# Algebra

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**Susie** on
.

Tara’s website, Garden Edibles, specializes

in the sale of herbs and flowers for colorful meals and garnishes. Tara

sells packets of nasturtium seeds for $0.95 each and packets of Johnny-jumpup

seeds for $1.43 each. She decides to offer a 16-packet spring-garden combination,

combining packets of both types of seeds at $1.10 per packet. How

many packets of each type of seed should be put in her garden mix?

1. Familiarize. To familiarize ourselves with the problem situation, we

make a guess and do some calculations. The total number of packets of

seed is 16. Let’s try 12 packets of nasturtiums and 4 packets of Johnnyjump-

ups.

The sum of the number of packets is , or 16.

The value of these seed packets is found by multiplying the cost per

packet by the number of packets and adding:

, or $17.12.

The desired cost is $1.10 per packet. If we multiply $1.10 by 16, we get

, or $17.60. This does not agree with $17.12, but these calculations

give us a basis for understanding how to translate.

We let the number of packets of nasturtium seeds and

the number of packets of Johnny-jump-up seeds. Next, we organize the

information in a table, as follows.

a b

16$1.10

$0.9512 $1.434

12 4

$ 0.95

$ 1.10

$ 1.43

575

8.4 Solving Applied Problems:

Two Equations

a b 16

17.60

0.95a 1.43b

NASTURTIUM

JOHNNYJUMP-

UP SPRING

Number

of Packets a b 16

Price

per Packet $0.95 $1.43 $1.10

Value

of Packets 0.95a 1.43b or 17.60

16 1.10,

ISBN:0-536-47742-6

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.

Copyright ©2007 by Pearson Education, Inc.

2. Translate. The total number of packets is 16, so we have one equation:

.

The value of the nasturtium seeds is 0.95a and the value of the Johnnyjump-

up seeds is 1.43b. These amounts are in dollars. Since the total

value is to be , or $17.60, we have

.

We can multiply by 100 on both sides of this equation in order to clear the

decimals. Thus we have the translation, a system of equations:

, (1)

. (2)

3. Solve. We decide to use substitution, although elimination could be

used as we did in Example 1. When equation (1) is solved for b, we

get . Substituting for b in equation (2) and solving

gives us

Substituting

Using the distributive law

Subtracting 2288 and collecting

like terms

.

We have . Substituting this value in the equation , we

obtain , or 5.

4. Check. We check in a manner similar to our guess in the Familiarize

step. The total number of packets is , or 16. The value of the packet

mixture is

, or $17.60.

Thus the numbers of packets check.

5. State. The spring garden mixture can be made by combining 11 packets

of nasturtium seeds with 5 packets of Johnny-jump-up seeds.

Do Exercise 2.

EXAMPLE 3 Student Loans. Jed’s student loans totaled $16,200. Part was

a Perkins loan made at 5% interest and the rest was a Stafford loan made at 4%

interest. After one year, Jed’s loans accumulated $715 in interest. What was the

amount of each loan?

1. Familiarize. Listing the given information in a table will help. The

columns in the table come from the formula for simple interest: .

We let the number of dollars in the Perkins loan and the number

of dollars in the Stafford loan.

x y

I Prt

$0.9511 $1.435

11 5

b 16 11

a 11 b 16 a

a 11

48a 528

95a 2288 143a 1760

95a 14316 a 1760

b 16 a 16 a

95a 143b 1760

a b 16

0.95a 1.43b 17.60

16$1.10

a b 16

2. Blending Coffees. The Coffee

Counter charges $9.00 per

pound for Kenyan French Roast

coffee and $8.00 per pound for

Sumatran coffee. How much of

each type should be used to

make a 20-lb blend that sells

for $8.40 per pound?

Answer on page A-36

Sumatran

Coffee

$8.00 lb

Kenyan French

Roast

$9.00 lb

HOUSE

BLEND

$8.40 lb

576

CHAPTER 8: Systems of Equations

x y 16,200

0.05x 0.04y 715

PERKINS

LOAN

STAFFORD

LOAN TOTAL

Principal x y $16,200

Rate of

Interest 5% 4%

Time 1 yr 1 yr

Interest 0.05x 0.04y $715

ISBN:0-536-47742-6

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.

Copyright ©2007 by Pearson Education, Inc.

2. Translate. The total of the amounts of the loans is found in the first row

of the table. This gives us one equation:

.

Look at the last row of the table. The interest totals $715. This gives us a

second equation:

, or .

After we multiply on both sides to clear the decimals, we have

.

3. Solve. Using either elimination or substitution, we solve the resulting

system:

,

.

We find that and .

4. Check. The sum is , or $16,200. The interest from $6700

at 5% for one year is , or $335. The interest from $9500 at 4% for

one year is , or $380. The total interest is , or $715.

The numbers check in the problem.

5. State. The Perkins loan was for $6700 and the Stafford loan was for

$9500.

Do Exercise 3.

EXAMPLE 4 Mixing Fertilizers. Yardbird Gardening carries two kinds of

fertilizer containing nitrogen and water. “Gently Green” is 5% nitrogen and

“Sun Saver” is 15% nitrogen. Yardbird Gardening needs to combine the two

types of solution to make 90 L of a solution that is 12% nitrogen. How much

of each brand should be used?

1. Familiarize. We first make a drawing and a guess to become familiar

with the problem.

