Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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College Algebra G&M—
SECTION 7.1 Solving Linear Systems Using Matrices and Row Operations
KEY CONCEPTS
•
A matrix is a rectangular arrangement of numbers. An matrix has m rows and n columns.
•
An augmented matrix is derived from a system of linear equations by augmenting the coefficient matrix (formed
by the variable coefficients) with the matrix of constants.
•
One matrix method for solving systems of equations is by triangularizing the augmented matrix.
•
An inconsistent system with no solutions will yield a contradictory statement such as . A dependent system
with infinitely many solutions will yield an identity statement such as .
EXERCISES
1. Write an example of the following matrices:
a. b. c. , in triangular form
Solve by triangularizing the augmented matrix. If the system is linearly dependent, state the answer using a parameter.
Use a calculator for Exercise 5.
2. 3. 4. 5.
SECTION 7.2 The Algebra of Matrices
KEY CONCEPTS
•
The entries of a matrix are denoted a
ij
, where i gives the row and j gives the column of its location.
•
Two matrices A and B of equal size (or order) are equal if corresponding entries are equal.
•
The sum or difference of two matrices of equal order is found by combining corresponding entries:
•
The identity matrix for addition is an matrix whose entries are all zeroes.
•
To perform scalar multiplication, take the product of the constant with each entry in the matrix, forming a new
matrix of like size. For matrix A: .
•
Matrix multiplication is performed as . For an matrix and an
matrix , AB is possible if . The result will be an matrix , where p
ij
is the product of
the ith row of A with the jth column of B.
•
When technology is used to perform operations on matrices, carefully enter each matrix into the calculator. Then
double check that each entry is correct and appraise the results to see if they are reasonable.
EXERCISES
Compute the operations indicated below (if possible), using the following matrices.
6. 7. 8. 9. 8A 10. BA
11. 12. 13. BC 14. 15. CD⫺4DD ⫺ CC ⫹ D
C ⫺ BB ⫺ AA ⫹ B
D ⫽ £
2 ⫺30
0.5 1 ⫺1
4 0.1 5
§C ⫽ £
⫺134
5 ⫺20
6 ⫺32
§B ⫽ c
⫺76
1 ⫺2
dA ⫽ c
⫺1
4
⫺3
4
⫺1
8
⫺7
8
d
P ⫽ 3p
ij
4m ⫻ tn ⫽ sB ⫽ 3b
ij
4
s ⫻ tA ⫽ 3a
ij
4m ⫻ nrow entry ⫻ column entry
kA ⫽ 3ka
ij
4
m ⫻ n
A ⫹ B ⫽ 3a
ij
⫹ b
ij
4
μ
2w ⫹ x ⫹ 2y ⫺ 3z ⫽⫺19
w ⫺ 2x ⫺ y ⫹ 4z ⫽ 15
x ⫹ 2y ⫺ z ⫽ 1
3w ⫺ 2x ⫺ 5z ⫽⫺60
•
2x ⫺ y ⫹ 2z ⫽⫺1
x ⫹ 2y ⫹ 2z ⫽⫺3
3x ⫺ 4y ⫹ 2z ⫽ 1
•
x ⫺ 2y ⫹ 2z ⫽ 7
2x ⫹ 2y ⫺ z ⫽ 5
3x ⫺ y ⫹ z ⫽ 6
e
x ⫺ 2y ⫽ 6
4x ⫺ 3y ⫽ 4
3 ⫻ 33 ⫻ 22 ⫻ 3
0 ⫽ 0
0 ⫽ 1
m ⫻ n
SUMMARY AND CONCEPT REVIEW
700 CHAPTER 7 Matrices and Matrix Applications 7–64
cob19545_ch07_700-706.qxd 10/21/10 12:16 PM Page 700
7–65 Summary and Concept Review 701
College Algebra G&M—
SECTION 7.3 Solving Linear Systems Using Matrix Equations
KEY CONCEPTS
•
The identity matrix for multiplication I, has 1’s on the main diagonal and 0’s for all other entries. For any
matrix A, the identity matrix is also an matrix I
n
, where .
•
For an (square) matrix A, the inverse matrix for multiplication is a matrix B such that . For
matrix A the inverse is denoted . Only square matrices have inverses.
