrix.
DATE
3/23/09 3:46:02 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 6
Reduced Row-Echelon Form
6-1
NAME
DATE
PERIOD
2 3⎤
to find AB,
4 -1 ⎦
2 3⎤ ⎡ 4(-1) + (-2)(-2)
=⎢
4 -1⎦ ⎣
2 3⎤ ⎡ 4(-1) + (-2)(-2) 4(2) + (-2)(4)
=⎢
4 -1⎦ ⎣
⎤
⎦
⎤
⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 11
Chapter 6
⎡ -17 -9 ⎤
⎡ -2
2⎤
; BA = ⎢
⎣
⎣ 3 -13⎦
6 2⎦
AB = ⎢
⎡-1 5⎤
⎡ 2 4⎤
1. A = ⎢
, B = ⎢
⎣ 0 -2⎦
⎣-3 -1⎦
11
2 3⎤ ⎡ 0 0 14⎤
=⎢
4 -1⎦ ⎣ -5 10 -6⎦
Find AB and BA, if possible.
Exercises
⎡ 4 -2⎤ ⎡ -1
⎢
·⎢
⎣ -1 3⎦ ⎣ -2
Lesson 6-2
3/23/09 3:46:06 PM
Glencoe Precalculus
⎡ -7 -7 6 ⎤
; BA is undefined.
⎣ 0 -14 4 ⎦
AB = ⎢
⎡-1 3⎤
⎡-2 4 0⎤
2. A = ⎢
, B = ⎢
⎣ –3 –1 2⎦
⎣-3 2⎦
2 3⎤ ⎡ 4(-1) + (-2)(-2) 4(2) + (-2)(4) 4(3) + (-2)(-1) ⎤
=⎢
4 -1⎦ ⎣ (-1)(-1) + 3(-2) (-1)(2) + 3(4) (-1)(3) + 3(-1) ⎦
Then simplify each sum.
⎡ 4 -2⎤ ⎡ -1
⎢
·⎢
⎣ -1 3⎦ ⎣ -2
Continue multiplying each row by each column to find the sum for each entry.
⎡ 4 -2⎤ ⎡ -1
⎢
·⎢
⎣ -1 3⎦ ⎣ -2
Follow this same procedure to find the entry for row 1, column 2 of AB.
⎡ 4 -2⎤ ⎡ -1
⎢
·⎢
⎣ -1 3⎦ ⎣ -2
To find the first entry in AB, write the sum of the products of the entries in
row 1 of A and in column 1 of B.
A is a 2 × 2 matrix and B is a 2 × 3 matrix. Because the number of columns
for A is equal to the number of rows for B, the product AB exists.
2 3⎤
4 -1⎦
⎡ 4 -2 ⎤
⎡ -1
Use matrices A = ⎢
and B = ⎢
⎣ -1 3 ⎦
⎣ -2
⎡ 4 -2⎤ ⎡ -1
AB = ⎢
·⎢
⎣ -1 3⎦ ⎣ -2
if possible.
Example
To multiply matrix A by matrix B, the number of
columns in A must be equal to the number of rows in B. If A has dimensions
m × r and B has dimensions r × n, their product, AB, is an m × n matrix.
If the number of columns in A does not equal the number of rows in B, the
matrices cannot be multiplied.
⎡ a b ⎤ ⎡ e f ⎤ ⎡ae + bg af + bh ⎤
⎢
=⎢
·⎢
⎣ c d ⎦ ⎣ g h ⎦ ⎣ ce + dg cf + dh⎦
Matrix Multiplication, Inverses, and Determinants
Study Guide and Intervention
Multiply Matrices
6-2
NAME
Answers (Lesson 6-1 and Lesson 6-2)
Chapter 6
DATE
(continued)
PERIOD
Matrix Multiplication, Inverses, and Determinants
Study Guide and Intervention
7(4) + (-4)(7) ⎤ ⎡1 0⎤
or ⎢
-5(4) + 3(7)⎦ ⎣0 1⎦
A5
no; AB ≠ BA ≠ I2
yes; AB = BA = I2
12
Glencoe Precalculus
10/23/09 12:02:43 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 12
Chapter 6
⎡3 2⎤
4. Find the determinant of A = ⎢
. Then find A–1, if it exists.
⎣1 -1⎦
⎡ 0.2 0.4⎤
det(A) = -5, A-1 = ⎢
⎣ 0.2 -0.6⎦
⎡ 5 -1⎤
3. Find the determinant of A = ⎢
. Then find A-1, if it exists. det(A) = 0;
⎣-10 2⎦
A-1 does not exist.
⎡3 2⎤
⎡1 2⎤
2. A = ⎢
, B = ⎢
⎣4 1⎦
⎣4 3⎦
⎡11 5⎤
⎡ 1 -5⎤
1. A = ⎢
, B = ⎢
⎣ 2 1⎦
⎣-2 11⎦
Determine whether A and B are inverse matrices. Explain your reasoning.
Exercises
Since det(A) ≠ 0, A is invertible.
= 2(-6) - 3(-2) or -6
⎡
1⎤
⎢1 -−
3
=⎢
1
1
⎢− - −
3⎦
⎣2
1 ⎡-6 2⎤
A-1 = - −
⎢
6 ⎣-3 2⎦
⎡ 2 -2⎤
Find the determinant of A = ⎢
. Then find A–1, if it exists.
⎣ 3 -6⎦
⎢2 -2⎢
det(A) = ⎢
⎢
⎢3 -6⎢
Example 2
⎡3 4⎤ ⎡ 7 -4⎤ ⎡3(7) + 4(-5) 3(-4) + 4(3) ⎤ ⎡1 0⎤
BA = ⎢
·⎢
=⎢
or ⎢
⎣5 7⎦ ⎣-5 3⎦ ⎣ 5(7) + 7(-5) 5(-4) + 7(3)⎦ ⎣0 1⎦
Because AB = BA = I, B = A-1 and A = B-1.
⎡ 7 -4⎤ ⎡3 4⎤ ⎡7(3) + (-4)(5)
AB = ⎢
·⎢
=⎢
⎣-5 3⎦ ⎣5 7⎦ ⎣ -5(3) + 3(5)
If A and B are inverse matrices, then AB = BA = I.
Example 1
⎡ 7 -4⎤
⎡3 4⎤
Determine whether A = ⎢
and B = ⎢
are
⎣-5 3⎦
⎣5 7⎦
inverse matrices.
The identity matrix is an n × n matrix consisting of
all 1s on its main diagonal, from upper left to lower right, and 0s for all other elements. Let
In be the identity matrix of order n and let A be an n × n matrix. If there exists a matrix B
such that AB = BA = In, then B is called the inverse of A and is written as A–1. If a matrix
has an inverse, it is invertible. The determinant of a 2 × 2 matrix can be used to
determine whether or not a matrix is invertible.
⎡a b ⎤
⎡ d -b⎤
1
If A = ⎢
, det(A) = ad - cb. If ad - cb ≠ 0, then A-1 = −
⎢
.
ad - cb ⎣-c a⎦
⎣ c d⎦
Inverses and Determinants
6-2
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
210
B
400
520
3-Wood
450
310
5-Wood
400
300
Putter
150
120
170
Putter
210
3-Wood
5-Wood
Club Value ($)
Club
1-Wood
⎡
⎤
⎡
⎢ ⎢ ⎢
no
⎡5 -2⎤
⎡-1 0⎤
7. A = ⎢
, B = ⎢
⎣4 -3⎦
⎣ 2 -8⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 13
Chapter 6
⎡ 3 -27 12⎤
⎢
⎣-9
4 -2⎦
10. AB + C
Evaluate.
