Stewart J. College Algebra: Concepts and Contexts

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624 CHAPTER 7

■

Systems of Equations and Data in Categories

example

Solving a System Using a Matrix Inverse

A pet store owner feeds his hamsters and gerbils different mixtures of three types of

rodent food: KayDee Food, Pet Pellets, and Rodent Chow. The protein, fat, and car-

bohydrates content (in mg) in one gram of each brand is given in the table below.

■

Hamsters need 340 mg of protein, 280 mg of fat, and 440 mg of carbohy-

drates each day.

■

Gerbils need 480 mg of protein, 360 mg of fat, and 680 mg of carbohydrates

each day.

The pet store owner wishes to feed his animals the correct amount of each brand to

satisfy their daily requirements exactly. How many grams of each food should the

storekeeper feed his hamsters and gerbils daily to satisfy their nutrient requirements?

KayDee Food Pet Pellets Rodent Chow

Protein (mg) 10 0 20

Fat (mg) 10 20 10

Carbohydrates (mg) 510 30

Solution

We let x, y, and be the respective amounts (in grams) of KayDee Food, Pet Pellets,

and Rodent Chow that the hamsters should eat and r, s, and t be the corresponding

amounts for the gerbils. Then we want to solve the matrix equations

Hamster Equation

Gerbil Equation

Let

Then we can write these matrix equations as

Hamster Equation

Gerbil Equation

AY = C

AX = B

A = £

10 0 20

10 20 10

51030

§B = £

340

280

440

§C = £

480

360

680

§X = £

§Y = £

10 0 20

10 20 10

51030

§£

§= £

480

360

680

10 0 20

10 20 10

51030

§£

§= £

340

280

440

■ Modeling with Matrix Equations

Suppose we need to solve several systems of equations with the same coefficient ma-

trix. Then expressing the systems as matrix equations provides an efficient way to

obtain the solutions, because we need to find the inverse of the coefficient matrix

only once. This procedure is particularly convenient if we use a graphing calculator

to perform the matrix operations, as in the next example.

SECTION 7.6

■

Matrix Equations: Solving a Linear System 625

Solving for X and Y, we have

Hamster Equation

Gerbil Equation

Using a graphing calculator, we find matrices X and Y.

Y = A

-1

X = A

-1

Thus each hamster should be fed 10 g of KayDee Food, 3 g of Pet Pellets, and 12 g

of Rodent Chow, and each gerbil should be fed 8 g of KayDee Food, 4 g of Pet

Pellets, and 20 g of Rodent Chow daily.

■ NOW TRY EXERCISE 27 ■

[A]

[B]

[[10]

[3 ]

[12]]

[A]

[C]

[[8 ]

[4 ]

[20]]

7.6 Exercises

Fundamentals

1. (a) The matrix is called an _______ matrix.

(b) If A is a matrix, then

_______ and _______.

_______ of A.

2. (a) Write the following system as a matrix equation .

System Matrix equation

(b) The inverse of A is

(d) The solution of the system is , .

3–6

■ Calculate the products AB and BA to verify that matrix B is the inverse of matrix A.

4. A = c

2 - 3

4 - 7

B = c

2 - 1

A = c

B = c

2 - 1

- 7 4

y = ______x = ______

d= c dc d= c d

X = A

-1

X = A

-1

= c

c dc d= c de

5x + 3y = 4

3x + 2y = 3

X = B

AX = B

AB = I2 * 2

I * A =A * I =2 * 2

I = c

1 0

0 1

CONCEPTS

SKILLS

626 CHAPTER 7

■

Systems of Equations and Data in Categories

7–18

■ Use a calculator that can perform matrix operations to find the inverse of the

matrix, if it exists.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19–26

■ Solve the system of linear equations by expressing the system as a matrix

equation and using the inverse of the coefficient matrix. Use the inverses you

found in Exercises 7–10, and 13, 14, 17, and 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. Sales Commissions An encyclopedia salesperson works for a company that offers

three different grades of bindings for its encyclopedias: standard, deluxe, and leather.

For each set that she sells, she earns a commission based on the set’s binding grade.

One week she sells one standard set, one deluxe set, and two leather sets and makes

$675 in commission. The next week she sells two standard sets, one deluxe set, and one

leather set for a $600 commission. The third week she sells one standard set, two deluxe

sets, and one leather set, earning $625 in commission.