•
Any system of equations can be written as a matrix equation and solved (if a unique solution exists) using
an inverse matrix. The system
.
•
Every square matrix has a real number associated with it, called its determinant. For matrix
.
•
If the determinant of a matrix is zero, the matrix is said to be singular or noninvertible. If the coefficient matrix of
a matrix equation is noninvertible, the system is either inconsistent or dependent.
EXERCISES
Complete Exercises 16 through 18 using the following matrices:
16. Exactly one of the matrices given is singular. Compute each determinant to identify it.
17. Show that What can you conclude about matrix A?
18. Show that What can you conclude about matrix C?
Use a graphing calculator to complete Exercises 19 through 21, using the matrices given:
19. Exactly one of the matrices is singular. Determine which one.
20. Compute the products FG and GF. What can you conclude about matrix G?
21. Verify that and . What can you conclude?
Solve manually using a matrix equation. Solve using a matrix equation and your calculator.
22.
23.
SECTION 7.4 Applications of Matrices and Determinants:
Cramer’s Rule, Partial Fractions, and More
KEY CONCEPTS
•
Cramer’s rule uses a ratio of determinants to solve systems of equations (if solutions exist).
•
The determinant of the matrix is
•
To compute the value of and larger determinants, a calculator is generally used.
•
Determinants can be used to find the area of a triangle in the plane if the vertices of the triangle are known, and as
a test to see if three points are collinear.
•
A system of equations can be used to write a rational expression as a sum of its partial fractions.
3 ⫻ 3
a
11
a
22
⫺ a
21
a
12
.`
a
11
a
12
a
21
a
22
`2 ⫻ 2
•
0.5x ⫺ 2.2y ⫹ 3z ⫽⫺8
⫺0.6x ⫺ y ⫹ 2z ⫽⫺7.2
x ⫹ 1.5y ⫺ 0.2z ⫽ 2.6
e
2x ⫺ 5y ⫽ 14
⫺3y ⫹ 4x ⫽⫺14
EF ⫽ FEEG ⫽ GE
G ⫽ £
⫺10⫺1
01 0
21 1
§F ⫽ £
1 ⫺11
010
⫺21⫺1
§E ⫽ £
1 ⫺23
⫺21⫺5
⫺1 ⫺1 ⫺2
§
BC ⫽ CB ⫽ I.
AB ⫽ BA ⫽ B.
D ⫽ c
10 ⫺6
⫺15 9
dC ⫽ c
2 ⫺1
31
dB ⫽ c
0.2 0.2
⫺0.6 0.4
dA ⫽ c
10
01
d
A ⫽ c
a
11
a
12
a
21
a
22
d, det1A2⫽ a
11
a
22
⫺ a
21
a
12
2 ⫻ 2
e
2x ⫹ 3y ⫽ 7
x ⫺ 4y ⫽⫺2
is written as c
23
1 ⫺4
dc
x
y
d⫽ c
7
⫺2
d
n ⫻ n
A
⫺1
AB ⫽ BA ⫽ I
n
n ⫻ n
AI
n
⫽ I
n
A ⫽ An ⫻ n
n ⫻ n
cob19545_ch07_700-706.qxd 10/21/10 12:17 PM Page 701
College Algebra G&M—
702 CHAPTER 7 Matrices and Matrix Applications 7–66
EXERCISES
Solve using Cramer’s rule. Use a graphing calculator for Exercise 26.
24. 25. 26.
27. Find the area of a triangle whose vertices have the coordinates (6, 1), , and
28. Find the partial fraction decomposition for
SECTION 7.5 Matrix Applications and Technology Use
KEY CONCEPTS
•
In studies involving numerous constraints, many variables, and/or large amounts of data, matrix methods have a
distinct advantage over other solution methods.
•
Square matrices with integer coefficients rarely have an inverse that also has integer coefficients. However these
can be carefully constructed.
•
A matrix [A] can be used to encode a written message, with the matrix [A]
⫺1
being used to decode it.
