6.2. One-Step Equations with Rational Coefficients
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1 LESSON 6.2 One-Step Equations with Rational Coefficients Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations to solve problems by reasoning about the quantities. ESSENTIAL QUESTION How do you use one-step equations with rational coefficients to solve problems? One-Step Equations You have written and solved one-step equations involving whole numbers. Now you will learn to work with equations containing negative numbers. Use inverse operations to solve each equation. Use inverse operations to solve each equation z = 9 2. r 17.1 = c = 36 Lesson
2 Writing and Solving One-Step Addition and Subtraction Equations Negative numbers often appear in real-world situations. For example, elevations below sea level are represented by negative numbers. When you increase your elevation, you are moving in a positive direction. When you decrease your elevation, you are moving in a negative direction. A scuba diver is exploring at an elevation of 12.2 meters. As the diver rises to the surface, she plans to stop and rest briefly at a reef that has an elevation of 4.55 meters. Find the vertical distance that the diver will travel. Write an equation. Let x represent the vertical distance between her initial elevation and the elevation of the reef. Solve the equation using an inverse operation. The diver will travel a vertical distance of 7.65 meters. Reflect 4. Make a Prediction Explain how you know whether the diver is moving in a positive or a negative direction before you solve the equation. 5. An airplane descends 1.5 miles to an elevation of 5.25 miles. Find the elevation of the plane before its descent. 180 Unit 3
3 Writing and Solving One-Step Multiplication and Division Problems Temperatures can be both positive and negative, and they can increase or decrease during a given period of time. A decrease in temperature is represented by a negative number. An increase in temperature is represented by a positive number. Between the hours of 10 P.M. and 6 A.M., the temperature decreases an average of of a degree per hour. How many minutes will it take for the temperature to decrease by 5 F? Write an equation. Let x represent the number of hours it takes for the temperature to decrease by 5 F. Solve the equation using an inverse operation. Why is multiplying by the inverse of multiplying by STEP 3 Convert the number of hours to minutes. It takes 400 minutes for the temperature to decrease by 5 F. 6. The value of a share of stock decreases in value at a rate of $1.20 per hour during the first 3.5 hours of trading. Write and solve an equation to find the decrease in the value of the share of stock during that time. 7. After a power failure, the temperature in a freezer increased at an average rate of 2.5 F per hour. The total increase was 7.5 F. Write and solve an equation to find the number of hours until the power was restored. Lesson
4 Guided Practice The table shows the average temperature in Barrow, Alaska, for three months during one year. 1. How many degrees warmer is the average temperature in November than in January? Write an equation. Let x represent Month January June November Average Temperature ( F) x +, or x Solve the equation. Show your work. The average temperature in November is warmer. 2. Suppose that during one period of extreme cold, the average daily temperature decreased F each day. How many days did it take for the temperature to decrease by 9 F? Write an equation. Let x represent Solve the equation. Show your work. It took days for the temperature to decrease by 9 F. Use inverse operations to solve each equation. 3. 2x = y 3.5 = ESSENTIAL QUESTION CHECK-IN 6. How does writing an equation help you solve a problem? 182 Unit 3
5 6.2 Independent Practice The table shows the elevation in feet at the peaks of several mountains. Use the table for 7 9. The table shows the average temperatures in several states from January through March. Use the table for Mountain Mt. McKinley K2 Tupungato Dom Elevation (feet) 20, , , , Mt. Everest is 8, feet higher than Mt. McKinley. What is the elevation of Mt. Everest? 8. Liam descended from the summit of K2 to an elevation of 23, feet. How many feet did Liam descend? What was his change in elevation? State Florida Minnesota Montana Texas Average Temperature ( C) Image Credits: Ilene MacDonald/Alamy Images 9. K2 is 11, feet higher than Mt. Kenya. Write and solve an equation to find the elevation of Mt. Kenya. 10. A hot air balloon begins its descent at a rate of feet per minute. How long will it take for the balloon's elevation to change by -315 feet? 11. During another part of its flight, the balloon in Exercise 10 had a change in elevation of -901 feet. What was its rate of descent? 12. Write and solve an equation to find how much warmer Montana's average 3-month temperature is than Minnesota's. 13. How much warmer is Florida's average 3-month temperature than Montana's? 14. How would the average temperature in Texas have to change to match the average temperature in Florida? 15. A football team has a net yardage of yards on a series of plays. The team needs a net yardage of 10 yards to get a first down. How many yards do they have to get on their next play to get a first down? Lesson
6 16. A diver begins at sea level and descends vertically at a rate of feet per second. How long does the diver take to reach 15.6 feet? 17. Analyze Relationships In Exercise 16, what is the relationship between the rate at which the diver descends, the elevation he reaches, and the time it takes to reach that elevation? 18. Check for Reasonableness Jane withdrew money from her savings account in each of 5 months. The average amount she withdrew per month was $ How much did she withdraw in all during the 5 months? Show that your answer is reasonable. 19. Justify Reasoning Consider the two problems below. Which values in the problems are represented by negative numbers? Explain why. (1) A diver below sea level ascends 25 feet to a reef at 35.5 feet. What was the elevation of the diver before she ascended to the reef? (2) A plane descends 1.5 miles to an elevation of 3.75 miles. What was the elevation of the plane before its descent? 20. Analyze Relationships How is solving 4x = 4.8 different from solving How are the solutions related? 21. Communicate Mathematical Ideas Flynn opens a savings account. In one 3-month period, he makes deposits of $75.50 and $ He makes withdrawals of $25.15 and $ His balance at the end of the 3-month period is $ Explain how you can find his initial deposit amount. 184 Unit 3
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