⎡-1 5⎤
A =⎢
⎣ 3 0⎦
Answers
13
Lesson 6-2
10/23/09 12:03:23 PM
Glencoe Precalculus
⎡-1 0 -4⎤
C =⎢
⎣ 3 -2 1⎦
⎡-12 -17 7⎤
⎢
⎣ -9
6 9⎦
11. A(B - C)
⎡-4 2 -1⎤
B =⎢
⎣ 0 -5 3⎦
Find the determinant of each matrix. Then find its inverse, if it exists.
⎡ 1 -2.5⎤
⎡6 5⎤
⎡-2
4⎤
8. ⎢
9. ⎢
2; ⎢
0; singular
⎣2 2⎦
⎣ 3 -6⎦
⎣-1
3⎦
yes
⎡1 2⎤
⎡ 3 -2⎤
6. A = ⎢
, B = ⎢
⎣1 3⎦
⎣-1 1⎦
Determine whether A and B are inverse matrices.
⎢
⎢ ⎢ 3x1 - 2x2 + 8x3 = 28
3 ⎤ ⎡ x1 ⎤
4
1 -2
2 1 2⎤ ⎡ x1 ⎤ ⎡ 11 ⎤
x
·
=
(4,
-3,
-2)
5 3 -1
-5 -1 4 · x2 = 1 (2, 1, 3)
13
2
x
4
4
-1
11
⎣
⎦
⎣ 3 -2 8⎦ ⎣ x3 ⎦ ⎣ 28 ⎦
⎦ ⎣ 3⎦ ⎣
⎡
-5x1 - x2 + 4x3 = 1
4x1 - x2 + 4x3 = 11
5. 2x1 + x2 + 2x3 = 11
5x1 + 3x2 - x3 = 13
4. x1 - 2x2 + 3x3 = 4
Write each system of equations as a matrix equation, AX = B. Then use
Gauss-Jordan elimination on the augmented matrix to solve for X.
600
1-Wood
A
Company
Club Type and Quantity
⎡ 9 15⎤
⎡ 4 10⎤
20 -4⎤
AB = ⎢
; BA = ⎢
⎣ 10 -12 3⎦
⎣-7 -5⎦
⎣–6 0⎦
3. GOLF The number of golf clubs manufactured daily by two different companies is
shown, as well as the selling price of each type of club. Use this information to
determine which company’s daily production has the highest retail value. How much
greater is the value? Company A; $69,300
AB is undefined; BA = ⎢
⎡-14
⎡-1 6 0⎤
⎡ 2 -4⎤
1. A = ⎢
, B = ⎢
⎣ 3 -2 1⎦
⎣ -1 3⎦
⎡ 3 0⎤
⎡ 3 5⎤
2. A = ⎢
, B = ⎢
⎣-1 2⎦
⎣ -2 0⎦
Matrix Multiplication, Inverses, and Determinants
Practice
Find AB and BA, if possible.
6-2
NAME
Answers (Lesson 6-2)
A6
300
4 HP
250
Mower Type
Retail Value ($)
2
350
5 HP
3
4
3
C
6
3
2
Holly
Joelle
Luisa
4
5
4
C
I
6
1
2
20%
50%
30%
SS
C
I
30%
30%
40%
System B
c. Luisa system A
b. Joelle system A
a. Holly system B
Use matrices to determine which system
favors each skater.
System A
Criteria
One of two weighted systems shown
below is used.
SS
Skater
2. ICE SKATING Holly, Joelle, and Luisa
are competitive skaters. Their routines
are judged on skating skill (SS),
choreography (C), and interpretation (I).
In a recent competition, they received the
following scores.
Store A; $4600
4.5 HP
7
5 HP
5
3
4.5 HP
4
B
Store
5
A
4 HP
Mower Type
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 14
Chapter 6
PERIOD
14
Charge per unit ($)
B
40
C
30
⎢ ⎢
Glencoe Precalculus
25 units from vendor A,
75 units from vendor B,
50 units from vendor C
c. Solve the system that you found in
part b to determine how many units
of beads were purchased from each of
the vendors.
⎢
⎡ 1 1 1⎤ ⎡ A ⎤ ⎡ 150 ⎤
35 40 30 · B = 5375
⎣ –2 0 1⎦ ⎣ C ⎦ ⎣
0⎦
b. Write the system of equations that
you found in part a as a matrix
equation, DX = E.
A + B + C = 150
35A + 40B + 30C = 5375
-2A + C = 0
The total delivery cost was $5375. The
store ordered twice the number of units
of beads from vendor C than it ordered
from vendor A.
a. Write a system of equations
representing this situation.
A
35
Vendor
4. CRAFTS A craft store orders beads from
three different vendors, A, B, and C. One
month, the store ordered a total of 150
units of beads from these vendors. The
shipping charges are as shown.
7 trips by the 10-ton truck,
13 trips by the 12-ton truck
3. LANDSCAPING Two dump trucks have
capacities of 10 tons and 12 tons. They
make a total of 20 round trips to haul
226 tons of topsoil for a landscaping
project. How many round trips does each
truck make?
Matrix Multiplication, Inverses, and Determinants
Word Problem Practice
DATE
3/23/09 3:46:23 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 6
1. INVENTORY A hardware company keeps
three types of lawnmowers in stock at
each of its three stores. The current
inventory and retail price for each mower
is shown. Determine which store’s
inventory has the greatest value. What is
this value?
6-2
NAME
Enrichment
DATE
⎢
P1
P2 P3
⎡ 0 2 3⎤
2 0 1
⎣ 3 1 0⎦
1
1
PERIOD
P2 P3
P1
⎢
⎢
P2 P3
⎢
P1
P2
P3
3
1
6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 15
Chapter 6
15
travel from P2 to P1 by going through P3.
Lesson 6-2
3/23/09 3:46:27 PM
Glencoe Precalculus
2. What does the product of row 2 and column 1 indicate? There are three ways to
travel from P1 to P3 by going through P2.
1. What does the product of row 1 and column 3 indicate? There are two ways to
Exercises
The result indicates that there are six ways to travel from P3 to P2 by going through P1.
3
2
1
0
0
1
×
+
×
+
×
=
P2→P2
P3→P3
P3→P2
P3→P1
P1→P2
P3→P2
Similarly, consider the product of row 3 and column 2.
The result indicates that there are three ways to travel from P1 to P2 by going through P3.
0
2
2
0
3
1
×
+
×
+
×
=
P1→P1
P1→P2
P1→P2
P2→P2
P1→P3
P3→P2
Consider the product of row 1 and column 2.
P1
⎡ 0 2 3⎤ P1 ⎡ 0 2 3⎤ P1 ⎡13 3 2⎤
M2 = P2 2 0 1 P2 2 0 1 = P2 3 5 6
P3 ⎣ 3 1 0⎦ P3 ⎣ 3 1 0⎦
P3 ⎣ 2 6 10⎦
P1
It can be shown that if we square matrix M, we can determine how many
ways there are to travel from one park to another by traveling through
the third park.
Confirm the numbers in each cell. For example, the element in row 2,
column 3 indicates that there is only one way to travel from P2 to P3 without
going through P1. The element in row 1, column 3 indicates there are three
ways to travel from P1 to P3 without going through P2.
P3
M = P2
P1
Suppose three state parks P1, P2, and P3 are connected by
roads as shown in the figure. As you can see, there are only
two ways to travel from P1 to P2 without going through P3.
There are only three ways to travel from P1 to P3 without
going through P2. The matrix M represents the number of
ways to go from one park to another without traveling
through the third park.