29. J.P. Sailing and Co. hand builds two types of classic sailing boats made entirely of wood. Both
the 13-foot Laser class dinghy and 25-foot Bermuda sloop yacht are constructed primarily with
three types of wood: Douglas fir, Brazilian jequitiba, and, of course, Asian teak. Each dingy is
built using 120 ft
2
of fir, 132 ft
2
of jequitiba, and 50 ft
2
of teak. The construction process for the
yacht uses 270 ft
2
of fir, 260 ft
2
of jequitiba, and 108 ft
2
of teak. In their first year of production,
J.P. Sailing built 11 dinghies and 5 yachts. For the following 4 yr, their dingy production
increased by 3 per year, while the yacht production increased by 2 per year. Use your calculator
and a matrix equation to determine how much of each type of wood they used during each of
their first five years of production as well as the 5-yr totals.
30. In addition to being one of the world’s first celebrity chefs, Marie-Antoine Carême (1784–1833)
was a master pastry chef. Had he used matrices to encode his favorite culinary math secret,
perhaps his message would have been:
If Carême used the cipher on page 696 (Example 4) and the matrix
for the encoding process, decode the message to find the
unconventional culinary secret that is actually familiar to students
of mathematics.
11, 10, 25, ⫺25, ⫺6, ⫺59, 33, 13, 76, 34, 43, 69, ⫺3, 0, ⫺6
7x
2
⫺ 5x ⫹ 17
x
3
⫺ 2x
2
⫹ 3x ⫺ 6
.
1⫺6, 22.1⫺1, ⫺62
•
2x ⫹ y ⫽⫺2
⫺x ⫹ y ⫹ 5z ⫽ 12
3x ⫺ 2y ⫹ z ⫽⫺8
•
2x ⫹ y ⫺ z ⫽⫺1
x ⫺ 2y ⫹ z ⫽ 5
3x ⫺ y ⫹ 2z ⫽ 8
e
5x ⫹ 6y ⫽ 8
10x ⫺ 2y ⫽⫺9
PRACTICE TEST
Solve each system by triangularizing the augmented
matrix and using back-substitution.
1.
2. •
3x ⫺ y ⫹ 5z ⫽ 1
3x ⫹ y ⫹ 4z ⫽ 4
x ⫹ y ⫹ z ⫽
7
3
e
3x ⫹ 8y ⫽⫺5
x ⫹ 10y ⫽ 2
3.
4. Given matrices A and B, compute:
a. b. c. AB d. e.
B ⫽ c
33
⫺3 ⫺5
dA ⫽ c
⫺3 ⫺2
54
d
冟
A
冟
A
⫺1
2
5
BA ⫺ B
•
4x ⫺ 5y ⫺ 6z ⫽ 5
2x ⫺ 3y ⫹ 3z ⫽ 0
x ⫹ 2y ⫺ 3z ⫽ 5
A ⫽ £
123
131
247
§
cob19545_ch07_700-706.qxd 11/29/10 11:39 AM Page 702
7–67 Practice Test 703
College Algebra G&M—
5. Given matrices C and D, use a calculator to find:
a. b. c. DC d. e.
6. Use matrices to find three different solutions of the
dependent system:
7. Solve using Cramer’s rule:
8. Solve using a calculator and Cramer’s rule:
9. Solve using a matrix equation and your calculator:
10. Solve using a matrix equation and your calculator:
11. Use the equality of matrices to write and solve a
system that gives values of x, y, and z so that .
and
12. Given matrix X is a solution to for the
matrix A given, find matrix B.
