Chapter 4 Section 4. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Applications of Linear Systems Solve problems about unknown numbers

Chapter 4

Section 4

Objectives1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Applications of Linear Systems

Solve problems about unknown numbers.

Solve problems about quantities and their costs.

Solve problems about mixtures.

Solve problems about distance, rate (or speed), and time.

4.4

2

3

4


Applications of Linear Systems

Recall from Section 2.4 the six step method for solving applied problems. These slightly modified steps allow for two variables and two equations.

Solving an Applied Problem with Two VariablesStep 1: Read the problem carefully. What information is given? What

are you asked to find?

Step 2: Assign variables to represent the unknown values. Use a sketch, diagram, or table, as needed. Write down what each variable represents.

Step 3: Write two equations using both variables.

Step 5: State the answer. Label it appropriately. Does it seem reasonable?

Step 4: Solve the system of two equations.

Step 6: Check the answer in the words of the original problem.

Slide 4.4-3


Objective 1

Solve problems about unknown numbers.

Slide 4.4-4


In 2008, spending on sporting equipment and recreational transport totaled $51,879 million. Spending on recreational transport exceeded spending on sporting equipment by $2113 million. How much was spent on each? (Source: National Sporting Goods Association.)

Solution:

51,879x y 2113x y

2113( ) 51,879xx 2 2113 51,879x

2 49

2 2

,766x

2,6996y

24,883x

There was $24,883 million spent on sporting equipment and $26,996 spent on recreational transport.

24,883 2,113 y

Slide 4.4-5

EXAMPLE 1 Solving a Problem about Two Unknown Numbers

Let x = sales of sporting equipment in millions of dollars.Let y = sales of recreational transport in millions of dollars.


Objective 2

Solve problems about quantities and their costs.

Slide 4.4-6


There were 5 main floor tickets and 14 mezzanine tickets bought.

For a production of Wicked at the Pantages Theatre in Los Angeles, main floor tickets cost $96 and mid-priced mezzanine tickets cost $58. If a group of 18 people attended the show and spent a total of $1234 for their tickets, how many of each kind of ticket did they buy? (Source: www.ticketmaster.com)Solution:Let x = the number of main floor ticketsand y = the number of mezzanine tickets.

18x y 96 58 1234x y

104458 58x y

13y

5x

Slide 4.4-7

EXAMPLE 2 Solving a Problem about Quantities and Costs

Number

of Price per Ticket Total Value

Tickets (in dollars) (in dollars)

Main Floor x 96 96x

Mezzanine y 58 58y

Total 18XXXXXXXXXX

X 1234

5 18y

96 58 1234x y 3

38 38

8 190x


Objective 3

Solve problems about mixtures.

Slide 4.4-8


How many liters of a 25% alcohol solution must be mixed with a 12% solution to get 13 L of a 15% solution?

Solution:

13x y .12 .25 1.95x y

.12 .25 1100 100.95x y

12 25 19513 yy

156 12 25 15 195 566 1y y 13 1

3

3

1 39y 3 31x

To make 13 L of a 15% solution, 3 L of 25% solution, and 10 L of 12% solution must be used.

Liters of Percent

(as Liters of

Solution a decimal)pure

alcohol

x .12 .12x

y .25 .25y

13 .15 1.95

3y 10x

Slide 4.4-9

EXAMPLE 3 Solving a Mixture Problem Involving Percent


Objective 4

Solve problems about distance, rate (or speed), and time.

Slide 4.4-10


Two cars that were 450 mi apart traveled towards each other. They met after 5 hr. If one car traveled twice as fast as the other, what were their rates?Solution:Let x = the rate of the slower car,and y = the rate of the faster car.

5 5 450x y 2y x

5 45025 xx

The faster car travels 60 mph and the slower car travels 30 mph.

1

15 15

5 450x

60y 30x

Slide 4.4-11

EXAMPLE 4 Solving a Problem about Distance, Rate, and Time

r t d

Faster Car y 5 5y

Slower Car x 5 5x

5 10 450x x 150 5 450y 4305 5 50y

5 0

5 5

30y

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Chapter 4 Section 4. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Applications of Linear Systems Solve problems about unknown numbers