Travel
6-2
NAME
Answers (Lesson 6-2)
Chapter 6
DATE
A7
7⎤
3⎦
⎢ ⎢ ⎢
x - z = -4
3x - y = -10
16
(2, -2, 5)
10/23/09 12:09:41 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 16
Chapter 6
(6, -11)
y + 4z = 18
2x - 3y + 2z = 20
4. x + y - z = -5
3. 3x + y = 7
-2x - 5y = 43
(-1, 0, 3)
(-2, 4)
3x + 2y + z = 0
2. x - y + 2z = 5
1. -2x + 5y = 24
Glencoe Precalculus
⎢ ⎢ ⎡ 4 7 2⎤ ⎡ 0⎤ ⎡ 1⎤
= 11 17 5 · 9 = -2
⎣-2 -3 -1⎦ ⎣-3⎦ ⎣ 4⎦
So, the solution of the system is
(1, -2, 4).
Multiply A-1 by B to solve the system.
· B
X=
A-1
⎢
⎡-2 1 1⎤ ⎡x⎤ ⎡ 0⎤
9
1 0 2 · y =
⎣ 1 -2 -9⎦ ⎣ z⎦ ⎣-31⎦
Use a graphing calculator to find A-1.
b. -2x + y + z = 0
x + 2z = 9
x - 2y - 9z = -31
Write the system in matrix form.
A
· X = B
Use an inverse matrix to solve each system of equations, if possible.
Exercises
So, the solution of the system is
(-24, -8).
Multiply A-1 by B to solve the system.
X =
A-1 · B
⎡-24⎤
⎡-2 -7 ⎤ ⎡-16⎤
= ⎢
·⎢
= ⎢
⎣ -8⎦
⎣-1 -3 ⎦ ⎣ 8⎦
⎡2
1
= −−
⎢
(2)(3) - (-7)(-1) ⎣1
⎡ d -b⎤
1
A-1 = −
⎢
ad - cb ⎣-c a⎦
Use the formula for an inverse of a 2 × 2
matrix to find the inverse A-1.
⎡-16⎤
⎡ 3 -7 ⎤ ⎡x⎤
⎢
·⎢ = ⎢
⎣ 8⎦
⎣-1 2 ⎦ ⎣y⎦
Write the system in matrix form.
A · X = B
a. 3x - 7y = -16
-x + 2y = 8
Use an inverse matrix to solve each system of equations, if possible.
A square system has the same
number of equations as variables. If a square matrix has an
inverse, the system has one unique solution.
Example
PERIOD
Solving Linear Systems Using Inverses and Cramer’s Rule
Study Guide and Intervention
Use Inverse Matrices
6-3
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
(continued)
PERIOD
|A |
|A|
|A|
|A |
|A |
|A|
⎪-25 -21 ⎥ = (-2)(-2) - 5(1) or -1
⎪-717 -21⎥
-1
|A|
-7(-2) - 17(1)
-1
-1
⎥
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 17
Chapter 6
-x + 2y = 14
3. 3x + y = 21
-2x - 5y = -8
1. x - 2y = -5
(4, 9)
(-1, 2)
Answers
17
x + 3y = -3
4. -2x - 4y = 2
-x + 4y = 9
2. 3x - 3y = -18
Lesson 6-3
10/23/09 12:11:22 PM
Glencoe Precalculus
(3, -2)
(-5, 1)
Use Cramer’s Rule to find the solution of each system of linear
equations, if a unique solution exists.
Exercises
Therefore, the solution is x1 = 3 and x2 = -1 or (3, -1).
⎪
-2 -7
|A2|
-2(17) - 5(-7)
5 17
1
x2 = −
=−= − =−
or -1
-1
-1
-1
|A|
-3
1
x1 = −
=−= − =−
or 3
|A |
Because the determinant of A does not equal zero, you can apply
Cramer’s Rule.
|A| =
⎡-2 1⎤
The coefficient matrix is A = ⎢
. Calculate the determinant of A.
⎣ 5 -2⎦
Example
Use Cramer’s Rule to find the solution of the system
of linear equations, if a unique solution exists.
-2x1 + x2 = -7
5x1 - 2x2 = 17
where Ai is the matrix obtained by replacing the ith column of A with the
column of constants B. If det(A) = 0, then AX = B has either no solution or
infinitely many solutions.
|A|
3
n
1
2
x1 = −
, x2 = −
, x3 = −
, … , xn = −
,
|A |
Let A be the coefficient matrix of a system of n linear equations in n
variables given by AX = B. If det(A) ≠ 0, then the unique solution of the
system is given by
Another method, known as Cramer’s Rule, can be used
to solve a square system of equations.
Solving Linear Systems Using Inverses and Cramer’s Rule
Study Guide and Intervention
Use Cramer’s Rule
6-3
NAME
Answers (Lesson 6-3)
PERIOD
)
x + 2y + z = 8
2x + 3y - z = 1
no solution
-4x + 5y - z = 18
5x - 3y = -11
(-1, 2, -4)
A8
y + 2z = 20 (-5, 6, 7)
3x - z = -22
7. x + y + z = 8
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 18
Chapter 6
18
Payton delivers 100; Santiago delivers 20; Queisha delivers 10
8. PAPER ROUTE Payton, Santiago, and Queisha each have a paper
route. Payton delivers 5 times as many papers as Santiago. Santiago
delivers twice as many papers as Queisha. If 20 papers were added to
Payton’s route, he would then deliver four times the number of papers
that Santiago and Queisha deliver. How many papers does each
person deliver?
-2x - 3y = -1 (-4, 3)
6. -4x - 5y = 1
Use Cramer’s Rule to find the solution of each system of linear
equations, if a unique solution exists.
Manuel: 3 hours, Harry: 2 hours, Ellen: 6 hours
5. TELEVISION During the summer, Manuel watches television M hours
per day, Monday through Friday. Harry watches television H hours per
day, Friday through Sunday. Ellen watches television E hours per day,
Friday and Saturday. Altogether, they watch television 33 hours each
week. On Fridays, they watch a total of 11 hours of television. If the
number of hours Ellen spends watching television on any given day is
twice the number of hours that Manuel spends watching television on
any given day, how many hours of television does each of them watch
each day?
4. x + y - 2z = 5
(-2, 5)
4x + 3y = 7
2. -2x - 8y = -36
3. x - 2y + 7z = -33
1
−
, -4
2
(
-6x + 2y = -11
1. 4x - 7y = 30
Glencoe Precalculus
Solving Linear Systems Using Inverses and Cramer’s Rule
Practice
DATE
3/23/09 3:46:41 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 6
Use an inverse matrix to solve each system of equations, if possible.
6-3
NAME
D
C
:
9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 19
Chapter 6
Marty hit 37 home runs;
Carlos hit 36 home runs;
Andrew hit 35 home runs.
2. BASEBALL In one season, Marty,
Carlos, and Andrew hit a total of
108 home runs. Marty and Andrew hit
twice as many home runs as did Carlos,
although Carlos had one more home run
than Andrew. How many home runs did
each player hit?
a = 16 centimeters,
b = 30 centimeters,
c = 34 centimeters
;
B
8
DATE
PERIOD
19
0.2
0.4
0.4
X
Y
Z
0.4
0
0.3
Almonds
0.2
0.6
0.5
Hazelnuts
⎢ ⎢
Lesson 6-3
3/23/09 3:46:44 PM
Glencoe Precalculus
8000 X gift boxes, 9500 Y gift
boxes, 11,500 Z gift boxes
c. Determine how many of each gift box
the company has.
⎢
⎡ 0.2 0.4 0.4 ⎤ ⎡ X ⎤ ⎡ 10,000 ⎤
0.3 0 0.4 · Y = 7000
⎣ 0.5 0.6 0.2 ⎦ ⎣ Z ⎦ ⎣ 12,000 ⎦
b. Solve the system of equations that you
wrote in part a as a matrix equation,
AX = B.
