3 4 solving real life problems big ideas math

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3.4 Solving Real-Life Problems

How can you use a linear equation in two variables to model and solve a real-life problem?

1 EXAMPLE: Writing a Story

Write a story that uses the graph at the right.

In your story, interpret the slope of the line, the y-intercept, and the x-intercept.

Make a table that shows data from the graph.

Label the axes of the graph with units. Draw pictures for your story.

There are many possible stories. Here is one about a reef tank.

250 225 200 175 150 125 100

75 50 25

0 0123456789

Tom works at an aquarium shop on Saturdays. One Saturday, when Tom gets to work, he is asked to clean a 175-gallon reef tank.

His first job is to drain the tank. He puts a hose into the tank and starts a siphon. Tom wonders if the tank will finish draining before he leaves work.

He measures the amount of water that is draining out and finds that 12.5 gallons drain out in 30 minutes. So, he figures that the rate is 25 gallons per hour. To see when the tank will be empty, Tom makes a table and draws a graph.

x-intercept: number of

y 225

hours to empty the tank

200

Water (gallons)

175

150

x0 1 2 3 4 5 6 7

125

y 175 150 125 100 75 50 25 0

100

75

50

y -intercept: amount

25

of water in full tank

0 0 1 2 3 4 5 6 7 8x

Time (hours)

From the table and also from the graph, Tom sees that

the tank will be empty after 7 hours. This will give him

1 hour to wash the tank before going home.

126 Chapter 3 Writing Linear Equations and Linear Systems

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2 ACTIVITY: Writing a Story

Work with a partner. Write a story that uses the graph of a line.

In your story, interpret the slope of the line, the y-intercept, and the x-intercept.

Make a table that shows data from the graph. Label the axes of the graph with units. Draw pictures for your story.

3 ACTIVITY: Drawing Graphs

Work with a partner. Describe a real-life problem that has the given rate and intercepts. Draw a line that represents the problem.

a. Rate: -30 feet per second y-intercept: 150 feet x-intercept: 5 seconds

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b. Rate: -25 dollars per month y-intercept: $200 x-intercept: 8 months

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75 50 25

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4. IN YOUR OWN WORDS How can you use a linear equation in two variables to model and solve a real-life problem? List three different rates that can be represented by slopes in real-life problems.

Use what you learned about solving real-life problems to complete Exercises 4 and 5 on page 130.

Section 3.4 Solving Real-Life Problems 127

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Spanish

3.4 Lesson

Lesson Tutorials

EXAMPLE 1 Real-Life Application

The percent y (in decimal form) of battery power remaining x hours after you turn on a laptop computer is y = -0.2 x + 1. (a) Graph the equation. (b) Interpret the x- and y-intercepts. (c) After how many hours is the battery power at 75%?

a. Use the slope and the y-intercept to graph the equation.

y = -0.2x + 1

slope

y-intercept

The y-intercept is 1. So, plot (0, 1).

y

1 (0, 1) -0.2

0.8

(1, 0.8)

0.6

Use the slope to plot another point, (1, 0.8). Draw a line through the points.

0.4

0.2

75% Remaining

-1

1 2 3 4 5 6x

b. To find the x-intercept, substitute 0 for y in the equation.

y = -0.2x + 1

Write the equation.

0 = -0.2x + 1

Substitute 0 for y.

5 = x

Solve for x.

The x-intercept is 5. So, the battery lasts 5 hours. The y-intercept is 1. So, the battery power is at 100% when you turn on the laptop.

c. Find the value of x when y = 0.75.

y = -0.2x + 1 Write the equation.

0.75 = -0.2x + 1 Substitute 0.75 for y.

1.25 = x

Solve for x.

The battery power is at 75% after 1.25 hours.

Exercise 6

1. The amount y (in gallons) of gasoline remaining in a gas tank after driving x hours is y = -2x + 12. (a) Graph the equation. (b) Interpret the x- and y-intercepts. (c) After how many hours are there 5 gallons left?

128 Chapter 3 Writing Linear Equations and Linear Systems

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EXAMPLE 2 Real-Life Application

The graph relates temperatures y (in degrees Fahrenheit) to temperatures x (in degrees Celsius). (a) Find the slope and y-intercept. (b) Write an equation of the line. (c) What is the mean temperature of Earth in degrees Fahrenheit?

a.

slope

=

change in y --

=

54 --

=

9 --

change in x 30 5

?F y 90

75

60

(30, 86)

(0, 32)

15 ?C

-30 -10 10 20 30 40 x

The line crosses the y-axis at (0, 32). So, the y-intercept is 32.

