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

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1 2.7 Solving Real-Life Problems How can ou use a linear equation in two variables to model and solve a real-life problem? EXAMPLE: Writing a Stor Write a stor that uses the graph at the right. In our stor, interpret the slope of the line, the -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 our stor. There are man possible stories. Here is one about a reef tank Tom works at an aquarium shop on Saturdas. One Saturda, when Tom gets to work, he is asked to clean a -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 2.5 gallons drain out in 3 minutes. So, he figures that the rate is gallons per hour. To see when the tank will be empt, Tom makes a table and draws a graph. COMMON CORE Writing Equations In this lesson, ou will solve real-life problems involving linear equations. Appling Standards 8.F.4 A.CED.2 F.IF.4 x-intercept: number of hours to empt the tank x intercept: amount of water in full tank From the table and also from the graph, Tom sees that the tank will be empt after 7 hours. This will give him hour to wash the tank before going home. Water (gallons) x Time (hours) 86 Chapter 2 Graphing and Writing Linear Equations

2 2 ACTIVITY: Writing a Stor Math Practice Label Axes What information is needed to label the axes? How do ou know where to place the labels? Work with a partner. Write a stor that uses the graph of a line. In our stor, interpret the slope of the line, the -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 our stor. 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: 3 feet per second -intercept: 5 feet x-intercept: 5 seconds b. Rate: dollars per month -intercept: $2 x-intercept: 8 months IN YOUR OWN WORDS How can ou use a linear equation in two variables to model and solve a real-life problem? List three different rates that can be represented b slopes in real-life problems. Use what ou learned about solving real-life problems to complete Exercises 4 and 5 on page 9. Section 2.7 Solving Real-Life Problems 87

3 2.7 Lesson Lesson Tutorials EXAMPLE Real-Life Application The percent (in decimal form) of batter power remaining x hours after ou turn on a laptop computer is =.2 x +. (a) Graph the equation. (b) Interpret the x- and -intercepts. (c) After how man hours is the batter power at %? a. Use the slope and the -intercept to graph the equation. =.2x + slope -intercept The -intercept is. So, plot (, ). (, ).8.2 (,.8) Use the slope to plot another point, (,.8). Draw a line through the points x b. To find the x-intercept, substitute for in the equation. =.2x + Write the equation. =.2x + Substitute for. 5 = x Solve for x. The x-intercept is 5. So, the batter lasts 5 hours. The -intercept is. So, the batter power is at % when ou turn on the laptop. c. Find the value of x when =.. =.2x + Write the equation.. =.2x + Substitute. for.. = x Solve for x. % Remaining The batter power is at % after. hours. Exercise 6. The amount (in gallons) of gasoline remaining in a gas tank after driving x hours is = 2x + 2. (a) Graph the equation. (b) Interpret the x- and -intercepts. (c) After how man hours are there 5 gallons left? 88 Chapter 2 Graphing and Writing Linear Equations

EXAMPLE 2 Real-Life Application The graph relates temperatures (in degrees Fahrenheit) to temperatures x (in degrees Celsius). (a) Find the slope and -intercept. (b) Write an equation of the line.

4 EXAMPLE 2 Real-Life Application The graph relates temperatures (in degrees Fahrenheit) to temperatures x (in degrees Celsius). (a) Find the slope and -intercept. (b) Write an equation of the line. (c) What is the mean temperature of Earth in degrees Fahrenheit? change in a. slope = change in x = 54 3 = 9 5 The line crosses the -axis at (, 32). So, the -intercept is 32. F 9 6 (, 32) 5 3 (3, 86) C x The slope is 9 and the -intercept is b. Use the slope and -intercept to write an equation. slope -intercept Mean Temperature: 5 C The equation is = 9 5 x c. In degrees Celsius, the mean temperature of Earth is 5. To find the mean temperature in degrees Fahrenheit, find the value of when x = 5. = 9 x + 32 Write the equation. 5 = 9 (5) + 32 Substitute 5 for x. 5 = 59 Simplif. The mean temperature of Earth is 59 F. Exercise 7 2. The graph shows the height (in feet) of a flag x seconds after ou start raising it up a flagpole. a. Find and interpret the slope. b. Write an equation of the line. c. What is the height of the flag after 9 seconds? (, 3) (2, 6) x Section 2.7 Solving Real-Life Problems 89

2.7 Exercises Help with Homework. REASONING Explain how to find the slope, -intercept, and x-intercept of the line shown. 2. OPEN-ENDED Describe a real-life situation that uses a negative slope. 3.

Draw a line that represents the problem. 4. Rate:.6 gallons per hour 5. Rate: 3 F per hour -intercept: 6 gallons -intercept: 2 F x-intercept: hours x-intercept: 7 hours 2 6.

5 2.7 Exercises Help with Homework. REASONING Explain how to find the slope, -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? x 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-)= Describe a real-life problem that has the given rate and intercepts. Draw a line that represents the problem. 4. Rate:.6 gallons per hour 5. Rate: 3 F per hour -intercept: 6 gallons -intercept: 2 F x-intercept: hours x-intercept: 7 hours 2 6. DOWNLOAD You are downloading a song. The percent (in decimal form) of megabtes remaining to download after x seconds is =.x +. a. Graph the equation. b. Interpret the x- and -intercepts. c. After how man seconds is the download 5% complete? 7. HIKING The graph relates temperature (in degrees Fahrenheit) to altitude x (in thousands of feet). a. Find the slope and -intercept. b. Write an equation of the line. c. What is the temperature at sea level? Altitude Change Temperature ( F) (, 59) (7, 33.8) x Altitude (thousands of feet) 9 Chapter 2 Graphing and Writing Linear Equations

Distance from St. Louis (miles) 8. REASONING Your famil is driving from Cincinnati to St. Louis. The graph relates our distance from St. Louis (in miles) and travel time x (in hours).

6 Distance from St. Louis (miles) 8. REASONING Your famil is driving from Cincinnati to St. Louis. The graph relates our distance from St. Louis (in miles) and travel time x (in hours). Driving Distance Springfield St. Louis x Time (hours) Indianapolis 7 a. Interpret the x- and -intercepts. b. What is the slope? What does the slope represent in this situation? c. Write an equation of the line. How would the graph and the equation change if ou were able to travel in a straight line? Daton 7 Cincinnati 9. PROJECT Use a map or the Internet to find the latitude and longitude of our 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 -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 -intercept represent?. A band is performing at an auditorium for a fee of $5. In addition to this fee, the band receives 3% of each $2 ticket sold. The maximum capacit of the auditorium is 8 people. a. Write an equation that represents the band s revenue R when x tickets are sold. b. The band needs $5 for new equipment. How man tickets must be sold for the band to earn enough mone to bu the new equipment? Solve the equation. Check our solution. (Section.2). h 7h + 3 = (k ) 4 = (2q 3) = 4. MULTIPLE CHOICE Which equation is the slope-intercept form of 24x 8 = 56? (Section 2.4) A = 3x + 7 B = 3x 7 C = 3x 7 D = 3x + 7 Section 2.7 Solving Real-Life Problems 9

REVIEW UNIT 4 TEST LINEAR FUNCTIONS

Name: Date: Page 1 of REVIEW UNIT 4 TEST LINEAR FUNCTIONS 1. Use the graph below to answer the following questions. a. Match each equation with line A, B, or C from the graph: A!!! =!! 1 B!! = 2! 2 = 3(!