In Section 2.3, we introduced the fi ve-step method for solving applied problems. Here, we modify the method for problems with two unknowns and two equations. Procedure Solving an Applied Problem Using a System of Equations Step 1: Read the problem carefully, more than once if necessary. Draw a picture, if applicable. Identify what you are being asked to fi nd. Step 2: Choose variables to represent the unknown quantities. Label any pictures with the variables. Step 3: Write a system of equations using two variables. It may be helpful to begin by writing the equations in words. Step 4: Solve the system. Step 5: Check the answer in the original problem, and interpret the solution as it relates to the problem. Be sure your answer makes sense in the context of the problem. 1 Solve Problems Involving General Quantities EXAMPLE 1 In-Class Example 1 As of the end of 2011, the Beatles had more of their singles hit number one on the U.S. charts than any other artist. In second place was Mariah Carey. The Beatles had two more songs hit number one, and together the artists had a total of 38 number one singles. How many of each of their singles topped the U.S. charts? (www.billboard.com) Answer: The Beatles: 20; Mariah Carey: 18 Write a system of equations, and solve. Pink Floyd’s album, The Dark Side of the Moon, spent more weeks on the Billboard 200 chart for top-selling albums than any other album in history. It was on the chart 251 more weeks than the second-place album, Johnny’s Greatest Hits, by Johnny Mathis. If they were on the charts for a total of 1231 weeks, how many weeks did each album spend on the Billboard 200 chart? (www.billboard.com) Solution Step 1: Read the problem carefully, and identify what we are being asked to fi nd. We must fi nd the number of weeks each album was on the chart. Step 2: Choose variables to represent the unknown quantities. x the number of weeks The Dark Side of the Moon was on the Billboard 200 chart y the number of weeks Johnny’s Greatest Hits was on the Billboard 200 chart Step 3: Write a system of equations using two variables. First, let’s think of the equations in English. Then we will translate them into algebraic equations. To get one equation, use the information that says these two albums were on the Billboard 200 chart for a total of 1231 weeks. Write an equation in words, then translate it into an algebraic equation. English: Number of weeks Pink Floyd was on Billboard 200 chart plus Number of weeks Johnny Mathis was on Billboard 200 chart equals Total weeks T T T T T Equation: x y 1231 The fi rst equation is x y 1231. 274 CHAPTER 4 Linear Equations and Inequalities in Two Variables www.mhhe.com/messersmith
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