The

slope

is

9 --

and

the

y-intercept

is

32.

5

b. Use the slope and y-intercept to write an equation.

Mean Temperature: 15?C

slope

The

equation

is

y

=

9 --

x

+

32.

5

y-intercept

c. In degrees Celsius, the mean temperature of Earth is 15?. To find the mean temperature in degrees Fahrenheit, find the value of y when x = 15.

y

=

9 --

x

+

32

5

= --9(15) + 32

5

= 59

Write the equation. Substitute 15 for x. Simplify.

The mean temperature of Earth is 59?F.

Exercise 7

2. The graph shows the height y (in feet) of a flag x seconds

after you start raising it up a flagpole.

y

a. Find and interpret the slope.

15

12

b. Write an equation of the line.

9

c. What is the height of the flag after 9 seconds?

6

(0, 3)

-1

(2, 6)

1 2 3 4 5x

Section 3.4 Solving Real-Life Problems 129

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3.4 Exercises

Help with Homework

1. REASONING Explain how to find the slope, y-intercept, and x-intercept of the line shown.

2. OPEN-ENDED Describe a real-life situation that uses a negative slope.

3. REASONING In a real-life situation, what does the slope of a line represent?

y 6

3

-2 -1

1 2 3 4x

-3

-12

93++4(-+(6-9(3)-=+)9=3()-=1)=

Describe a real-life problem that has the given rate and intercepts. Draw a line that represents the problem.

4. Rate: -1.6 gallons per hour

5. Rate: -45 pesos per week

y-intercept: 16 gallons x-intercept: 10 hours

y-intercept: 180 pesos x-intercept: 4 weeks

1 6. DOWNLOAD You are downloading a song. The percent y (in decimal form) of megabytes remaining to download after x seconds is y = -0.1x + 1.

a. Graph the equation. b. Interpret the x- and y-intercepts. c. After how many seconds is the download 50% complete?

2 7. HIKING The graph relates temperature y (in degrees Fahrenheit) to altitude x (in thousands of feet).

a. Find the slope and y-intercept. b. Write an equation of the line. c. What is the temperature at sea level?

Temperature (?F)

Altitude Change

y

70

(0, 59)

60

50

40

(7, 33.8)

30

20

10

0 2 4 6 8 10 12 14 16

-10

20 22 x

Altitude (thousands of feet)

130 Chapter 3 Writing Linear Equations and Linear Systems

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Distance from St. Louis (miles)

8. TRAVEL Your family is driving from Cincinnati to St Louis. The graph relates your distance from St Louis y (in miles) and travel time x (in hours).

Driving Distance

y 360 320

Springfield

55

St. Louis

57

Indianapolis

70 65

Dayton

75 71

74

Cincinnati

71

280

a. Interpret the x- and y-intercepts.

240

200

b. What is the slope? What does the slope

160

represent in this situation?

120

80

c. Write an equation of the line. How would

40

the graph and the equation change if you

0

0

1

2

3

4

5

6x

were able to travel in a straight line?

Time (hours)

9. PROJECT Use a map or the Internet to find the latitude and longitude of your school to the nearest whole number. Then find the latitudes and longitudes of: Antananarivo, Madagascar; Denver, Colorado; Brasilia, Brazil; London, England; and Beijing, China.

a. Plot a point for each of the cities in the same coordinate plane. Let the positive y-axis represent north and the positive x-axis represent east.

b. Write an equation of the line that passes through Denver and Beijing. c. In part (b), what geographic location does the y-intercept represent?

10.

A band is performing at an

auditorium for a fee of $1500. In addition to

this fee, the band receives 30% of each $20

ticket sold. The maximum capacity of the

auditorium is 800 people.

a. Write an equation that represents the band's revenue R when x tickets are sold.

b. The band needs $5000 for new equipment. How many tickets must be sold for the band to earn enough money to buy the new equipment?

Tell whether the system has one solution, no solution, or infinitely many solutions. (SeSctEiConTI2O.N5 a2n.5d SectioSnE2C.6T)ION 2.6

11. y = -x + 6

12. y = 3x - 2

13. -9x + 3y = 12

-4(x + y) = -24

-x + 2y = 11

y = 3x - 2

14. MULTIPLE CHOICE Which equation is the slope-intercept form of 24x - 8y = 56? (SeScEtiCoTnIO2.N3)2.3

A y = -3x + 7 B y = 3x - 7

C y = -3x - 7

D y = 3x + 7

Section 3.4 Solving Real-Life Problems 131

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