Transcript Slide 1

Chapter 4
Section 4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
4.4
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2
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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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
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 4: Solve the system of two equations.
Step 5: State the answer to the problem. Is the answer
reasonable?
Step 6: Check the answer in the words of the original problem.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Objective 1
Solve problems about unknown
numbers.
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EXAMPLE 1
Solving a Problem about Two
Unknown Numbers
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.)
Solution:
Let x = gross of Lilo and Stitch in millions,
and y = gross of The Santa Clause 2 in millions.
x  y  284.2
x  7.4  y
7.4  y   y  7.4  284.2  7.4
2 y 276.8

2
2
y  138.4
x  7.4  138.4
x  145.8
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
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Objective 2
Solve problems about quantities
and their costs.
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EXAMPLE 2
Solving a Problem about
Quantities and Costs
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?
Number of Price per Ticket
Total Value
Solution:
x  y  36
8x  10 y  298
Tickets
(in dollars)
(in dollars)
Paris
x
8
8x
Geneva
y
10
10y
Total
36
XXXXXXXX
298
836  y  10 y  298
288  8 y  10 y  288  298  288
2 y 10

2
2
y5
x  36  5
x  31
There were 5 people from Geneva, and 31 people from
Paris that went to see The Rookie.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4.4 - 7
Objective 3
Solve problems about mixtures.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4.4 - 8
EXAMPLE 3
Solving a Mixture Problem
Involving Percent
How many liters of a 25% alcohol solution must
be mixed with a 12% solution to get 13 L of a
15% solution?
Liters of
Percent (as
Liters of
Solution:
x  y  13
.12 x  .25 y  1.95
100.12x  .25 y   1.95100
12 13  y   25 y  195
156  12 y  25 y  156  195  156
Solution
a decimal)
pure alcohol
x
.12
.12x
y
.25
.25y
13
.15
1.95
13 y 39

13 13
y3
x  13  3
x  10
To make 13 L of a 15% solution, 3 L of 25% solution,
and 10 L of 12% solution must be used.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4.4 - 9
Objective 4
Solve problems about distance,
rate (or speed), and time.
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Slide 4.4 - 10
EXAMPLE 4
Solving a Problem about
Distance, Rate, and Time
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.
Solution:
Let x = Abby’s speed in still water in mph,
and y = the water speed of the current in mph.
x  y  10
 2  y   y  2  10  2
x  24
x y 2
2y 8

2 2
y4
x6
Abby’s speed in still water is 6 mph, and the speed of
the current is 4 mph.
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