Stewart J. College Algebra: Concepts and Contexts

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584 CHAPTER 7

■

Systems of Equations and Data in Categories

To find the complete solution of the system, we begin by solving for y in terms

of , using the new second equation.

Equation 2

Multiply by

Solve for y

Then we solve for x in terms of , using the first equation.

Substitute into Equation 1

Simplify

Solve for x

To describe the complete solution, we let t represent any real number. The solution is

We can also write the solution as the ordered triple .

■ NOW TRY EXERCISE 27 ■

In the solution of Example 4 the variable t is called a parameter. To get a spe-

cific solution, we give a specific value to the parameter t. For instance, if we set

, we get

Thus is a solution of the system. Here are some other solutions of the sys-

tem obtained by substituting other values for the parameter t.

1- 6, 6, 22

z = 2

y = 212 2+ 2 = 6

x =-3122=-6

t = 2

1- 3t, 2t + 2, t2

z = t

y = 2t + 2

x =-3t

x =-3z

x + 3z - 2 =-2

y = 2z + 2 x - 12z + 22+ 5z =-2

y = 2z + 2

y - 2z = 2

3 y - 6z = 6

Parameter t Solution ( )ⴚ3t, 2t ⴙ 2, t

- 1

13, 0, - 12

10, 2, 0 2

1- 9, 8, 32

1- 30, 22, 102

You should check that each of these points satisfies the original equations.

There are infinitely many choices for the parameter t, so the system has infinitely

many solutions.

■ Modeling with Linear Systems

Linear systems are used to model situations that involve several varying quantities.

In the next example we consider an application of linear systems to finance.

SECTION 7.2

■

Systems of Linear Equations in Several Variables 585

example

Modeling a Financial Problem Using a Linear System

Jason receives an inheritance of $50,000. His financial advisor suggests that he in-

vest this in three mutual funds: a money-market fund, a blue-chip stock fund, and a

high-tech stock fund. The advisor estimates that the money-market fund will return

5% over the next year, the blue-chip fund 9%, and the high-tech fund 16%. Jason

wants a total first-year return of $4000. To avoid excessive risk, he decides to invest

three times as much in the money-market fund as in the high-tech stock fund. How

much should he invest in each fund?

Solution

Let

We convert each fact given in the problem into an equation.

Multiplying the second equation by 100 and rewriting the third gives the following

system, which we solve using Gaussian elimination.

Interchanging Equations 2 and 3, we get:

Now that the system is in triangular form, we use back-substitution to find that

, , and . This means that Jason should investz = 10,000y = 10,000x = 30,000

x + y + z = 50,000

y + 4z = 50,000

z = 10,000

x + y + z = 50,000

z = 10,000

y + 4z = 50,000

x + y + z = 50,000

- 5z =-50,000

- y - 4z =-50,000

x + y + z = 50,000

4y + 11z = 150,000

- y

- 4z =-50,000

x + y + z = 50,000

5x + 9y + 16z = 400,000

x - 3z = 0

x + y + z = 50,000

0.05x + 0.09y + 0.16z = 4000

x = 3z

z = amount invested in the high-tech stock fund

y = amount invested in the blue-chip stock fund

x = amount invested in the money-market fund

Total amount invested is $50,000

Total investment return is $4000

Money-market amount = 3 * high-tech amount

Equation 1

Equation 3 + 1- 12* Equation 1 = new Equation 3

Equation 2 + 1- 52* Equation 1 = new Equation 2

Equation 1

Equation 3

Equation 2 + 4 * Equation 3 = new Equation 2

Equation 1

1- 12* Equation 3

2* Equation 2

Equation 1

Equation 2

Equation 3

Equation 1

Equation 2

Equation 3

586 CHAPTER 7

■

Systems of Equations and Data in Categories

$30,000 in the money-market fund

$10,000 in the blue-chip stock fund

$10,000 in the high-tech stock fund

■ NOW TRY EXERCISE 31 ■

7.2 Exercises

Fundamentals

1. The following system of equations is in triangular form.

(a) From the third equation we know that

_______.

(b) Replacing by 3 in the second equation, we get y ⫽

_______.

_______ and y by _______ in the first equation, we find that

x ⫽

_______.

(d) The solution of the system is (

____, ____, ____).

2. Consider the following system of linear equations.

(a) If we add 2 times the first equation to the second equation, the second equation

becomes

______________ ⫽ _______.

(b) To eliminate x from the third equation, we add

_______ times the first equation to

the third equation. The third equation becomes

______________ ⫽ _______.

Think About It

3–4 ■ Consider the following system of linear equations.