13. Use matrices and a graphing calculator to determine
which three of the following four points are collinear:
( , 4), (1, 3), (2, 1), (4, )⫺1⫺1
A ⫽ £
1 ⫺22
2 ⫺63
34⫺1
§X ⫽ £
⫺1
⫺3
2
2
§
AX ⫽ B
B ⫽ c
z ⫺ 13
2y ⫹ 5 y ⫹ 8
d
A ⫽ c
2x ⫹ y 3
x ⫹ z 3x ⫹ 2z
d
A ⫽ B
•
x ⫺ 2y ⫹ 2z ⫽ 7
2x ⫹ 2y ⫺ z ⫽ 5
3x ⫺ y ⫹ z ⫽ 6
e
2x ⫺ 5y ⫽ 11
4x ⫹ 7y ⫽ 4
•
2x ⫹ 3y ⫹ z ⫽ 3
x ⫺ 2y ⫺ z ⫽ 4
x ⫺ y ⫺ 2z ⫽⫺1
e
2x ⫺ 3y ⫽ 2
x ⫺ 6y ⫽⫺2
•
2x ⫺ y ⫹ z ⫽ 4
3x ⫺ 2y ⫹ 4z ⫽ 9
x ⫺ 2y ⫹ 8z ⫽ 11
D ⫽ £
0.5 0.1 0.2
⫺0.1 0.1 0
0.3 0.4 0.8
§
C ⫽ £
0.5 0 0.2
0.4 ⫺0.5 0
0.1 ⫺0.4 ⫺0.1
§
冟
D
冟
D
⫺1
⫺0.6DC ⫺ D
14. A farmer plants a triangular field with wheat. The
first vertex of the triangular field is 1 mi east and
1 mi north of his house. The second vertex is 3 mi
east and 1 mi south of his house. The third vertex is
1 mi west and 2 mi south of his house. What is the
area of the field?
15. For and given, find r
and s.
Create a system of equations to model each exercise,
then solve using any matrix method.
16. Dr. Brown and Dr. Stamper graduate from medical
school with $155,000 worth of student loans. Due to
her state’s tuition reimbursement plan, Dr. Brown
owes one fourth of what Dr. Stamper owes. How
much does each doctor owe?
17. Justin is rehabbing two old houses simultaneously.
He calculates that last week he spent 23 hr working
on these houses. If he spent 8 more hours on one of
the houses, how many hours did he spend on each
house?
18. In his first month as assistant principal of
Washington High School, Mr. Johnson gave out
20 detentions. They were either for 1 day, 2 days,
or 5 days. He recorded a total of 38 days of
detention served. He also noted that there were
twice as many 2-day detentions as 5-day
detentions. How many of each type of detention
did Mr. Johnson give out?
19. The city of Cherrywood has approved a $1,800,000
plan to renovate its historic commercial district. The
money will be coming from three separate sources.
The first is a federal program that charges a low
2% interest annually. The second is a municipal
bond offering that will cost 5% annually. The third is
a standard loan from a neighborhood bank, but it
will cost 8.5% annually. In the first year, the city will
not make any repayment on these loans and will
accrue $94,500 more debt. The federal program and
bank loan together are responsible for $29,500 of
this interest. How much money was originally
provided by each source?
20. Decompose the expression into partial fractions:
4x
2
⫺ 4x ⫹ 3
x
3
⫺ 27
A
2
⫽ c
10 ⫺2
⫺37
dA ⫽ c
r 2
3 s
d
cob19545_ch07_700-706.qxd 11/29/10 11:40 AM Page 703
College Algebra G&M—
704 CHAPTER 7 Matrices and Matrix Applications 7–68
STRENGTHENING CORE SKILLS
Augmented Matrices and Matrix Inverses
The formula for finding the inverse of a matrix has its roots in the more general method of computing the inverse
of an matrix. This involves augmenting a square matrix M with its corresponding identity I
n
on the right (forming
an matrix), and using row operations to transform M into the identity. In some sense, as the original matrix is
transformed, the “identity part” keeps track of the operations we used to convert M and we can use the results to “get
back home,” so to speak. We’ll illustrate with the matrix from Section 7.3, Example 2B, where we found that
was the inverse matrix for We begin by augmenting with the identity matrix.
As you can see, the identity is automatically transformed into the inverse matrix when this method is applied.
Performing similar row operations on the general matrix results in the formula given earlier.c
ab
cd
d
c
6510
01⫺13
d⫺5R2 ⫹ R1 S R1c
60 6 ⫺15
01⫺13
d
R1
6
S R1c
10 1 ⫺2.5
01⫺13
d
c
6510
2201
d⫺3R2 ⫹ R1 S R2c
6510
0 ⫺11⫺3
d⫺1R2 S R2c
6510
01⫺13
d
2 ⫻ 2c
65
22
dc
65
22
d.c
1 ⫺2.5
⫺13
d
2 ⫻ 2
n ⫻ 2n
n ⫻ n
2 ⫻ 2
CALCULATOR EXPLORATION AND DISCOVERY
Cramer’s Rule
In Section 7.4, we saw that one interesting application of matrices is Cramer’s rule. You may have noticed that when
technology is used with Cramer’s rule, the chances of making an error are fairly high, as we need to input the entries
for numerous matrices. However, as we mentioned in the chapter introduction, one of the advantages of matrices is
that they are easily programmable, and we can actually write a very simple program that will make Cramer’s rule a
more efficient method.
