3.4 Solving Real-Life Problems

English3.4SpanishSolving Real-Life ProblemsHow can you use a linearequation in two variables to model and solve a real-life problem?1EXAMPLE: Writing a StoryWrite a story that uses the graph at the right.250 In your story, interpret the slope of theline, the y-intercept, and the x-intercept.225 Make a table that shows data fromthe graph.150 Label the axes of the graph with units. Draw pictures for your story.200175125100755025There are many possible stories. Here isone about a reef tank.00123456789Tom works at an aquarium shop on Saturdays.One Saturday, when Tom gets to work, he is askedto clean a 175-gallon reef tank.His first job is to drain the tank. He puts a hose intothe tank and starts a siphon. Tom wonders if thetank will finish draining before he leaves work.He measures the amount of water that isdraining out and finds that 12.5 gallonsdrain 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.y225x-intercept: number ofhours to empty the tank01234567y1751501251007550250Water (gallons)x2001751501251007550y -intercept: amountof water in full tank250012345 6Time (hours)From the table and also from the graph, Tom sees thatthe tank will be empty after 7 hours. This will give him1 hour to wash the tank before going home.126Chapter 3Writing Linear Equations and Linear Systems78 x

EnglishSpanish2ACTIVITY: Writing a StoryWork with a partner. Write a story that uses the graph of a line.3 In your story, interpret the slope of the line, the y-intercept, andthe x-intercept. Make a table that shows data from the graph. Label the axes of the graph with units. Draw pictures for your story.ACTIVITY: Drawing GraphsWork with a partner. Describe a real-life problem that has the given rate andintercepts. Draw a line that represents the problem.a. Rate: 30 feet per second250y-intercept: 150 feet225x-intercept: 5 seconds2001751501251007550250b. Rate: 25 dollars per month01234567890123456789250y-intercept: 200225x-intercept: 8 months20017515012510075502504. IN YOUR OWN WORDS How can you use a linear equation in two variablesto model and solve a real-life problem? List three different rates that canbe represented by slopes in real-life problems.Use what you learned about solving real-life problems to completeExercises 4 and 5 on page 130.Section 3.4Solving Real-Life Problems127

English3.4SpanishLessonLesson TutorialsEXAMPLE1Real-Life ApplicationThe percent y (in decimal form) of battery power remaining x hoursafter you turn on a laptop computer is y 0.2 x 1. (a) Graph theequation. (b) Interpret the x- and y-intercepts. (c) After how manyhours is the battery power at 75%?a. Use the slope and the y-intercept to graph the equation.y 0.2x 1slopeThe y-intercept is1. So, plot (0, 1).y-intercepty1(0, 1)Use the slope to plot anotherpoint, (1, 0.8). Draw a linethrough the points. 0.20.8(1, 0.8)0.60.40.2 1123456 xb. To find the x-intercept, substitute 0 for y in the equation.y 0.2x 1Write the equation.0 0.2x 1Substitute 0 for y.5 xSolve for x.The x-intercept is 5. So, the battery lasts 5 hours. The y-interceptis 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 175% RemainingExercise 6128Chapter 3Write the equation.0.75 0.2x 1Substitute 0.75 for y.1.25 xSolve for x.The battery power is at 75% after 1.25 hours.1. The amount y (in gallons) of gasoline remaining in a gas tankafter driving x hours is y 2x 12. (a) Graph the equation.(b) Interpret the x- and y-intercepts. (c) After how many hoursare there 5 gallons left?Writing Linear Equations and Linear Systems

EnglishSpanishEXAMPLE2Real-Life Application F y90The graph relates temperatures y (in degreesFahrenheit) to temperatures x (in degreesCelsius). (a) Find the slope and y-intercept.(b) Write an equation of the line. (c) What isthe mean temperature of Earth in degreesFahrenheit?change in ychange in x543075(0, 32)15 3095(30, 86)60 C 1010 20 30 40 xa. slope — — —The line crosses the y-axis at (0, 32).So, the y-intercept is 32.95The slope is — and the y-intercept is 32.b. Use the slope and y-intercept to write an equation.slopey-intercept95The equation is y — x 32.Mean Temperature:15íCc. In degrees Celsius, the mean temperature of Earth is 15 . To findthe mean temperature in degrees Fahrenheit, find the value of ywhen x 15.95Write the equation. —(15) 3295Substitute 15 for x. 59Simplify.y — x 32The mean temperature of Earth is 59 F.Exercise 72. The graph shows the height y (in feet) of a flag x secondsafter you start raising it up a flagpole.ya. Find and interpret the slope.1512b. Write an equation of the line.c. What is the height of the flagafter 9 seconds?Section 3.496(2, 6)(0, 3) 11234Solving Real-Life Problems5 x129

EnglishSpanishExercises3.4Help with Homework1. REASONING Explain how to find the slope, y-intercept,and x-intercept of the line shown.y632. OPEN-ENDED Describe a real-life situation that uses anegative slope. 2 1 33. REASONING In a real-life situation, what does the slope ofa line represent?12 126) 39 (- 3) 3 (- 9) 4 (- 1)9 (-Describe a real-life problem that has the given rate and intercepts. Draw a linethat represents the problem.4. Rate: 1.6 gallons per hour15. Rate: 45 pesos per weeky-intercept: 16 gallonsy-intercept: 180 pesosx-intercept: 10 hoursx-intercept: 4 weeks6. DOWNLOAD You are downloading a song. The percent y (in decimal form) ofmegabytes 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?27. 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?Altitude ChangeTemperature ( F)y7060(0, 59)5040(7, 33.8)3020100 102468 10 12 14 1620 22 xAltitude (thousands of feet)130Chapter 3Writing Linear Equations and Linear Systems34 x

EnglishSpanish8. TRAVEL Your family is driving from Cincinnati to St Louis. The graph relates yourdistance from St Louis y (in miles) and travel time x (in hours).SpringfieldDistance from St. Louis(miles)Driving DistanceDayton75y360742805771Cincinnati65St. Louis32071a. Interpret the x- and y-intercepts.240200b. What is the slope? What does the sloperepresent in this situation?16012080400Indianapolis70550123456xc. Write an equation of the line. How wouldthe graph and the equation change if youwere able to travel in a straight line?Time (hours)9. PROJECT Use a map or the Internet to find the latitude and longitude of yourschool to the nearest whole number. Then find the latitudes and longitudesof: 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 thepositive 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 anauditorium for a fee of 1500. In addition tothis fee, the band receives 30% of each 20ticket sold. The maximum capacity of theauditorium is 800 people.a. Write an equation that represents theband’s revenue R when x tickets are sold.b. The band needs 5000 for new equipment.How many tickets must be sold for theband to earn enough money to buy the new equipment?Tell whether the system has one solution, no solution, or infinitely many solutions.(Section2.5 and2.6)SECTION2.5 SectionSECTION2.611. y x 6 4(x y) 2412. y 3x 213. 9x 3y 12 x 2y 11y 3x 214. MULTIPLE CHOICE Which equation is the slope-intercept form of 24x 8y 56?(Section2.3)2.3SECTIONA y 3x 7 B y 3x 7 C y 3x 7 Section 3.4D y 3x 7 Solving Real-Life Problems131

Section 3.4 Solving Real-Life Problems 127 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. 2 ACTIVITY: Writing a Story Work with a partner.