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This word equation is called a verbal model. U SING A P ROBLEM S OLVING P LAN The verbal model is then used to write a mathematical statement, which is called an algebraic model. W RITE A VERBAL MODEL. A SSIGN LABELS. W RITE AN ALGEBRAIC MODEL. It is helpful when solving real-life problems to first write an equation in words before you write it in mathematical symbols. S OLVE THE ALGEBRAIC MODEL. A NSWER THE QUESTION.

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Writing and Using a Formula The Bullet Train runs between the Japanese cities of Osaka and Fukuoka, a distance of 550 kilometers. When it makes no stops, it takes 2 hours and 15 minutes to make the trip. What is the average speed of the Bullet Train?

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r 550 2.25 = Write algebraic model. Divide each side by 2.25. Use a calculator. r 244 Writing and Using a Formula L ABELS V ERBAL M ODEL Distance = Rate Time 550Distance = (kilometers) 2.25Time = (hours) rRate = (kilometers per hour) A LGEBRAIC M ODEL You can use the formula d = r t to write a verbal model. The Bullet Train’s average speed is about 244 kilometers per hour. d = rt r550(2.25) =

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Writing and Using a Formula You can use unit analysis to check your verbal model. 550 kilometers 244 kilometers hour 2.25 hours UNIT ANALYSIS

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U SING O THER P ROBLEM S OLVING S TRATEGIES When you are writing a verbal model to represent a real-life problem, remember that you can use other problem solving strategies, such as draw a diagram, look for a pattern, or guess, check and revise, to help create a verbal model.

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Drawing a Diagram R AILROADS In 1862, two companies were given the rights to build a railroad from Omaha, Nebraska to Sacramento, California. The Central Pacific Company began from Sacramento in 1863. Twenty-four months later, the Union Pacific company began from Omaha. The Central Pacific Company averaged 8.75 miles of track per month. The Union Pacific Company averaged 20 miles of track per month. The companies met in Promontory, Utah, as the 1590 miles of track were completed. In what year did they meet? How many miles of track did each company build?

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Write algebraic model. 1590=8.75+(t – 24)20t Union Pacific time = (months) t – 24 Union Pacific rate =20 (miles per month) Central Pacific time = (months) t Central Pacific rate =8.75 (miles per month) Total miles of track =1590 (miles) Drawing a Diagram A LGEBRAIC M ODEL L ABELS V ERBAL M ODEL Total miles of track = + Number of months Miles per month Central Pacific Number of months Miles per month Union Pacific

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Divide each side by 28.75. 72 = t The construction took 72 months (6 years) from the time the Central Pacific Company began in 1863. They met in 1869. 1590=8.75t+20(t – 24) A LGEBRAIC M ODEL Write algebraic model. Drawing a Diagram 1590 = 8.75 t + 20 t – 480 2070 = 28.75 t Distributive property Simplify.

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Drawing a Diagram The number of miles of track built by each company is as follows: Central Pacific: Union Pacific: 72 months (72 – 24) months 8.75 miles 20 miles month = 630 miles = 960 miles The construction took 72 months (6 years) from the time The Central Pacific Company began in 1863.

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12 Looking for a Pattern The table gives the heights to the top of the first few stories of a tall building. Determine the height to the top of the 15th story. After the lobby, the height increases by 12 feet per story. S OLUTION Look at the differences in the heights given in the table. Story Height to top of story (feet) Lobby12 3 4 20 324456 68 12 2032 44 12 4456 68

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You can use the observed pattern to write a model for the height. Substitute 15 for n. Write algebraic model. Simplify. = + h2012n Height to top of a story =h (feet) Height per story =12 (feet per story) Height of lobby =20 (feet) A LGEBRAIC M ODEL = 200 Height to top of a story = Height per story Story number Height of lobby + = 20 + 12 (15) Story number =n (stories) L ABELS V ERBAL M ODEL The height to the top of the 15th story is 200 feet. Looking for a Pattern

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Example 3 Solving an Equation Using Addition The solution is 17.5. ANSWER Original equation 13=4.5c– Add 4.5 to each side. (Addition property of equality)

Example 3 Solving an Equation Using Addition The solution is 17.5. ANSWER Original equation 13=4.5c– Add 4.5 to each side. (Addition property of equality)

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