x + 2y ⫹ 3w = 0

y + z + w = 1

y + w = 2

x + 2y + 2w = 3

- 2y + 2z = 12

3x + y + 3z =-2

x - 2y + 3z = 8

5x + 7y + 4z = 1

3x - y + 3z = 1

6x + 7

y + 5z = 1

2x + 4y + z = 7

- x + y - z = 0

x + 4y =-2

- 7x + 4y = 0

8x - 5y = 100

2x + 5y = 2

- 5x - 13y = 20

3x + 4y = 10

7x + 9y = 20

- 3x - 5y = 4

2x + 3y = 0

≥

1203

0111

0101

1202

¥£

0 - 22

3 13

1 - 23

210

114

212

§£

1 2 3

4 5 - 1

1 - 1 - 10

5 74

3 - 13

6 75

§£

2 4 1

- 11- 1

1 4 0

6 - 3

- 8 4

- 7 4

8 - 5

2 5

- 5 - 13

- 3 - 5

2 3

A = £

32 4

11- 6

21 12

§B = £

9 - 10 - 8

- 12 14 11

A = £

1 3 - 1

1 4 0

- 1 - 3 2

B = £

8 - 3 4

- 2 1 - 1

1 0 1

CONTEXTS

CHAPTER 7

■

Review 627

(a) Let x, y, and represent the commission she earns on standard, deluxe, and leather

sets, respectively. Translate the given information into a system of equations.

(b) Express the system of equations you found in part (a) as a matrix equation of the

form .

in part (b). How much commission does the salesperson earn on a set of

encyclopedias in each grade of binding?

28. Nutrition A nutritionist is studying the effects of the nutrients folic acid, choline, and

inositol. He has three types of food available, and each type contains the amounts of

these nutrients per ounce shown in the table.

AX = B

Type A Type B Type C

Folic acid (mg) 313

Choline (mg) 424

Inositol (mg) 324

Let A be the matrix

(a) Find the inverse of matrix A.

(b) How many ounces of each food should the nutritionist feed his laboratory rats if he

wants each rat’s daily diet to contain 10 mg of folic acid, 14 mg of choline, and

13 mg of inositol?

wants each rat’s daily diet to contain 9 mg of folic acid, 12 mg of choline, and

10 mg of inositol?

A = £

313

424

324

CHAPTER 7

REVIEW

CONCEPT CHECK

Make sure you understand each of the ideas and concepts that you learned in this chapter,

as detailed below section by section. If you need to review any of these ideas, reread the

appropriate section, paying special attention to the examples.

7.1 Systems of Linear Equations in Two Variables

A system of equations is a set of equations in which each equation involves the same

variables. A solution of a system is a set of values for the variables that makes each

equation true.

A system of two equations in two variables can be solved by the substitution

method or the elimination method. (The details of these methods are described on

pages 569 and 570.)

CHAPTER 7

628 CHAPTER 7

■

Systems of Equations and Data in Categories

The graph of a system of two linear equations in two variables is a pair of lines;

the solution of the system corresponds to the point of intersection of the lines. A two-

variable linear system has either:

■

One solution if the lines have different slopes.

■

No solution if the lines are different but have the same slope. (That is, the

lines are parallel.)

■

Infinitely many solutions if the lines are the same.

7.2 Systems of Linear Equations in Several Variables

To solve a linear system involving several equations and variables, we use Gaussian

elimination to put the system in triangular form and then use back-substitution

to get the solution. The following elementary row operations are used in the

Gaussian elimination process:

■

Add a nonzero multiple of one equation to another.

■

Multiply an equation by a nonzero constant.

■

Interchange the positions of two equations.

A system of linear equations can have one solution, no solution, or infinitely

many solutions. A system with no solution is said to be inconsistent, and a system

with infinitely many solutions is said to be dependent. If we apply Gaussian elimi-

nation to a system and arrive at a false equation, then the system is inconsistent. The

solution of a dependent system can be described by using one or more parameters.

7.3 Using Matrices to Solve Systems of Equations

A matrix is a rectangular array of numbers. For instance, here is a matrix with three

rows and four columns:

A matrix with m rows and n columns is said to have dimension ; the

above matrix has dimension . The numbers in the matrix are its entries; the

(i, j) entry is the number in row i and column j. So in the above matrix the (2, 3) en-

try is 1, and the (3, 2) entry is .

The augmented matrix of a system of linear equations is the matrix that is ob-

tained by writing the coefficients of the variables and the constants in each equation

in matrix form, omitting the symbols for the variables and the equal signs. For ex-

ample, the above matrix is the augmented matrix for the system

To solve a linear system, we apply elementary row operations to the augmented

matrix of the system, transforming it into an equivalent system in row-echelon form.

A matrix is in row-echelon form if it satisfies the following conditions:

■

The first nonzero number (called the leading entry) is 1.

6x - 2y + 4z = 0

- 5x + 7y + z =-3

9x - 4y +

z = 8

- 4

3 * 4

m * n

6 - 24 0

- 571- 3

9 - 4

CHAPTER 7

■

Review 629

■

The leading entry in each row is to the right of the leading entry in the row

above it.

■

All rows that consist entirely of zeros are at the bottom of the matrix.

If the matrix also satisfies the following condition, then it is in reduced row-

echelon form:

■

Every number above and below each leading entry is 0.