0.2X + 0.4Y + 0.4Z = 10,000
0.3X + 0.4Z = 7000
0.5X + 0.6Y + 0.2Z = 12,000
a. Write a system of equations
representing this situation.
The company has 10,000 pounds of
cashews, 7000 pounds of almonds,
and 12,000 pounds of hazelnuts in
its gift boxes.
Cashews
Gift
Box
4. NUTS A nut company makes three types
of one-pound gift boxes: X, Y,
and Z. The table shows the amount of
each nut in each box.
$750 at 4%, $2000 at 5%,
$1250 at 3.5%
3. INVESTING A total of $4000 is invested
in three accounts paying 4%, 5%, and
3.5% simple interest. The combined
annual interest is $173.75. If the interest
earned at 5% is $70 more than the
interest earned at 4%, how much money
is invested in each account?
Solving Linear Systems Using Inverses and Cramer’s Rule
Word Problem Practice
1. PERIMETER The perimeter of rectangle
WXYZ is 92 centimeters. The perimeter
of triangle WXZ is 80 centimeters. If the
−−
length of XZ is two more than twice the
−−−
length of WZ, what are the values
of a, b, and c?
6-3
NAME
Answers (Lesson 6-3)
Chapter 6
Enrichment
DATE
PERIOD
A9
⎢ ⎢
2
⎢ ⎢
20
18
11
rectangle
triangle
6
12
1
n
10.5
20
4
A
Glencoe Precalculus
8
3/23/09 3:46:49 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 20
Chapter 6
See students’ work.
2. Verify Pick’s Theorem using three simple
polygons of your choice.
⎢
⎡ 11 7 1 ⎤ ⎡ x ⎤ ⎡ 11.5 ⎤
4 3 1 y = 4
⎣ 11 3 1 ⎦ ⎣ z ⎦ ⎣ 7.5 ⎦
1. Write the matrix equation that would be
used to verify Pick’s Theorem using the
polygons at the right.
Exercises
1
Solving this system, we can see that x = −
, y = 1, and z = -1.
⎢
Use the table to write a system and matrix equation.
8x + y + z = 4
⎡ 8 1 1⎤ ⎡x⎤ ⎡ 4 ⎤
18x + 12y + z = 20
18 12 1 y = 20
11x + 6y + z = 10.5
⎣11 6 1⎦ ⎣ z⎦ ⎣10.5⎦
b
Figure
square
Start by drawing three simple polygons on square dot paper similar to the ones shown
below. Be sure that the number of boundary points, interior points, and the area of the
figures are different. The table shown below summarizes the information.
You can use systems of equations and matrices to verify Pick’s Theorem. To verify that the
1
coefficients in the equation for A are −
, 1, and -1, you can write a system of three equations
2
of the form A = bx + ny + z, where the values of A, b, and n vary from polygon to polygon.
2
10
+ 2 - 1 or 6 square units.
and n = 2. Therefore, A = −
2
b
interior points n minus 1, or A = −
+ n - 1. In the figure, b = 10
Consider the simple polygon drawn on square dot paper, shown at
the right. Pick’s Theorem states that the area of the polygon A is
equal to half the number of boundary points b plus the number of
Pick’s Theorem
6-3
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Spreadsheet Activity
DATE
PERIOD
A
x=
y=
= A1*C2 - C1*A2
= C1*B2 - B1*C2
= A1*B2 - B1*A2
6
5
B
= (A6/A4)
= (A8/A4)
3
1
-12
8
C
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 21
Chapter 6
(0.4, 0.2)
Answers
21
(-4, 4)
7. 3y = 4x + 28
5x + 7y = 8
(2, -3)
(4, -12)
6. 0.3x + 1.6y = 0.44
0.4x + 2.5y = 0.66
5. 5x - 3y = 19
7x + 2y = 8
4. 6x + 3y = -12
5x + y = 8
Use the spreadsheet to solve each system of equations.
C1 B1
A1 C1
⎪
⎪
A2 C2⎥
C2 B2⎥
x = −; y = −
B1
A1 B1
⎪A2 B2⎥ ⎪A1
A2 B2⎥
3. Explain how the values of x and y are found using
Cramer’s Rule.
⎡C1 B1⎤ ⎡A1 C1⎤
⎢
; ⎢
⎣C2 B2⎦ ⎣A2 C2⎦
2. Write matrices whose determinants are found using the
formulas in cells A6 and A8.
⎡A1 B1⎤
⎢
⎣A2 B2⎦
Lesson 6-3
3/23/09 3:46:54 PM
Glencoe Precalculus
To use the spreadsheet to solve a
system of equations, write each
equation in the form below.
ax + by = c
In the spreadsheet, the values of
a, b, and c for the first equation are
entered in cells A1, B1, and C1,
respectively. The values of a, b, and c
for the second equation are entered in
cells A2, B2, and C2, respectively. The
values for the system 6x + 3y = -12
and 5x + y = 8 are shown.
1. Study the formula in cell A4. Write a matrix whose
determinant is found using this formula.
Exercises
1
2
3
4
5
6
7
8
9
10
11
12
You can use a spreadsheet to solve systems of equations with Cramer's Rule.
Cramer’s Rule
6-3
NAME
Answers (Lesson 6-3)
Partial Fractions
Study Guide and Intervention
DATE
x-1
2x - 1
x + 11
.
Find the partial fraction decomposition of −
x2 - 3x - 4
x-4
x+1
A10
·
⎡1 1 ⎤
⎢
⎣1 -4⎦
⎡A⎤
⎢ ⎣B⎦
X
=
=
⎡ 1⎤
⎢ ⎣ 11 ⎦
D
-4
x-7
-3
x+2
2
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 22
Chapter 6
2x(x - 1)
2x
1
(x - 1)
22
x (x - 1)
x
x
Glencoe Precalculus
x-1
2
3
1
-x-1 2
− + −2 + −
−
4. 5x
2
-7x + 13
x - 5x - 14
x +1
1
3. −2 − + −2
-2
x-4
2. −
−+−
2
x - x - 12
5x - 34
1. −
−+−
2
7
x+3
Find the partial fraction decomposition of each rational expression.
Exercises
x + 11
3
-2
Solving for X yields A = 3 and B = -2. Therefore, −
=−
+−
.
x-4
x+1
x2 - 3x - 4
A+B=1
→
A + (-4B) = 11
·
C
Equate the coefficients on the left and right side of the equation to obtain a
system of two equations. To solve the system, write it in matrix form CX = D
and solve for X.
Group like terms.
Distributive Property
1x + 11 = (A + B)x + (A + (-4B))
Multiply each side by the LCD, x2 - 3x - 4.
x + 11 = Ax + A + Bx - 4B
Form a partial fraction decomposition.
x + 11 = A(x + 1) + B(x - 4)
A
B
−
=−
+−
2
x + 11
x - 3x - 4
Rewrite the equation as partial fractions with constant numerators, A and B,
and denominators that are the linear factors of the original denominator.
Example
Each fraction in the sum is a partial fraction. The sum of these partial
fractions make up the partial fraction decomposition of the original
rational function. If the denominator of a rational expression contains a
repeated linear factor, the partial fraction decomposition must include a
partial fraction with its own constant numerator for each power of this factor.
2x - 3x + 1
3x - 1
2
-1
=−
+−
g(x) = −
2
The function g(x) shown below can be written
as the sum of two fractions with denominators that are linear
factors of the original denominator.
PERIOD
10/23/09 12:20:21 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 6
Linear Factors
6-4
NAME
DATE
Partial Fractions
Study Guide and Intervention
(continued)
PERIOD
4x4 - 2x3 - 13x2 + 7x + 9
x(x - 3)
Find the partial fraction decomposition of −−
.