We choose two numbers that total 90 L—say, 40 L of Gently Green

and 50 L of Sun Saver—for the amounts of each fertilizer. Will the

resulting mixture have the correct percentage of nitrogen? To find out,

we multiply as follows:

and .

Thus the total amount of nitrogen in the mixture is 2 L 7.5 L, or 9.5 L.

5%40 L 2 L of nitrogen 15%50 L 7.5 L of nitrogen

5

5% nitrogen

Gently

Green

Sun

Saver

g liters s liters 90 liters

1 15% nitrogen 12% nitrogen

4%$9500 $335 $380

5%$6700

$6700 $9500

x 6700 y 9500

5x 4y 71,500

x y 16,200

5x 4y 71,500

5%x 4%y 715 0.05x 0.04y 715

x y 16,200

3. Client Investments. Kaufman

Financial Corporation makes

investments for corporate

clients. It makes an investment

of $3700 for one year at simple

interest, yielding $297. Part of

the money is invested at 7%

and the rest at 9%. How much

was invested at each rate?

Do the Familiarize and

Translate steps by completing

the following table. Let the

number of dollars invested at

7% and the number of

dollars invested at 9%.

Answer on page A-36

y

x

577

8.4 Solving Applied Problems:

Two Equations

FIRST

INVESTMENT

SECOND

INVESTMENT TOTAL

Principal, P x $3700

Rate of

Interest, r 9%

Time, t 1 yr 1 yr

Interest, I 0.07x $297

x 3700

0.07x 297

ISBN:0-536-47742-6

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.

Copyright ©2007 by Pearson Education, Inc.

The final mixture of 90 L is supposed to be 12% nitrogen. Now

.

Since 9.5 L and 10.8 L are not the same, our guess is incorrect. But these

calculations help us to become familiar with the problem and to make

the translation.

We let the number of liters of Gently Green and the number

of liters of Sun Saver.

The information can be organized in a table, as follows.

2. Translate. If we add g and s in the first row, we get 90, and this gives us

one equation:

.

If we add the amounts of nitrogen listed in the third row, we get 10.8, and

this gives us another equation:

, or .

After clearing the decimals, we have the following system:

(1)

. (2)

3. Solve. We solve the system using elimination. We multiply equation (1)

by 5 and add the result to equation (2):

Multiplying equation (1) by 5

Adding

; Dividing by 10

Substituting in equation (1) of the system

g 27. Solving for g

g 63 90

s 63

10s 630

5g 15s 1080

5g 5s 450

5g 15s 1080

g s 90,

5%g 15%s 10.8 0.05g 0.15s 10.8

g s 90

g s

12%90 L 10.8 L

578

CHAPTER 8: Systems of Equations

g s 90

0.05g 0.15s 10.8

GENTLY

GREEN

SUN

SAVER MIXTURE

Number

of Liters g s 90

Percent

of Nitrogen 5% 15% 12%

Amount

of Nitrogen 0.05g 0.15s or 10.8 liters

0.12 90,

ISBN:0-536-47742-6

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.

Copyright ©2007 by Pearson Education, Inc.

4. Check. Remember that g is the number of liters of Gently Green, with

5% nitrogen, and s is the number of liters of Sun Saver, with 15% nitrogen.

Total number of liters of mixture:

Amount of nitrogen:

Percentage of nitrogen in mixture:

The numbers check in the original problem.

5. State. Yardbird Gardening should mix 27 L of Gently Green and 63 L of

Sun Saver.

Do Exercise 4.

Motion Problems

When a problem deals with speed, distance, and time, we can expect to use

the following motion formula.

THE MOTION FORMULA

TIPS FOR SOLVING MOTION PROBLEMS

1. Draw a diagram using an arrow or arrows to represent distance

and the direction of each object in motion.

2. Organize the information in a table or chart.

3. Look for as many things as you can that are the same, so you can

write equations.

EXAMPLE 5 Auto Travel. Your brother leaves on a trip, forgetting his suitcase.

You know that he normally drives at a speed of 55 mph. You do not

discover the suitcase until 1 hr after he has left. If you follow him at a speed

of 65 mph, how long will it take you to catch up with him?

1. Familiarize. We first make a drawing. From the drawing, we see that

when you catch up with your brother, the distances from home are the

same. We let the distance, in miles. If we let the time, in hours, for

you to catch your brother, then the time traveled by your brother

at a slower speed.

55 mph

t + 1 hours d miles

65 mph

t hours d miles

Cars

meet

here

Your car

Your brother's car

t 1

d t

d rt

Distance Rate or speed Time

10.8

90

0.12 12%

5%27 15%63 1.35 9.45 10.8 L

g s 27 63 90

4. Mixing Cleaning Solutions.

King’s Service Station uses two

kinds of cleaning solution

containing acid and water.

“Attack” is 2% acid and “Blast”

is 6% acid. They want to mix

the two to get 60 qt of a

solution that is 5% acid. How

many quarts of each should

they use?

Do the Familiarize and

Translate steps by completing

the following table. Let the

number of quarts of Attack

and the number of quarts

of Blast.

Review examples 2, 3, and 4 in section 8.4 of the text. How does the author determine what the first equation should be? What about the second equation? How are these examples similar? How are they different? Find a problem in the text that is similar to examples 2, 3, and 4. Post the problem for your classmates to solve.

Consider responding to your classmates by asking clarifying questions or by expanding a classmate’s response. Also, help students solve the problem you posted by providing feedback or hints if necessary. You may also want to provide an explanation for your solution after a sufficient number of students have replied.