To begin, press the key, and then the right arrow twice and to create a name for our program. At the
prompt, we’ll enter CRAMER2. As we write the program, note that the needed commands (ClrHome, Disp, Pause,
Prompt, Stop) are all located in the submenus of the key, and the sign is found under the TEST menu,
accessed using the keys (the arrows “S” are used to indicate the key.
The following program takes the coefficients and constants of a 2 ⫻ 2 linear system, and returns the ordered pair
solution in the form of x ⫽ h and y ⫽ k (for constants h and k). Even with minimal programming experience, reading
through the program will help you identify that Cramer’s rule is being used.
STO
MATH2nd
=
PRGM
ENTER
PRGM
ClrHome
Disp ''2
2 SYSTEMS''
Pause
Disp ''AXBY C''
Disp ''DXEY F''
Disp ''''
Disp ''ENTER THE VALUES''
Disp ''FOR A,B,C,D,E,F''
Disp ''''
Prompt A,B,C,D,E,F
Exercise 1: Use the program to check the answers to Exercises 1 and 7 of the Practice Test.
Exercise 2: Create systems of your own that are (a) consistent, (b) inconsistent, and (c) dependent. Then verify
results using the program.
Exercise 3: Use the box on page 681 of Section 7.4 to write a similar program for systems. Call the program
CRAMER3, and repeat parts (a), (b), and (c) from Exercise 2.
3 ⫻ 3
2 ⫻ 2
(CE–BF)/(AE–BD)SX
(AF–DC)/(AE–BD)SY
ClrHome
Disp ''THE SOLUTION IS''
Disp ''''
Disp ''X''
Disp X
Disp ''Y''
Disp Y
Stop
cob19545_ch07_700-706.qxd 1/8/11 3:24 PM Page 704
7–69 Cumulative Review Chapters R–7 705
College Algebra G&M—
As you might imagine, attempting this on a general matrix is problematic at best, and instead we simply apply
the augmented matrix method to find for the matrix shown in blue.
To verify, we show ✓ ( A ⫽ I also checks).
Exercise 1: Use the preceding inverse and a matrix equation to solve the system
•
2x ⫹ y ⫽⫺2
⫺x ⫹ 3y ⫺ 2z ⫽⫺15
3x ⫺ y ⫹ 2z ⫽ 9
.
A
⫺1
£
210
⫺13⫺2
3 ⫺12
§ £
1 ⫺0.5 ⫺0.5
⫺11 1
⫺2 1.25 1.75
§⫽ £
100
010
001
§AA
⫺1
⫽ I:
£
100 1 ⫺0.5 ⫺0.5
010⫺11 1
001⫺2 1.25 1.75
§.
R1
⫺14
S R1
R2
7
S R2
£
⫺1400⫺14 7 7
070⫺77 7
001⫺2 1.25 1.75
§
4R3 ⫹ R2 S R2
4R3 ⫹ R1 S R1
£
⫺14 0 ⫺4 ⫺62 0
07⫺41 2 0
001⫺2 1.25 1.75
§
R3
8
S R3£
⫺14 0 ⫺4 ⫺620
07⫺41 20
008⫺16 10 14
§
R2 ⫺ 7R1 S R1
5R2 ⫹ 7R3 S R3
£
210100
07⫺4120
0 ⫺54⫺302
§
R1 ⫹ 2R2 S R2
⫺3R1 ⫹ 2R3 S R3
£
210100
⫺13⫺2 010
3 ⫺12001
§
3 ⫻ 3A
⫺1
3 ⫻ 3
→
→
→
CUMULATIVE REVIEW CHAPTERS R–7
1. Perform the operations indicated.
a.
b.
c.
d.
2. Solve for h.
Solve the following equations. Verify solutions using a
graphing calculator.
3.
4.
5.
6.