2
2
2
x
x -3
(x - 3)
x
x +5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 23
Chapter 6
2x
x +1
3-x
(x + 1)
3. −
+−
−
2
2
2
2
2
2x3 + x + 3
(x + 1)
x + 5x
5
1. −
−-−
3
2
1
x
23
x+2
x +1
x-1
(x - 3)
Lesson 6-4
11/17/09 4:38:50 PM
Glencoe Precalculus
-x
(x + 1)
3x - 2
x -3
4. −
+ −
−
2
2
2
2
2
x3 + 2x2 + 2
(x + 1)
3x3 - 2x2 - 8x + 5
(x - 3)
2. −−
+−
−
2
2
2
2
2
Find the partial fraction decomposition of each rational expression.
Exercises
x(x - 3)
4x - 2x - 13x + 7x + 9
2x + 1
3x - 2
1
Therefore, −−
=−
+−
+−
.
2
2
2
2
2
1
=
E
9A = 9
3
=
D
-3C + E = 7
4
= -2
2
=
C
3
1
B
→
=
C = -2
A
-6A - 3B + D = -13
A+B=4
Write and solve the system of equations obtained by equating coefficients.
4x4 - 2x3 -13x2 + 7x + 9 = (A + B)x4 + Cx3 + (-6A - 3B + D)x2 + (-3C + E)x + 9A
4x4 - 2x3 -13x2 + 7x + 9 = Ax4 + Bx4 + Cx3- 6Ax2 - 3Bx2 + Dx2 - 3Cx + Ex + 9A
4x4 - 2x3 -13x2 + 7x + 9 = A(x2 - 3)2 + (Bx + C)x(x2 - 3) + (Dx + E)x
4x4 - 2x3 - 13x2 + 7x + 9
Bx + C
Dx + E
A
−−
=−
+−
+−
x2 - 3
x(x2 - 3)2
(x2 - 3)2
x
This expression is proper. The denominator has one linear factor and one
irreducible factor of multiplicity 2.
Example
Not all rational expressions
can be written as the sum of partial fractions using only linear
factors in the denominator. If the denominator of a rational expression
contains an irreducible quadratic factor, the partial fraction decomposition
must include a partial fraction with a linear numerator of the form Bx + C
for each power of this factor.
Irreducible Quadratic Factors
6-4
NAME
Answers (Lesson 6-4)
Chapter 6
Partial Fractions
Practice
DATE
PERIOD
-2
x-3
x+1
2x - 9x + 9
1
x-3
x
x+2
-1
2x - 3
x-1
x
4. −
−+−
2
x + x - 2x
2
-2
7
- 10x - 2 1
−
2. 6x
−+−+−
3
2
x+2
A11
x+1
-5x2 - 11x + 54
x + 2x - 8
x+4
2x - 3
(2x - 3)
5
1
-2 - −
+−
2
-8x2 + 22x - 10
8. −
(2x - 3)2
x-2
-3
2
-5 + −
+−
6. −
2
2
4x4 + 8x3 + 6x2 + 6x + 5
(3x + 2)(x + 1)
x+2
-x
1
−
+−
+−
3x + 2
x2 + 1
(x2 + 1)2
1
−
50 - x
x
50x
50(50 - x)
24
Glencoe Precalculus
3/23/09 3:47:07 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 24
Chapter 6
50
50
1
1
−
or −
+−
+−
1
−
partial decomposition of f(x).
decompose these functions into the sum of its partial fractions. Find the
necessary to work with functions of the form f(x) = − and to
1
x(50 - x)
13. GROWTH When working with exponential growth in calculus, it is often
–23
5
13
−
+−
+−
9x
9(x + 3)
3(x + 3)2
12. −−
2
2
2x2 + 5
x + 6x + 9x
11. −
3
2
3
x-5
2x - 1
−
+−
+−
x-3
x2 - 2
(x2 - 2)2
5x - 7x - 12x + 6x + 21
(x - 3)(x - 2)
3
10. −−
2
2
4
-4
2
-1
−
+−
+−
x-5
x
(x - 5)2
9. −−
3
2
-2x + 29x - 100
x - 10x + 25x
2
Find the partial fraction decomposition of each rational expression with
repeated factors.
x
9
2
6+−
+−
6x2 + 17x + 2
7. −
x2 + x
x
1
-2
3+−
+−
5. −
2
3x2 + 5x + 2
x + 2x
Find the partial fraction decomposition of each improper rational expression.
x + 3x + 2
6
9x + 15
3. −
+−
−
2
3
x+2
3x - 7
1. −
−+−
2
5
x - 7x + 12 x - 4
Find the partial fraction decomposition of each rational expression.
6-4
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Partial Fractions
50(25 + x)
50 + x
x + 50
350 - x
y
2
6
60
x(x + 3)2
4
f(x) =
8
x
10
–20
−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 25
Chapter 6
x+3
x
(x + 3)
20
–20
–20
−
+−
+−
3x
3(x + 3)
(x + 3)2
25
PERIOD
x+4
x-4
10
-10
−
+−
decomposition of f(x).
x - 16
Lesson 6-4
10/23/09 12:32:31 PM
Glencoe Precalculus
-80
f(x) = −
. Find the partial fraction
2
5. KAYAKING The total time it takes
for a kayaker to travel 10 miles upstream
and 10 miles downstream with a
paddling rate of 4 miles per hour in
still water is given by the function
x - 11
-3
4
−
−
x +
c. Find the partial fraction decomposition
of the rational expression.
⎡ 1 1⎤ ⎡ A⎤ ⎡ 1⎤
⎢
·⎢ =⎢
⎣-11 0⎦ ⎣B⎦ ⎣ -44⎦
b. Write the matrix form AX = B for the
system of equations found in part a.
A+B=1
-11A + 0B = -44
a. Write the system of equations
obtained by equating the coefficients.
[0, 10] scI: 1 by [-2, 10] scI: 1
x = 2, and x = 8. To find the area, first
find the partial fraction decomposition
of f(x).
x - 11x
x - 44
by the graphs of f(x) = −
, y = 0,
2
Answers
–20
3
3
−
+−
+−
or
2
20
−
Suppose you were to revolve the graph
of f(x) around the x-axis, creating a
three-dimensional object. Using calculus,
you could find the volume of the object.
But first, you would need to find the
partial fraction decomposition of f(x).
Find the partial decomposition of f(x).
0
2
4
6
8
10
0 ≤ x ≤ 10 for the graph of f(x)
shown below.
351(x + 1)
351(350 - x)
3. VOLUME Consider the domain
1
1
−
+ −
x+1
− + − or
plays a role when studying the spread of
an infection in certain populations. Find
the partial fraction decomposition of
g(x) when a = 350.
1
1
−
−
351
351
(x + 1)(a - x)
1
, where a > 0 often
g(x) = −
2. INFECTIONS A function of the form
1250
50 - −
how much acid she must mix with a 25%
acid solution to achieve the desired
percentage. Find the partial fraction
decomposition of f(x).
function f(x) = − to determine
DATE
4. AREA Calculus can be used to find
the area of the shaded region shown
below. The shaded region is bounded
Word Problem Practice
1. CHEMISTRY A chemist uses the
6-4
NAME
Answers (Lesson 6-4)
Enrichment
PERIOD
x+4
x-7
A12
x+4
x-7
x-1
x+2
12
x-2
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 26
Chapter 6
26
x + 3x - 54
Glencoe Precalculus
-x - 114
6. −
−+−
2
7
x+9
-8
x-6
x
1
x+3
5
x
5. −
−+−+−
3
2
18x + 39x - 30
x - x - 6x
2
x + x - 2x
-9
3
-6x + 18
+−
4. −
−
x2 - 10x + 24 x - 4
x-6
5
x-6
7
6
13x - 51
+−
3. −
−
x2 - 8x + 15 x - 3
x-5
-4
x+3
2
-3
4
- 8x - 4 2
−
2. 3x
−+−+−
3
2
x + 39
x - 3x - 18
1. −
−+−
2
Use the Heaviside Method to write the partial fraction
decomposition of each rational expression.
