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OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 1 Applications of Linear Equations Learn procedures for solving applied problems. Use linear equations to solve applied problems. SECTION

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Procedure for Solving Applied Problems Step 1Read the problem as many times as needed to understand it thoroughly. Pay close attention to the questions asked to help identify the quantity the variable should represent. Step 2Assign a variable to represent the quantity you are looking for, and, when necessary, express all other unknown quantities in terms of this variable. Frequently, it is helpful to draw a diagram to illustrate the problem or to set up a table to organize the information. 2 © 2010 Pearson Education, Inc. All rights reserved

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Step 3Write an equation that describes the situation. Step 4Solve the equation. Step 5Answer the question asked in the problem. Step 6Check the answer against the description of the original problem (not just the equation solved in step 4). Procedure for Solving Applied Problems 3 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 1 Solving a Geometry Problem The length of a rectangle is 5 m more than twice its width. Find the dimensions of the rectangle assuming that its perimeter is 28 m. 4 © 2010 Pearson Education, Inc. All rights reserved To be solved in class

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EXAMPLE 2 Page 105 Problem # © 2010 Pearson Education, Inc. All rights reserved To be solved in class

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If a principal of P dollars is borrowed for a period of t years with interest rate r (expressed as a decimal) computed yearly, then the total interest paid at the end of years is Interest computed with this formula is called simple interest. When interest is computed yearly, the rate r is called an annual interest rate (or per annum interest rate). SIMPLE INTEREST 6 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 3 Page 105 Problem # 28 7 © 2010 Pearson Education, Inc. All rights reserved

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If an object moves at a rate (average speed) r, then the distance traveled d in time t is UNIFORM MOTION 8 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 4 Solving a Uniform-Motion Problem A motorcycle police officer is chasing a car that is speeding at 80 miles per hour. The police officer is 3 miles behind the car and is traveling 90 miles per hour. How long will it be before the officer overtakes the car? Solution We are asked to find the amount of time before the officer overtakes the car. Draw a sketch to help visualize the problem. Step 2 9 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 4 Solving a Uniform-Motion Problem Solution continued Let x = distance in miles the car travels before being overtaken x + 3 = distance in miles the motorcycle travels before overtaking the car Step 2 10 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 4 Solving a Uniform-Motion Problem Solution continued The time from the start of the chase to the interception point is the same for both the car and the motorcycle. Object d miles r mph hours Car Motorcycle 11 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 4 Solving a Uniform-Motion Problem Solution continued 12 © 2010 Pearson Education, Inc. All rights reserved Rest of the problem will be discussed in class

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The portion of a job completed per unit of time is called the rate of work. WORK RATE If a job can be completed in x units of time, then the portion of the job completed in one unit of time is. The portion of the job completed in t units of time is. When the portion of the job completed is 1, the job is done. 13 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 5 Page 106 Problem # © 2010 Pearson Education, Inc. All rights reserved To be solved in class

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EXAMPLE 6 Solving a Mixture Problem A full 6-quart radiator contains 75% water and 25% pure antifreeze. How much of this mixture should be drained and replaced by pure antifreeze so that the resulting 6-quart mixture is 50% pure antifreeze? Solution Step 1Find the quantity of the radiator mixture that should be drained (25% pure antifreeze) and replaced by pure (100%) antifreeze. 15 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 6 Solving a Mixture Problem Solution continued Let x = quarts 25% antifreeze drained. x = quarts pure antifreeze added. 0.25x = quarts pure antifreeze drained. Step 2 Pure antifreeze final mix Pure antifreeze original mix Pure antifreeze drained Pure antifreeze added =–+ (50% of 6) = (25% of 6) – (25% of x) + x Step 3 16 © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 6 Solving a Mixture Problem Solution continued Step 3 Step 4 Step 5Drain 2 quarts of mixture from the radiator. Step 6Drain 2 qt from the 6, leaves 4 qt: 1 qt pure antifreeze, 3 qt water. Add 2 qt antifreeze, now have 3 qt antifreeze, 3 qt water, so it’s 50% pure antifreeze solution. 17 © 2010 Pearson Education, Inc. All rights reserved

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