7. Find an equation of the line that passes through the
points (2, ) and ( , 5).⫺3⫺2
12x ⫹ 11
⫺ x ⫽ 6
9x
2
⫹ 1 ⫽ 6x
x ⫹ 6
x ⫹ 2
⫺
1
x
⫽
12
x
2
⫹ 2x
2x ⫺ 413x ⫹ 12⫽ 5 ⫺ 4x
S ⫽ 2rh ⫹ 2r
2
i
49
8 ⫺ i
2 ⫹ i
15 ⫺ 3i2
2
13 ⫺ 2i212 ⫹ i2
8. Find an equation of the line
perpendicular to the line
shown, with the same
y-intercept.
9. Given
and ,
find
a. f(3) b. g(23)
c. d.
Graph the following by using transformations of the
parent function.
10. 11.
12. 13.
Solve the following inequalities. Express your answer
in interval notation.
14. 15. x
2
⫹ 5 7 6x21x ⫺ 22⫹ 3 ⱕ 8
y ⫽ 2⫺x
⫺ 2y ⫽⫺1x ⫹ 22
3
y ⫽
1
2
冟
x
冟
⫺ 3y ⫽ 1x ⫺ 32
2
g1⫺32f 1⫺22
g1x2⫽
0
4 ⫺ 1x ⫹ 2
0
f 1x2⫽ 3 ⫺ 4x ⫺ x
2
54321⫺5⫺4⫺3⫺2⫺1
⫺1
⫺2
⫺3
⫺4
⫺5
2
3
4
1
5
y
x
cob19545_ch07_700-706.qxd 12/1/10 6:24 PM Page 705
College Algebra G&M—
706 CHAPTER 7 Matrices and Matrix Applications 7–70
16. A long-distance calling card advertises one of the
lowest rates available. It claims to charge per
minute for a long distance call, after a
connection fee. The cost of a call is modeled by the
ceiling function , where m is
the length of the call in minutes and c is the cost of
the call in cents. Graph this function and explain
why this card may not be as inexpensive as
advertised.
Name interval(s) where the following functions are
increasing, decreasing, or constant. Write answers
using interval notation.
17. 18.
19. Compute the quotient .
20. Use the remainder theorem to find for
.
21. Find a cubic polynomial with real coefficients
having the roots . Assume a
lead coefficient of 1.
22. State the end-behavior and y-intercept of the
following functions.
a.
b.
c.
23. Solve the system of equations using any method.
24. Solve the system of inequalities by graphing. Verify
your solution using a graphing calculator.
25. Simon Legré, a notorious French criminal, is
planning to embezzle money from the Prefect of
Police, and believes he can remove 12% of their
e
y 7 x
2
⫺ 1
x
2
⫹ y
2
ⱕ 13
e
5x ⫹ 6y ⫽⫺30
⫺3x ⫹ 2y ⫽ 4
h1x2⫽ 10 ⫺ 2x
5
g1x2⫽⫺5x
6
⫹ 2x
3
⫺ 3x ⫹ 7
f 1x2⫽ 3x
3
⫺ 16
x ⫽⫺2 and x ⫽ 3 ⫺ i
P1x2⫽ x
3
⫹ 4x
2
⫺ 15
P1⫺32
14y
2
⫺ 3y ⫹ 102⫼ 12y ⫹ 12
108642⫺10⫺8⫺6⫺4⫺2
⫺2
⫺4
⫺6
⫺8
⫺10
4
6
8
2
10
y
x
654321⫺4⫺3⫺2⫺1
⫺1
⫺2
⫺3
⫺4
⫺5
2
3
4
1
5
y
x
g1x2⫽
1
1x ⫹ 32
⫹ 3y ⫽ f 1x2
c1m2⫽ 9l
1
3
mm⫹ 15
15¢
3¢
$5,000,000 budget without being detected. He
plans to use his ill-gotten gain to buy a legitimate
business, and purchase a condominium and a
delivery truck for the business. The business will
cost $20,000 more than twice the delivery truck,
and the condominium will cost $200,000 more than
the truck and business combined. How much will
each cost?
For Exercises 26 through 30, use of a graphing
calculator to solve.
26. Solve the system using a graphing calculator. Round
your answers to two decimal places.