3. Explain how the first two equations tell us that the system cannot have a solution.

4. Show that (1, 1, 0) is a solution of the second and third equations. Why is (1, 1, 0) not a

solution of the system?

5–8

■ Use back-substitution to solve the triangular system.

5. 6.

7. 8. c

x - 2y + 3z = 10

2y - z = 2

3z = 12

x + 2y + z = 7

- y + 3z = 9

2z = 6

x + y - 3z = 8

y - 3z = 5

z =-1

x - 2y + 4z = 3

y + 2z = 7

z = 2

x + y + z = 1

+ y + z = 2

2x + y - z = 3

x - y + z = 2

- x + 2y + z =-3

3x + y - 2z = 2

z =

x + 2y - z = 6

y - z =-2

z = 3

CONCEPTS

SKILLS

SECTION 7.2

■

Systems of Linear Equations in Several Variables 587

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30. c

2x + 4y - z = 3

x + 2y + 4z = 6

x + 2y - 2z = 0

x + 3y - 2z = 0

2x + 4z = 4

4x + 6y = 4

x - 2y + z = 3

2x - 5y + 6z = 7

2x - 3y - 2z = 5

x + y - z = 0

x + 2y - 3z =-3

x + 3y - 4z =-3

x - 2y - 3z = 5

2x + y - z = 5

4x - 3y - 7z = 5

2x + 3y - z = 1

x + 2y = 3

x + 3y + z = 4

- x + 2y + 5z = 4

x - 2z = 0

4x - 2y - 11z = 2

x + 2y - z =

2x + 3y - 4z =-3

3x + 6y - 3z = 4

2y + z = 3

5x + 4y + 3z =-1

x - 3y =-2

y - 2z = 0

2x + 3y = 2

- x - 2y + z =-1

2x + y - z =-8

- x + y + z = 3

- 2x

+ 4z = 18

2x + 4y - z = 2

x + 2y - 3z =-4

3x - y + z = 1

9–12

■ Perform an operation on the given system that eliminates the indicated variable.

Write the new equivalent system.

9. 10.

Eliminate the x-term from Eliminate the x-term from

the second equation. the second equation.

11. 12.

Eliminate the x-term from Eliminate the y-term from

the third equation. the third equation.

13–16

■ Find the solution of the linear system.

13. 14.

15. 16.

17–30 ■ Solve the system, or show that it has no solution. If the system has infinitely many

solutions, express them in the ordered triple form given in Example 4.

17. 18. c

x - y + 2z = 2

3x + y + 5z = 8

2x - y - 2z =-7

x - 4z = 1

2x - y - 6z = 4

2x + 3y - 2z = 8

x + y + z = 0

- x + 2y + 5z = 3

3x - y = 6

x + y + z =

x + 3y + 3z = 10

2x + y - z = 3

x - y + z = 0

y + 2z =-2

x + y - z = 2

x - y - z = 4

2y + z =-1

- x + y - 2z = 5

x - 4y + z = 3

y - 3z = 10

3y - 8z = 24

2x - y + 3z = 2

x + 2y - z = 4

- 4x + 5y + z = 10

x + y - 3z = 3

- 2x + 3y + z = 2

x - y + 2z = 0

x - 2y - z = 4

x - y + 3z = 0

2x + y + z = 0

588 CHAPTER 7

■

Systems of Equations and Data in Categories

31–32

■ An investor has $100,000 to invest in three types of bonds: short-term,

intermediate-term, and long-term. How much should she invest in each type to

satisfy the given conditions?

31. Finance Short-term bonds pay 4% annually, intermediate-term bonds pay 5%, and

long-term bonds pay 6%. The investor wishes to realize a total annual income of 5.1%,

with equal amounts invested in short- and intermediate-term bonds.

32. Finance Short-term bonds pay 4% annually, intermediate-term bonds pay 6%, and

long-term bonds pay 8%. The investor wishes to have a total return of $6700 on her

investment, with equal amounts invested in intermediate- and long-term bonds.

33. Agriculture A farmer has 1200 acres of land on which he grows corn, wheat, and

soybeans. It costs $45 per acre to grow corn, $60 to grow wheat, and $50 to grow

soybeans. Because of market demand the farmer will grow twice as many acres of

wheat as corn. He has allocated $63,750 for the cost of growing his crops. How many

acres of each crop should he plant?

CONTEXTS

36. Diet Program Nicole started a new diet that requires each meal to have 460 calories,

6 grams of fiber, and 11 grams of fat. The table shows the fiber, fat, and calorie content

of one serving of each of three breakfast foods. How many servings of each food should

Nicole eat to follow her diet?

