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Algebra: Equations and Inequalities
CHAPTER 6 Algebra: Equations and Inequalities
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Applications of Linear Equations
6.3 Applications of Linear Equations
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Objectives Use linear equations to solve problems. Solve a formula for a variable.
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Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the quantities in the problem. Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x. Step 3 Write an equation in x that models the verbal conditions of the problem. Step 4 Solve the equation and answer the problem’s question. Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
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Algebraic Translations of English Phrases
Addition Subtraction Multiplication Division sum more than increased by minus decreased by subtracted from difference between less than fewer than times product of percent of a number multiplied by twice divided by quotient reciprocal
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Example: Education Pays Off
The bar graph shows the ten most popular college majors with median, or middlemost, starting salaries for recent college graduates.
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Example: Education Pays Off
The median starting salary of a business major exceeds that of a psychology major by $8 thousand. The median starting salary of an English major exceeds that of a psychology major by $3 thousand. Combined, their median starting salaries are $116 thousand. Determine the median starting salaries of psychology majors, business majors, and English majors with bachelor’s degrees.
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Example 1: continued Step 1 Let x represent one of the unknown quantities. We know something about the median starting salaries of business majors and English majors: Business majors earn $8 thousand more than psychology majors and English majors earn $3 thousand more than psychology majors. We will let x = the median starting salary, in thousands of dollars, of psychology majors. x + 8 = the median starting salary, in thousands of dollars, of business majors. x + 3 = the median starting salary, in thousands of dollars, of English majors.
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Example: continued Step 3: Write an equation in x that models the conditions. x + (x + 8) + (x + 3) = 116 Step 4: Solve the equation and answer the question.
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Example: continued starting salary of psychology majors: x = 35 starting salary of business majors: x + 8 = = 43 starting salary of English majors: x + 3 = = 38. Step 5: Check the proposed solution in the wording of the problem. The solution checks.
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Example: Selecting Monthly Text Message Plan
You are choosing between two texting plans. Plan A has a monthly fee of $20.00 with a charge of $0.05 per text. Plan B has a monthly fee of $5.00 with a charge of $0.10 per text. Both plans include photo and video texts. For how many text messages will the costs for the two plans be the same? Step 1 Let x represent one of the unknown quantities. Let x the number of text messages for which the two plans cost the same.
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Example: continued Step 2 Represent other unknown quantities in terms of x. There are no other unknown quantities, so we can skip this step. Step 3 Write an equation in x that models the conditions. The monthly cost for plan A is the monthly fee, $20.00, plus the per-text charge, $0.05, times the number of text messages, x. The monthly cost for plan B is the monthly fee, $5.00, plus the per-text charge, $0.10, times the number of text messages, x.
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Example: continued Step 4 Solve the equation and answer the question. Because x represents the number of text messages for which the two plans cost the same, the costs will be the same for 300 texts per month.
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Example: continued Step 5 Check the proposed solution in the original wording of the problem. Cost for plan A = $20 + $0.05(300) = $20 + $15 = $35 Cost for plan B = $5 + $0.10(300) = $5 + $30 = $35. With 300 text messages, both plans cost $35 for the month. Thus, the proposed solution, 300 text messages, satisfies the problem’s conditions.
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Example: A Price Reduction on a Digital Camera
Your local computer store is having a terrific sale on digital cameras. After a 40% price reduction, you purchase a digital camera for $276. What was the camera’s price before the reduction? Step 1 Let x represent one of the unknown quantities. We will let x = the original price of the digital camera prior to the reduction. Step 2 Represent other unknown quantities in terms of x. There are no other unknown quantities to find, so we can skip this step.
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Example: continued Step 3 Write an equation in x that models the conditions. The camera’s original price minus the 40% reduction is the reduced price, $276. x − 0.4x = 276 Step 4 Solve the equation and answer the question. 0.6x = 276 x = 460 The camera’s price before the reduction was $460.
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Example: continued Step 5 Check the proposed solution in the original wording of the problem. The price before the reduction, $460, minus the 40% reduction should equal the reduced price given in the original wording, $276: 460 − 40% of 460 = 460 − 0.4(460) = 460 − 184 = 276. This verifies that the digital camera’s price before the reduction was $460.
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Example: Solving a Formula for a Variable
Solve the formula P = 2l + 2w for l. First, isolate 2l on the right by subtracting 2w from both sides. Then solve for l by dividing both sides by 2. P = 2l + 2w P − 2w = 2l + 2w − 2w P − 2w = 2l
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Example: Solving a Formula for One of its Variables
The total price of an article purchased on a monthly deferred payment plan is described by the following formula: T is the total price, D is the down payment, p is the monthly payment, and m is the number of months one pays. Solve the formula for p. T – D = D – D + pm T – D = pm m m T – D = p m
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