27. Use a calculator to compute the determinant of each
matrix. If the determinant is zero, write singular
matrix.
a.
b.
28. In Cuivre River State Park, there are two dominant
squirrel species: gray-tailed and red-tailed. The
function
models the population of gray-tailed squirrels from
1990 to 2010, while
models the red-tailed population during this
same 20-yr period. These two functions give the
respective number of squirrels (in hundreds) living
in this park during year t, where t ⫽ 0 corresponds
to 1990. Using integer-valued years and the TABLE
feature of your calculator, determine when these two
populations were closest in size. What was the
difference in populations that year?
29. Determine the intervals over which the function
is increasing.
30. Find all solutions (if they exist) using a graphing
calculator.
B
x
2
x ⫹ 1
⫽ e
x
2
x⫹ 1
f 1x2⫽ 4x
3
⫺ 4x
2
⫺ 32x ⫹ 17
1
|
⫹ 3
r1t2⫽ ln
|
⫺2t
3
⫹ 5t
2
⫺ 3t ⫹
g1t2⫽ ln
|
t
5
⫺ 3t
4
⫹ 2t
3
⫺ 5t
2
⫹ t ⫺ 3
|
≥
01⫺15
4 ⫺11⫺4
6 ⫺24 9
⫺97 6 3
¥
≥
1 ⫺53 2
032⫺3
1011
1057
¥
e
x ⫹ y
2
⫽ 10
3x
2
⫹ 2y
2
⫽ 30
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College Algebra G&M—
Analytic Geometry
and the Conic
Sections
CHAPTER OUTLINE
8.1 A Brief Introduction to Analytical Geometry 708
8.2 The Circle and the Ellipse 715
8.3 The Hyperbola 731
8.4 The Analytic Parabola;
More on Nonlinear Systems 744
CHAPTER CONNECTIONS
One of the most breathtaking, daredevil stunts
performed at air shows is the
power dive
. In
some cases, as the pilot dives toward the
ground and pulls out of the dive just in time,
the plane flys along a path that can be
modeled by a hyperpola (the third member of
the family of conic sections). If we consider a
given point on the ground and under the
grandstand as the origin (0, 0), we can use the
equation that models the hyperbolic path to
determine its minimum altitude as it passes
over the stands. This application appears as
Exercise 81 in Section 8.3.
Check out these other real-world connections:
䊳
Designing an Elliptical Garden with Fountains
(Section 8.2, Exercise 66)
䊳
The Design of a Lithotripter for Treating
Kidney Stones with Shockwaves
(Section 8.2, Exercise 68)
䊳
Locating a Ship Using Radar
(Section 8.3, Exercise 85)
䊳
Parabolic Shape of a Solar Furnace
(Section 8.4, Exercise 94)
707
cob19545_ch08_707-715.qxd 10/25/10 2:43 PM Page 707
8.1 A Brief Introduction to Analytical Geometry
Generally speaking, analytical geometry is a study of geometry using the tools of algebra
and a coordinate system. These tools include the midpoint and distance formulas; the
algebra of parallel, perpendicular, and intersecting lines; and other tools that help establish
geometric concepts. In this section, we’ll use these tools to verify certain relationships,
then use these relationships to introduce a family of curves known as the conic sections.
A. Verifying Relationships from Plane Geometry
For the most part, the algebraic tools used in this study were introduced in previous
chapters. As the midpoint and distance formulas play a central role, they are restated
here for convenience.
Algebraic Tools Used in Analytical Geometry
Given two points and in the xy-plane.
Midpoint Formula Distance Formula
The midpoint of line segment P
1
P
2
is The distance from P
1
to P
2
is
These formulas can be used to verify the conclusion of many theorems from Euclidean
geometry, while providing important links to an understanding of the conic sections.
EXAMPLE 1
䊳
Verifying a Theorem from Basic Geometry
A theorem from basic geometry states: The
midpoint of the hypotenuse of a right triangle
is an equal distance from all three vertices.
Verify this statement for the right triangle
formed by , and (4, 4).
Solution
䊳
After the plotting points and drawing a
triangle, we note the hypotenuse has endpoints
and (4, 4), with midpoint
Using the distance formula to find the distance from (0, 1) to (4, 4) gives
From the definition of midpoint, (0, 1) is also 5 units from .