Food Fiber Fat Calories

Toast 2 1 100

Cottage cheese 0 5 120

Fruit 2 0 60

34. Gas Station A gas station sells three types of gas: Regular for $3.00 a gallon,

Performance Plus for $3.20 a gallon, and Premium for $3.30 a gallon. On a particular

day 6500 gallons of gas were sold for a total of $20,050. Three times as many gallons of

Regular as Premium gas were sold. How many gallons of each type of gas were sold

that day?

35. Nutrition A biologist is performing an experiment on the effects of various

combinations of vitamins. She wishes to feed each of her laboratory rabbits a diet that

contains exactly 9 mg of niacin, 14 mg of thiamin, and 32 mg of riboflavin. She has

available three different types of commercial rabbit pellets; their vitamin content (per

ounce) is given in the table. How many ounces of each type of food should each rabbit

be given daily to satisfy the experiment requirements?

Vitamin (mg) Type A Type B Type C

Niacin 2 3 1

Thiamin 3 1 3

Riboflavin 8 5 7

SECTION 7.2

■

Systems of Linear Equations in Several Variables 589

37. Juice Blends The Juice Company offers three kinds of smoothies: Midnight Mango,

Tropical Torrent, and Pineapple Power. Each smoothie contains the amounts of juices

shown in the table. On a particular day the Juice Company used 820 ounces of mango

juice, 690 ounces of pineapple juice, and 450 ounces of orange juice. How many

smoothies of each kind were sold that day?

38. Appliance Manufacturing Kitchen Korner produces refrigerators, dishwashers, and

stoves at three different factories. The table gives the number of each product produced

at each factory per day. Kitchen Korner receives an order for 110 refrigerators, 150

dishwashers, and 114 ovens. How many days should each plant be scheduled to fill

this order?

Day Stock A Stock B Stock C

Monday $10 $25 $29

Tuesday $12 $20 $32

Wednesday $16 $15 $32

Appliance Factory A Factory B Factory C

Refrigerators 8 10 14

Dishwashers 16 12 10

Stoves 10 18 6

Smoothie

Mango

Juice (oz)

Pineapple

Juice (oz)

Orange

Juice (oz)

Midnight Mango 8 3 3

Tropical Torrent 6 5 3

Pineapple Power 2 8 4

39. Stock Portfolio An investor owns three stocks: A, B, and C. The closing prices of

the stocks on three successive trading days are given in the table. Despite the volatility

in the stock prices, the total value of the investor’s stocks remained unchanged at

$74,000 at the end of each of these three days. How many shares of each stock does the

investor own?

16 ⍀

4 V

8 ⍀

5 V

4 ⍀

I⁄

I¤

I‹

40. Electricity By using Kirchhoff’s Laws, it can be shown that the currents , , and

that pass through the three branches of the circuit in the figure satisfy the given linear

system. Solve the system to find , , and .

= 0

16I

- 8I

= 4

+ 4I

= 5

590 CHAPTER 7

■

Systems of Equations and Data in Categories

Here are some examples of matrices.

Matrix Dimension

2 rows by 3 columns

1 row by 4 columns

In the first matrix the (1, 2) entry is 3, the (2, 2) entry is 4, and the (2, 3) entry is .- 1

1 * 436 - 5014

2 * 3c

13 0

24- 1

7.3 Using Matrices to Solve Systems of Linear Equations

■

Matrices

■

The Augmented Matrix of a Linear System

■

Elementary Row Operations

■

Row-Echelon Form

■

Reduced Row-Echelon Form

■

Inconsistent and Dependent Systems

IN THIS SECTION… we study a method for solving systems of linear equations by first

expressing the system as a matrix. Matrix notation allows us to use a graphing calculator to

solve the system.

■ Matrices

A matrix is simply a rectangular array of numbers. Matrices* are used to organize

data into categories that correspond to the rows and columns of the matrix (see

Section 7.4). In this section we represent a linear system by a matrix, called the aug-

mented matrix of the system. We begin by defining the various elements that make

up a matrix.

Definition of a Matrix

■ The Augmented Matrix of a Linear System

An matrix is a rectangular array of numbers with m rows and n

columns. We say that the matrix has dimension . The numbers in the

array are the entries of the matrix. The (i, j) entry is the number in the ith

row and jth column.

m * n

*The plural of matrix is matrices.

We can write a system of linear equations as a matrix, called the augmented matrix of

the system, by writing only the coefficients and constants that appear in the equations.

SECTION 7.3

■

Using Matrices to Solve Systems of Linear Equations 591

Note that performing any of these operations on the augmented matrix of a sys-

tem does not change its solution. In the next example we compare the two ways of

writing systems of linear equations.