Exercises
x - 3x - 28
2x - 47
5
-3
−
+−
= −
2
Substitute A = 5 and B = -3 into the original equation to find the partial
fraction decomposition.
-3 = B
-33 = 11B
2(7) - 47 = A(7 - 7) + B(7 + 4)
To solve for B, let x = 7. This eliminates A.
5 = A
-55 = -11A
2(-4) - 47 = A(-4 - 7) + B(-4 + 4)
To solve for A, let x = -4. This eliminates B.
2x - 47 = A(x - 7) + B(x + 4)
Multiply both sides of the equation by the least common denominator (x + 4)(x - 7).
⎡ A
2x - 47
B ⎤
= (x + 4)(x - 7) ⎢−
(x + 4)(x - 7) −
+−
x2 -3x - 28
⎣x + 4 x - 7⎦
x - 3x - 28
2x - 47
A
B
−
=−
+−
2
Another method that can be used to find the partial fraction decomposition
of a rational expression is called the Heaviside Method. Consider the
equation shown below.
DATE
3/23/09 3:47:17 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 6
Heaviside Method
6-4
NAME
DATE
PERIOD
Write an objective function and a list of constraints to model the situation.
Graph the region corresponding to the solution of the system of constraints.
Find the coordinates of the vertices of the region formed.
Evaluate the objective function at each vertex to find the minimum
or maximum.
f(3, 0) = 18(3) + 12(0)
(3, 0)
63
54
← Maximum
0
(3, 0)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 27
Chapter 6
1. f(x, y) = 3x - 2y
2x + y ≤ 10
max at (5, 0) = 15,
x + 2y ≤ 8
min at (0, 4) = -8
x≥0
y≥0
27
x
Lesson 6-5
3/23/09 3:47:22 PM
Glencoe Precalculus
2. f(x, y) = x + 2y
x+y≤4
max at (3, 1) = 5,
x + 3y ≤ 6
min at (0, 0) = 0
x≥0
y≥0
Find the maximum and minimum values of the objective function f(x, y) and for
what values of x and y they occur, subject to the given constraints.
Exercises
Since f is greatest at (1.5, 3), the company will maximize profit if it makes and sells
1.5 belts for every 3 wallets.
f(1.5, 3) = 18(1.5) + 12(3)
(1.5, 3)
0
f(0, 4) = 18(0) + 12(4)
(0, 4)
48
Result
f(x, y) = 18x + 12y
f(0, 0) = 18(0) + 12(0)
Point
(0, 0)
Example
A leather company wants to add belts and wallets to its product
line. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets
require 3 hours of cutting time and 3 hours of sewing time. The cutting machine
is available 12 hours a week and the sewing machine is available 18 hours per
week. Belts will net $18 in profit and wallets will net $12. How much of each
product should be produced to achieve maximum profit?
Let x represent the number of belts and y represent the number of wallets.
The objective function is then given by f(x, y) = 18x + 12y.
Write the constraints.
x ≥ 0; y ≥ 0
Numbers of items cannot be negative.
2x + 3y ≤ 12
Cutting time
6x + 3y ≤ 18
Sewing time
Graph the system. The solution is the shaded region, including its
y
boundary segments. Find the coordinates of the four vertices by solving
the system of boundary equations for each point of intersection. The
coordinates are (0, 0), (0, 4), (1.5, 3), and (3, 0).
(1.5, 3)
(0, 4)
Evaluate the objective function for each ordered pair.
Step 1
Step 2
Step 3
Step 4
Linear programming is a process for finding
a minimum or maximum value for a specific quantity. The following steps can be used to
solve a linear programming problem.
Linear Optimization
Study Guide and Intervention
Linear Programming Applications
6-5
NAME
Answers (Lesson 6-4 and Lesson 6-5)
Chapter 6
A13
Linear Optimization
Study Guide and Intervention
DATE
(continued)
PERIOD
0
y
(3, 2)
(2, 4)
x
x≤8
y≥0
for 3 ≤ x ≤ 5;
min. at (0, 0) = 0
x≤5
x≥0
28
x≥0
point on y = -2x + 11
y≥0
2
Glencoe Precalculus
min. at (0, 0) = 0
for 4 ≤ x ≤ 8;
1
point on y = - −
x + 10
max. of 60 at every
11/17/09 4:39:09 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 28
Chapter 6
x + 2y ≤ 20
max. of 11 at every
x - y ≥ -4
2. f(x, y) = 3x + 6y
y≤5
2x + y ≤ 11
1. f(x, y) = 2x + y
Find the maximum and minimum values of the objective function
f(x, y) and for what values of x and y they occur, subject to the
given constraints.
Exercises
Because f(x, y) = 24 at (2, 4) and (3, 2), the problem has multiple optimal
solutions. An equation of the line through these two vertices is y = -2x + 8.
Therefore, f has a maximum value of 24 at every point on y = -2x + 8
for 2 ≤ x ≤ 3.
f(3, 0) = 6(3) + 3(0) or 18
f(3, 2) = 6(3) + 3(2) or 24
f(2, 4) = 6(2) + 3(4) or 24
f(0, 4) = 6(0) + 3(4) or 12
f(0, 0) = 6(0) + 3(0) or 0
Graph the region bounded by the given constraints. Find the value of the
objective function f(x, y) = 6x + 3y at each vertex.
Example
Find the maximum value of the objective function
f(x, y) = 6x + 3y and for what values of x and y it occurs, subject to
the following constraints.
2x + y ≤ 8
y≤4
x≤3
x≥0
y≥0
No or Multiple Optimal Solutions Linear programming models
can have one, multiple, or no optimal solutions. If the graph of the objective
function f to be optimized is coincident with one side of the region of feasible
solutions, f has multiple optimal solutions. If the region does not form
a polygon, but instead is unbounded, f may have no minimum value or
maximum value.
6-5
NAME
Linear Optimization
Practice
DATE
PERIOD
2x + 3y ≤ 18
max. at (0, 6) = 30,
min. at (0, 0) = 0
max. of 30 at every point on
3
y = -−
x + 12 for 2 ≤ x ≤4,
2
min. at (0, 0) = 0
max. at (7, 3.5) = 3.5,
min. at (4, 5) = -7
$40
Profit per Skate
$30
4 minutes
4 minutes
Ice Skates
120 minutes
200 minutes
Maximum Time Available
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 29
Chapter 6
Answers
29
Sample answer: If customers cannot get ice
skates, they might go somewhere else. They should
combine the math model with customer needs.
d. Describe why the company would choose a number of roller
skates and ice skates different from the answer in part c.
c. How many roller skates and ice skates should be
manufactured to maximize profit? What is the maximum
profit? 40 roller skates and no ice skates; $1600
b. Sketch a graph of the region determined by the constraints
from part a to find the set of feasible solutions for the
objective function.
0
10
20
30
40
50
60
y
x
Lesson 6-5
10/23/09 12:39:08 PM
Glencoe Precalculus
Roller Skates
(40,0)
(20,25)
10 20 30 40 50 60
(0,30)
f(x, y) = 40x + 30y; x ≥ 0; y ≥ 0; 5x + 4y ≤ 200; x + 4y ≤ 120
a. Write an objective function and list the constraints that model the
given situation.