Checking the distance from (0, 1) to the vertex gives
The midpoint of the hypotenuse is an equal distance from all three vertices
(see the figure).
Now try Exercises 7 through 12
䊳
5
125
24
2
132
2
d 214 02
2
12 12
2
14, 22
14, 22
5
125
2142
2
132
2
d 214 02
2
14 12
2
a
4 142
2
,
4 122
2
b 10, 12.
14, 22
14, 22, 14, 22
d 21x
2
x
1
2
2
1y
2
y
1
2
2
1x, y2 a
x
1
x
2
2
,
y
1
y
2
2
b
P
2
1x
2
, y
2
2P
1
1x
1
, y
1
2
LEARNING OBJECTIVES
In Section 8.1 you will see
how we can:
A. Verify theorems from
basic geometry involving
the distance between
two points
B. Verify that points (x, y)
are an equal distance
from a given point and a
given line
C. Use the defining
characteristics of a conic
section to find its
equation
708 8–2
College Algebra G&M—
55
5
5
y
x
R (4, 4)
M
(0, 1)
(4, 2)(4, 2)
Q
P
cob19545_ch08_707-715.qxd 10/25/10 2:43 PM Page 708
8–3 Section 8.1 A Brief Introduction to Analytical Geometry 709
College Algebra G&M—
Recall from Section 1.1 that a circle is the set
of all points that are an equal distance (called the
radius) from a given point (called the center). If all
three vertices of a triangle lie on the circumference
of a circle, we say the circle circumscribes the tri-
angle. Based on our earlier work, it appears we
could also state the theorem in Example 1 as For
any circle in the xy-plane whose center (h, k) is the
midpoint of the hypotenuse L of a right triangle,
the circle defined by ,
circumscribes the triangle. The circle and trian-
gle from Example 1 illustrate this theorem in
Figure 8.1, where the equation of the circle is
. See Exercises 13 through 20.
B. The Distance between a Point and a Line
In a study of analytical geometry, we are also interested in the distance d between a
point and a line. This is always defined as the perpendicular distance, or the length of
a line segment perpendicular to the given line, with the given point and the point of in-
tersection as endpoints (see Figure 8.2).
EXAMPLE 2
䊳
Locating Points That Are an Equal Distance from a Given Point and Line
In Figure 8.3, the origin (0, 0) is seen to be an equal distance from the point
(0, 2) and the line . Show that the following points are also an equal
distance from (0, 2) and :
a. b. (4, 2) c. (8, 8)
Solution
䊳
Since the given line is horizontal, the perpendicular distance from the line to each
point can be found by vertically counting the units. It remains to show that this is
also the distance from the given point to (0, 2) (see Figure 8.4).
a. The distance from to is 2.5 units. The distance from to
(0, 2) is
✓
b. The distance from (4, 2) to
is 4 units. The distance
from (4, 2) to (0, 2) is
✓
c. The distance d
2
from (8, 8) to
is 10 units. The distance d
1
from (8, 8) to (0, 2) is
✓
Now try Exercises 23 through 26
䊳
10
1100
2182
2
162
2
d
1
210 82
2
12 82
2
y 2
4
116
2142
2
102
2
d 210 42
2
12 22
2
y 2
2.5
16.25
2122
2
11.52
2
d 210 22
2
12 0.52
2
12,
1
2
2y 212,
1
2
2
12,
1
2
2
y 2
y 2
1x 02
2
1y 12
2
1
10
2
2
2
1x h2
2
1y k2
2
1
L
2
2
2
y
x
55
5
5
(4, 4)
(0, 1)
(4, 2)
(4, 2)
Figure 8.1
A. You’ve just seen how
we can verify theorems from
basic geometry involving the
distance between two points
d
Figure 8.2
55
3
5
y
(0, 0)
(0, 2)
y 2
x
Figure 8.3
Figure 8.4
55
3
5
y
(0, 0)
(2, 0.5)
(4, 2)
(8, 8)
(0, 2)
y 2
x
d
1
d
2
d
1
d
2
B. You’ve just seen how
we can verify that points (x, y)
are an equal distance from a
given point and a given line
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