Graphing calculators have commands that put matrices into row-echelon and re-

duced row-echelon form (ref and rref on the TI-83 calculator).

If an augmented matrix in row-echelon form has a row equivalent to the equa-

tion , then the system is inconsistent; this means the system has no solutions.

If the augmented matrix in row-echelon form is not inconsistent but has fewer

nonzero rows than variables, then the system is dependent; this means that the sys-

tem has infinitely many solutions.

7.4 Matrices and Data in Categories

Matrices can be used to organize and analyze data about two different categorical

characteristics of a population. For instance, if you gather data about the eye color

(brown, blue, or green) and the hair color (black, brown, blond, or red) of the mem-

bers of your class, then the numbers can be conveniently presented as a ma-

trix in which the rows represent eye color and the columns represent hair color.

(For example, the (3, 4) entry would be the number of green-eyed redheads in the

class.)

To combine analogous data matrices for two different populations, we add the

matrices by adding corresponding entries.

7.5 Matrix Operations: Getting Information From Data

Suppose that A and B are two matrices of the same dimension and that c is a

real number.

■

The sum is the matrix whose entries are the sums of the

corresponding entries of A and B.

■

The scalar multiple cA is the matrix obtained by multiplying each

entry in A by the number c.

The product AB of two matrices A and B is defined if the number of columns

of A is the same as the number of rows of B. (The procedure for multiplying two ma-

trices is described on pages 612–615.) Matrix products can be used to extract infor-

mation from sets of categorical data.

7.6 Matrix Equations: Solving a Linear System

The identity matrix is the square matrix for which each main diagonal en-

try (from the top left corner to the bottom right corner) is a 1 and every other entry

is a 0. For instance, the identity matrix is

= £

100

010

001

3 * 3

n * nI

m * n

m * nA + B

m * n

3 * 4

0 = 1

When a matrix is multiplied by an identity matrix I of the appropriate dimen-

sion, it remains unchanged:

If A is a square matrix and if there exists an matrix such that

then we say that is the inverse of A.

A system of equations can be written as a matrix equation of the form .

To solve a matrix equation, we multiply both sides by :

REVIEW EXERCISES

1–4 ■ Graph each linear system, either by hand or by using a graphing device. Use the

graph to determine whether the system has one solution, no solution, or infinitely

many solutions. If there is exactly one solution, use the graph to find it.

1. 2.

3. 4.

5–8

■ Solve the linear system, or show that it has no solution. (Use either the elimination

or the substitution method.)

5. 6.

7. 8.

9–16

■ Use Gaussian elimination to find the complete solution of the system, or show that

no solution exists.

9. 10.

11. 12.

13. 14.

15. 16.

17–18

■ The augmented matrix of a linear system is given. Use a graphing calculator to

put the matrix into reduced row-echelon form, and then find the complete solution

of the system. (Assume that the variables in the system are x, y, and .)z

x - y + z = 0

3x + 2y - z = 6

x + 4y - 3z = 3

x - 2y + 3z =-2

2x - y + z = 2

2x - 7y + 11z =-9

x + 2y - z = 1

2x + 3y - 4z =-3

3x + 6y - 3z = 4

x - y + z =

x + y + 3z = 6

3x - y + 5z = 10

x - y + z = 2

x + y + 3z = 6

2y + 3z = 5

x + 2y + 2z = 6

x - y =-1

2x + y + 3z = 7

x - 2y + 3z = 1

x - 3y - z = 0

2x - 6z = 6

x + y + 2z =

2x + 5z = 12

x + 2y + 3z = 9

6x + 9y = 5

4x + 6y = 3

3x - y = 12

- x +

y =-4

2x + 5y = 9

- x + 3y = 1

2x + 3y = 7

x - 2y = 0

2x - 7y = 28

y =

x - 4

6x - 8y = 15

x + 2y =-4

y = 2x + 6

y =-x + 3

3x - y = 5

2x + y = 5

X = A

-1

AX = B

-1

A = AA

-1

= I

-1

n * nn * n

AI = AandIB = B

CHAPTER

SKILLS

630 CHAPTER 7

■

Systems of Equations and Data in Categories

17. 18.

19–30

■ Let A, B, C, D, E, and F be the given matrices. Carry out the indicated operations,

or explain why they cannot be performed.

19. 20. 21.

22. 23. FA 24. AF

25. BC 26. CB 27. BF

28. FC 29. 30.

31–32

■ Verify that the matrices A and B are inverses of each other by calculating the

products AB and BA.

31. 32.

33–36

■ Use a graphing calculator to find the inverse of the matrix.

33. 34.

35. 36.

37–40

■ Express the system of linear equations as a matrix equation. Then solve the matrix

equation by multiplying each side by the inverse of the coefficient matrix.

37. 38.

