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Chapter Chapter 44Section Section 44
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Applications of Linear Systems
11
44
33
22
4.44.44.44.4Solve problems about unknown numbers.Solve problems about quantities and their costs.Solve problems about mixtures.Solve problems about distance, rate (or speed), and time.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Applications of Linear Systems
Slide 4.4 - 3
Recall from Section 2.4 the six step method for solving applied problems. These slightly modified steps allow for two variables and two equations.
Step 1: Read the problem carefully until you understand what is given and what is to be found.
Step 2: Assign variables to represent the unknown values, using diagrams or tables as needed. Write down what each variable represents.
Step 3: Write two equations using both variables.
Step 5: State the answer to the problem. Is the answer reasonable?
Step 4: Solve the system of two equations.
Step 6: Check the answer in the words of the original problem.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Slide 4.4 - 4
Solve problems about unknown numbers.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Two top-grossing Disney movies in 2002 were Lilo and Stitch and The Santa Clause 2. Together they grossed $284.2 million. The Santa Clause 2 grossed $7.4 million less than Lilo and Stitch. How much did each movie gross? (Source: Variety.)
EXAMPLE 1Solving a Problem about Two Unknown Numbers
Solution:Let x = gross of Lilo and Stitch in millions,and y = gross of The Santa Clause 2 in millions.
Slide 4.4 - 5
284.2x y 7.4x y
7.4 27 84.2. 7.44 y y 2 27
2 2
6.8y
138.4y
8.7.4 13 4x 145.8x
Lilo and Stitch grossed 145.8 million dollars and The Santa Clause 2 grossed 138.4 million dollars.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Solve problems about quantities and their costs.
Slide 4.4 - 6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
In 1997 – 1998, the average movie ticket (to the nearest U.S. dollar) cost $10 in Geneva and $8 in Paris. (Source: Parade, September 13, 1998.) If a group of 36 people from these two cities paid $298 for tickets to see The Rookie, how many people from each city were there?
EXAMPLE 2Solving a Problem about Quantities and Costs
Solution:
Slide 4.4 - 7
36x y 8 10 298x y
8 10 29836 yy 22 888 88 21 9 80 2 8 8y y
2 0
2 2
1y
5y 31x
There were 5 people from Geneva, and 31 people from Paris that went to see The Rookie.
Number of Price per Ticket Total Value
Tickets (in dollars) (in dollars)
Paris x 8 8x
Geneva y 10 10y
Total 36 XXXXXXXX 298
6 53x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Solve problems about mixtures.
Slide 4.4 - 8
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
How many liters of a 25% alcohol solution must be mixed with a 12% solution to get 13 L of a 15% solution?
EXAMPLE 3Solving a Mixture Problem Involving Percent
Slide 4.4 - 9
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
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 44
Solve problems about distance, rate (or speed), and time.
Slide 4.4 - 10
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
In one hour, Abby can row 2 mi against the current or 10 mi with the current. Find the speed of the current and Abby’s speed in still water.
EXAMPLE 4Solving a Problem about Distance, Rate, and Time
Solution:Let x = Abby’s speed in still water in mph,and y = the water speed of the current in mph.
Slide 4.4 - 11
10x y
2x y
22 1 20y y
Abby’s speed in still water is 6 mph, and the speed of the current is 4 mph.
2 8
2 2
y
4y
2 4x 6x