1 minute
5 minutes
Assembling
Checking and Packaging
Roller Skates
Manufacturer Information
5. SKATES A manufacturer produces roller skates and ice skates.
x + y ≤ 10
3x + 2y ≤ 24
y≥0
y≤5
y≥0
y≤8
x≤7
x + 2y ≥ 14
x≥0
x≥0
4. f(x, y) = 3x + 3y
x+y≤8
max. at (8, 0) = 32,
min. at (0, 2) = 6
x+y≤7
3. f(x, y) = 2x - 3y
y≥0
2x + 3y ≥ 6
y≥0
x≥0
2. f(x, y) = 4x + 3y
x≥0
1. f(x, y) = 2x + 5y
Find the maximum and minimum values of the objective function
f(x, y) and for what values of x and y they occur, subject to the given
constraints.
6-5
NAME
Ice Skates
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 6-5)
A14
Linear Optimization
300
340
Profit ($)
300
150
Soybeans
b
50
150
Acres of Corn
100
(40, 320)
(0, 360)
200
250
(200, 0) c
Glencoe Precalculus
005_032_PCCRMC06_893807.indd 30
Chapter 6
plant 40 acres of corn and 320
acres of soybeans; $109,600
c. How can Mr. Fields maximize his
profit? What is his maximum profit?
0
50
100
150
200
250
300
350
450
400
b. Sketch a graph of the region
determined by the constraints from
part a to find the feasible solutions for
the objective function.
f(c, b) = 340c + 300b
c + b ≤ 360
300c + 150b ≤ 60,000
c ≥ 0, b ≥ 0
a. Write an objective function and
list the constraints that model
this situation.
Corn
Per Acre
30
PERIOD
4
(5, 3)
2⎞
⎝0, 9 3⎠
12
x
(14, 0)
of X; about $77.33
3
Glencoe Precalculus
2
9−
pounds of Y and 0 pounds
c. How many pounds of each type of food
should be purchased to satisfy the
requirements at the minimum cost?
What is the minimum cost?
0
4
8
12 ⎛
y
b. Sketch a graph of the region
determined by the constraints from
part a to find the feasible solutions
for the objective function.
f(x, y) = 12x + 8y
10x + 30y ≥ 140
20x + 15y ≥ 145
x ≥ 0, y ≥ 0
a. Write an objective function and
list the constraints that model
this situation.
2. NUTRITION A certain diet recommends
at least 140 milligrams of Vitamin A and
at least 145 milligrams of Vitamin B
daily. These requirements can be
obtained from two types of food. Type X
contains 10 milligrams of Vitamin A and
20 milligrams of Vitamin B per pound.
Type Y contains 30 milligrams
of Vitamin A and 15 milligrams of
Vitamin B per pound. Type X costs $12
per pound. Type Y costs $8 per pound.
Word Problem Practice
Labor ($)
Acres of Soybeans
DATE
3/23/09 3:47:35 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 6
1. FARMING Mr. Fields owns a 360-acre
farm on which he plants corn and
soybeans. The table shows the cost of
labor and the profit per acre for each
crop. Mr. Fields can spend up to $60,000
for spring planting.
6-5
NAME
DATE
8
PERIOD
0
1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC06_893807.indd 31
Chapter 6
Sample answer: ax + by = f(P)
31
2. If f(Q) = f(P), find an equation of the line containing P and Q.
2
8
x
x
Lesson 6-5
3/23/09 3:47:40 PM
Glencoe Precalculus
−−
It produces alternate optimal solutions all along PQ; f(W) = f(P).
1. If f(Q) = f(P), what is true of f ? of f(W)?
2
8
3
−−
Let P and Q be vertices of a closed convex polygon, and let W lie on PQ.
Let f(x, y) = ax + by.
Exercises
Example
If f(x, y) = 3x + 2y, find the maximum value of the
function over the shaded region at the right.
y
The maximum value occurs at the vertex (6, 3). The minimum value
occurs at (0, 0). The values of f(x, y) at W 1 and W 2 are between the
maximum and minimum values.
f(Q) = f(6, 3) = 3(6) + 2(3) or 24
f(W 1) = f(2, 1) = 3(2) + 2(1) or 8
1
f(W 2) = f(5, 2.5) = 3(5) + 2(2.5) or 20
0
f(P) = f(0, 0) = 3(0) + 2(0) or 0
Then 0 < w < 1 and the coordinates of W are
((1 - w)x 1 + wx 2, (1 - w)y 1 + wy2). Now consider
the function f(x, y) = 3x - 5y.
f(W) = 3[(1 - w)x 1 + wx 2] - 5[(1 - w)y 1 + wy 2]
= (1 - w)(3x 1) + 3wx 2 + (1 - w)(-5y1) - 5wy 2
= (1 - w)(3x1 - 5y1) + w(3x 2 - 5y2)
= (1 - w)f(P) + wf(Q)
This means that f(W) is between f(P) and f(Q), or that the
greatest and least values of f(x, y) must occur at P or Q.
You have already learned that over a closed convex polygonal region, the
maximum and minimum values of any linear function occur at the vertices
of the polygon. To see why the values of the function at any point on the
boundary of the region must be between the values at the vertices, consider
the convex polygon with vertices P and Q.
−−−
Let W be a point on PQ.
y
PW
If W lies between P and Q, let −
= w.
PQ
Enrichment
Convex Polygons
6-5
NAME
Answers (Lesson 6-5)
Chapter 6 Assessment Answer Key
(Lessons 6-1 and 6-2)
Page 33
⎡
(-5, 1)
3.
(-1, 3, 2)
4.
x
1.
2 -5 7
-1 -1 2
1
⎣ -3 4 -1 10 ⎦
4.
x-1
x+2
x-3
-3
x-4
2
x
5
(x - 4)
− + − + −2
x
6.
B
2.
H
3.
D
4.
F
x +4
A
B
⎡5 7⎤
⎢
⎣2 3⎦
5.
1.
4
2x - 1
−
+−
2
5.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Page 35
6
-4
−
+−
2.
3.
Mid-Chapter Test
3
-2
−
+−
-9 ⎤
⎢
2.
(Lesson 6-4)
Page 34
⎡ 3 -2
1⎤
⎢
⎣ -5 1 -11 ⎦
1.
Quiz 3
Answers
Quiz 1
Quiz 4 (Lesson 6-5)
Quiz 2 (Lesson 6-3)
Page 34
Page 33
1.
(-4, 2)
1.
2.
A
(-2, -2, -2)
2.
3.
4.
(-16, 7)
max. at (3, 4) of 25;
min. at (0, 0) of 0
max. at (0, 0) of 0;
min. at (12, 0)
3. of -24
(1, 0, -3)
4a + 6p = 9.80
5a. 3a + 9p = 10.95;
5b.
$1.25, $0.80
0
6.
⎡
5.
Chapter 6
C
40 small and
4a. 6 large
$2670
4b.
A15
3⎤
-3
⎣ 3⎦
⎢ 7.
Glencoe Precalculus
Chapter 6 Assessment Answer Key
Vocabulary Test
Page 36
Form 1
Page 37
1.
2.
1.
Page 38
C
8.
G
9.
C
10.
H
11.
C
12.
J
13.
A
14.
G
15.
B
J
square system
2. Gaussian elimination
3.
D
3. inverse matrix
4. optimization
6. Cramer’s Rule
partial fraction
7. decomposition
4.
F
5.
B
8. Sample answer: the
multiplicative
identity for a
square matrix
6.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. invertible
F
9. Sample answer:
the function in a
linear programming
problem that is to
be optimized
Chapter 6
7.
⎡ 2 -3⎤
⎢
⎦
⎣
8
-5
B:
B
A16
Glencoe Precalculus
Chapter 6 Assessment Answer Key
Page 40
8.
1.
2.
G
1.
9.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
12.
4.
6.
7.
C
14.
5.
H
D
Chapter 6
6.
15.
B
B:
⎡1 0 ⎤
⎢
⎣0 1 ⎦
7.