39. 40. c

2x + 3z = 5

x + y + 6z = 0

3x - y + z = 5

2x + y + 5z =

x + 2y + 2z =

x + 3z =

6x - 5y = 1

8x - 7y =-1

12x - 5y = 10

5x - 2y = 17

1 2 3

- 102

2 1 2

§£

110

101

011

7 - 15

5 - 11

A = £

2 - 13

2 - 21

0 11

B = £

- 1 1 2

1 - 1 - 1

§A = c

2 - 5

- 2 6

dB = c

F12C - D21C + D2E

5B - 2C

2C + 3DC - DA + B

D = £

1 4

0 - 1

2 0

 E = c

2 - 1

d F = £

4 0 2

- 110

7 5 0

A = 320- 14 B = c

1 2 4

- 210

d C = £

- 21

1 0 - 23

1 1 - 34

2 - 2 54

§£

1210

2348

1027

CHAPTER 7

■

Review Exercises 631

CONNECTING

THE CONCEPTS

These exercises test your understanding by combining ideas from several sections in a

single problem.

41. Solving a Linear System in Six Different Ways We have learned several different

methods for solving linear systems of equations. In this problem we compare these

methods by solving the following system in six different ways:

x - 3y = 2

2x - 5y = 6

632 CHAPTER 7

■

Systems of Equations and Data in Categories

Tomatoes Onions Zucchini

Saturday 25 16 30

Sunday 14 12 16

CONTEXTS

43. Children’s Ages Eleanor has two children: Kieran and Siobhan. Kieran is 4 years

older than Siobhan, and the sum of their ages is 22.

(a) Let x be Kieran’s age, and let y be Siobhan’s. Find a system of linear equations that

models the facts given about the children’s ages.

(b) Solve the system. How old is each child?

44. Interest on Bank Accounts Clarisse invests $60,000 in money-market accounts at

three different banks. Bank A pays 2% interest per year, Bank B pays 2.5%, and Bank C

pays 3%. She decides to invest twice as much in Bank B as in the other two banks. After

one year, Clarisse has earned $1575 in interest.

(a) Let x, y, and be the amounts that Clarisse invests in Banks A, B, and C,

respectively. Find a system of linear equations that models the facts given about the

amounts invested in each bank.

(b) Solve the system. How much did Clarisse invest in each bank?

45. A Fisherman’s Catch A commercial fisherman fishes for haddock, sea bass, and red

snapper. He is paid $1.25 a pound for haddock, $0.75 a pound for sea bass, and $2.00 a

pound for red snapper. Yesterday he caught 560 pounds of fish worth $575. The

haddock and red snapper together are worth $320. How many pounds of each fish did

he catch?

46. A Vegetable Stand Rhonna and Ivan grow tomatoes, onions, and zucchini in their

backyard and sell them at a roadside stand on Saturdays and Sundays. They price

tomatoes at $1.50 per pound, onions at $1.00 per pound, and zucchini at 50 cents per

pound. The following table shows the number of pounds of each type of produce that

they sold during the last weekend in July.

(a) Solve the system by graphing both lines on a graphing calculator and then using

the feature to find the intersection point of the lines.

(b) Solve the system using substitution.

(d) Find the augmented matrix of the system, and use Gaussian elimination on the

matrix to solve the system.

(e) Find the reduced row-echelon form of your augmented matrix in part (d), and use it

to solve the system.

(f) Write the system as a matrix equation of the form , and solve the system by

multiplying both sides of the equation by .

(g) Which of these six methods did you find simplest to use? Which was the most

complicated? Check that you got the same answer with all six methods.

42. Solving a Linear System in Three Different Ways Not all of the six methods

described in Exercise 41, parts (a)–(f ), work for a linear system that has three equations

in three unknowns.

(a) Which methods do not apply? Why?

(b) Solve the following system by the three methods from Exercise 41 (a)–(f) that do

work in solving such a system. Which method do you prefer?

x + 2y - z = 1

x - y + 2z = 2

x + 2y + z = 4

-1

AX = B

TRACE

CHAPTER 7

■

Review Exercises 633

(a) Let

Compare these matrices to the data given in the problem, and describe what the

matrix entries represent.

(b) Only one of the products AB or BA is defined. Calculate the product that is defined,

and describe what its entries represent.

47. Automatic Teller Machine An ATM at a bank in Qualicum Beach, British Columbia,

dispenses $20 and $50 bills. Jo Ann withdraws $600 from this machine and receives a

total of 18 bills. Let x be the number of $20 bills, and let y be the number of $50 bills

that she receives.

(a) Find a system of two linear equations in x and y that expresses the information

given in the problem.

(b) Write your linear system as a matrix equation of the form .

each type did Jo Ann receive?

-1

AX = B

A = c

25 16 30

14 12 16

andB = £

1.50

1.00

0.50