Linear system Augmented matrix

The augmented matrix contains the same information as the system but in a simpler

form.

2 - 15

147

2x - y = 5

x + 4y = 7

example

Finding the Augmented Matrix of a Linear System

Write the augmented matrix of the system of equations.

Solution

First we write the linear system with the variables lined up in columns.

The augmented matrix is the matrix whose entries are the coefficients and the con-

stants in this system.

■ NOW TRY EXERCISE 1 ■

6 - 2 - 14

1 0 31

0 7 15

6x - 2y - z = 4

x + 3z = 1

7y + z = 5

6x - 2y - z = 4

x + 3z = 1

7y + z = 5

■ Elementary Row Operations

The operations that we used in Section 7.2 to solve a linear system correspond to op-

erations on the rows of the augmented matrix of the system. For example, adding a

multiple of one equation to another corresponds to adding a multiple of one row of

the augmented matrix to another.

Elementary Row Operations

1. Add a multiple of one row to another.

2. Multiply a row by a nonzero constant.

3. Interchange two rows.

Equation 1

Equation 2

592 CHAPTER 7

■

Systems of Equations and Data in Categories

example

Using Elementary Row Operations to Solve a Linear System

Solve the system of linear equations.

Solution

Our goal is to eliminate the x-term from the second equation and the x- and y-terms

from the third equation. For comparison we write both the system of equations and

its augmented matrix.

Now we use back-substitution to find that , , and . The solution is

(2, 7, 3).

■ NOW TRY EXERCISE 11 ■

z = 3y = 7x = 2

1 - 134

01- 21

0013

§c

x - y + 3z = 4

y - 2z = 1

z = 3

1 - 134

0013

01- 21

§c

x - y + 3z = 4

z = 3

y - 2z = 1

1 - 134

03- 56

01- 21

§c

x - y + 3z = 4

3y - 5z = 6

y - 2z = 1

1 - 134

03- 56

02- 42

§c

x - y + 3z = 4

3y - 5z = 6

2y - 4z = 2

1 - 13 4

12- 210

3 - 1514

§c

x - y + 3z = 4

x + 2y - 2z = 10

3x - y + 5z = 14

x - y + 3z = 4

x + 2y - 2z = 10

3x - y + 5z = 14

Back-substitution is explained in

Section 7.2.

■ Row-Echelon Form

In Example 2 we solved a system by using its augmented matrix and applying elementary

row operations to arrive at a matrix in a certain form. This form is called row-echelon form.

Row-Echelon Form of a Matrix

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

1. The first nonzero number in each row (reading from left to right) is 1.

This is called the leading entry.

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

row immediately above it.

3. All rows consisting entirely of zeros are at the bottom of the matrix.

Add 1- 32* Equation 1 to Equation 3

Add 1- 12* Equation 1 to Equation 2

Add 1- 32* Row 1 to Row 3

Add 1- 12* Row 1 to Row 2

Multiply Equation 3 by

Multiply Row 3 by

Add 1- 32* Equation 3 to Equation 2 Add 1- 32* Row 3 to Row 2

Interchange Equations 2 and 3 Interchange Rows 2 and 3

SECTION 7.3

■

Using Matrices to Solve Systems of Linear Equations 593

In the following matrices the first one is not in row-echelon form. The second

one is in row-echelon form

1 32- 3

001 6

000 0

§£

0 1 - 25

1 034

00 1 8

To put an augmented matrix into row-echelon form, we can use a graphing calcula-

tor to perform the required elementary row operations. (On the TI-83 the command

is ref.)

Leading 1’s do not

shift to the right in

successive rows.

Leading 1’s shift

to the right in

successive rows.

example

Solving a System Using Row-Echelon Form

Solve the system of linear equations by using row-echelon form.

Solution

Augmented matrix. We first write the augmented matrix of the system.

Row-echelon form. Using the ref command on a graphing calculator we put

the matrix into row-echelon form:

Calculator output Row-echelon form

12- 11

01 2- 3

00 1- 2

48- 44

38 5- 11

- 21 12 - 17

4x + 8y - 4z = 4

3x + 8y + 5z =-11

- 2x + y + 12z =-17

ref([A])

[[1 2 -1 1 ]

[0 1 2 -3]

[0 0 1 -2]]

We now have an equivalent matrix in row-echelon form, and the corresponding

system of equations is

Back-substitute. Now we use back-substitution to find that , , and

. The solution is .

■ NOW TRY EXERCISE 15 ■

1- 3, 1, - 22z =-2

y = 1x =-3

x + 2y - z = 1

y + 2z =-3

z =-2