A17
9.
C
10.
F
11.
D
12.
F
13.
C
14.
G
15.
B
H
C
G
J
D
J
J
13.
5.
3.
A
8.
F
J
B
Page 42
C
B
2.
11.
4.
F
A
10.
3.
Form 2B
Page 41
B
G
A
⎡1 0⎤
⎢
0 1⎦
⎣
B:
Glencoe Precalculus
Answers
Form 2A
Page 39
Chapter 6 Assessment Answer Key
Form 2C
Page 43
⎡ -4
⎣ 3
1.
Page 44
1 12 ⎤
-2 -14 ⎦
11.
$3.25
12.
±3
2. Sample answer:
⎡1
⎣1
-3 5 ⎤
0 -2 ⎦
(-1, 2, 0)
3.
-2
7
−
+−
x-5
13.
x+3
3
5
-2
−
+−
+−
2
14. x + 1
x
-72
6.
⎡ 4 -48 ⎤
⎣ -5 53 ⎦
7.
does not exist
8.
⎡ -1 6 ⎤
⎣ 3 -4 ⎦
9.
10.
Chapter 6
1
−
10
x +3
1
−
10
−+ −
or
100 - x
1
1
−
+ −
10x
10(100 - x)
16. x
17.
16
18.
12
19.
0, (0, 0)
20.
$10.80
B:
BC
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
(x - 4)
6
2x - 4
−
+−
2
15.
hooks: $1.10
bobbers: $1.25
4. sinkers: $0.90
x-4
(1, 6, -1)
(2, 3)
A18
Glencoe Precalculus
Chapter 6 Assessment Answer Key
Form 2D
Page 45
2.
⎡ 5 -2 -3 ⎤
⎣ 4 7 -11 ⎦
Sample answer:
⎡ 1 -3 5 ⎤
⎣ 0 1 -2 ⎦
3.
(2, -3, 1)
11.
$1.75
12.
±3
Answers
1.
Page 46
-2
4
+−
−
x-3
x
13.
3
-5
2
−
+−
+−
2
x+1
x+3
-4
−
+−
x-2
2x2 + 1
14. x
15.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
sports: $3.00
news: $4.50
4. educational: $5.00
5.
3
6.
⎡ 26 -4 ⎤
⎣ 49 -6 ⎦
7.
⎡1
1⎤
⎣ 5.5 6 ⎦
8.
⎡ -3 4 ⎤
⎣ -1 5 ⎦
9.
(4, 1, -7)
10.
(2, -3)
Chapter 6
x
1
−
11
16.
1
−
11
−+−
or
121 - x
x
1
1
−+ −
11x
11(121 - x)
17.
12
18.
24
19.
-15, (15, 0)
20.
$420
B:
C
A19
Glencoe Precalculus
Chapter 6 Assessment Answer Key
Form 3
Page 47
1.
Page 48
⎡ 3 -2 5 ⎤
⎣ 4 -1 -7 ⎦
11.
$1.10
12.
2, 3
2. Sample answer:
⎡1
⎣0
0 -2 ⎤
0 0⎦
x+4
x-6
2x - 5
13.
3
-5
6
−+−+−
x+2
(x + 2)2
14. x
-1
4
+−
4+−
x
+
1
x
15.
7
+−
−
2
(-3, 4, -5)
3.
1
−
1
−
13
13
−
or
+−
5.
-42
6.
16. x
169 - x
1
1
−
+ −
13x
13(169 - x)
17.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4.
tandem: $34
child: $19
adult: $26
30
x = -2, y = 4
⎡ 0.2 0.3 0.1 ⎤
-0.2 -0.8 0.4 7. ⎣ 0.2 -0.7 0.1 ⎦
8.
⎡ -3 1 ⎤
⎣ 0 -2 ⎦
9.
(-1, 1, -1, 1)
10.
(4, -3)
Chapter 6
14
min of 0 at (0, 0);
max of 68 for
every point on
4x + y = 34 where
6≤x≤8
19.
18.
20.
B:
A20
$2500
ce - bf
−
ae - bd
Glencoe Precalculus
Chapter 6 Assessment Answer Key
⎡0.40 0.25 0.15 0.35⎤
1a. 0.25 0.35 0.40 0.35
⎣0.35 0.40 0.45 0.30⎦
⎢
⎡
⎢2000
⎢3500
1b.
⎢3000
⎢
⎣2500
2500
4000
4200
2000
1500
2000
1800
1000
3a. The objective function for this situation
is f(E, S) = 30,000E + 25,000S. The
vertices of the feasibility region are
(0, 0), (6, 0), (6, 4), and (0, 10).
⎤
⎦
f(0, 0) = $0
f(6, 0) = $180,000
f(6, 4) = $280,000
f(0, 10) = $250,000
The point (6, 4) maximizes the builder’s
profit at $280,000, so the builder should
construct 6 Executive houses and 4
Suburban houses.
⎡0.40 0.25 0.15 0.35⎤
1c. 0.25 0.35 0.40 0.35 ×
⎣0.35 0.40 0.45 0.30⎦
⎢
⎡
⎢2000
⎢3500
⎢3000
⎢
⎣2500
2500
4000
4200
2000
⎤
1500
2000
=
1800
1000⎦
⎡3000 3330 1720⎤
3800 4405 2145
⎣4200 4965 2435⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎢
3b. Sample answer: He should max out the
number of Executive houses built, which
would be 6 houses, and then build
Suburban houses to complete his
construction for the year. This would
maximize his profit.
4405 people visited Big Mountain Park
in the afternoon while 4200 people
visited Lazy River Park in the morning.
More people visited Big Mountain Park.
⎡ 3 -1 0⎤ ⎡x⎤ ⎡-5⎤
2a. -1 0 2 · y = 3
⎣ 0 1 -1⎦ ⎣z ⎦ ⎣ 1⎦
⎢
⎢ ⎢ ⎡0.4 0.2 0.4⎤
2b. 0.2 0.6 1.2
⎣0.2 0.6 0.2⎦
⎢
⎡-1⎤
⎡0.4 0.2 0.4⎤ ⎡-5⎤
2c. 0.2 0.6 1.2 · 3 =
2
⎣ 1⎦
⎣0.2 0.6 0.2⎦ ⎣ 1⎦
⎢
Chapter 6
⎢ ⎢ A21
Glencoe Precalculus
Answers
Page 49, Extended-Response Test
Sample Answers
Chapter 6 Assessment Answer Key
Standardized Test Practice
Page 50
1.
A
B
C
F
G
H
J
3.
A
B
C
D
F
G
H
F
G
H
J
11.
A
B
C
D
12.
F
G
H
J
13.
A
B
C
D
14.
F
G
H
J
15.
A
B
C
D
16.
F
G
H
J
17.
A
B
C
D
J
A
B
C
D
6.
F
G
H
J
7.
A
B
C
D
8.
F
G
H
J
9.
A
B
C
D
A22
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
Chapter 6
10.
D
2.
4.
Page 51
Glencoe Precalculus
Chapter 6 Assessment Answer Key
Standardized Test Practice
(continued)
Answers
Page 52
-3π
3π
x=−
,x=−
2
18.
2
2
-4
−
+−
3x - 2
x
19.
3
1
±−
, ±1, ± −
, ±3
20.
21.
2
2
(x - 2)3
x
ln −
4
π π 3π 5π
−
, −, −, −
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
22. 3
2
2
3
1 π
, −, 2
3, 2π, −
23.
2π 2
24a.
0, 0
24b.
x=1
24c.
y=3
between -3 and -2
25a.
rel. max. at x ≈ -2
and x ≈ 0; rel. min.
at x ≈ -1 and x ≈ 1
25b.
Chapter 6
A23
Glencoe Precalculus