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CHAPTER 6

CHAPTER 6. Algebra: Equations and Inequalities. 6.3. Applications of Linear Equations. Objectives Use linear equations to solve problems. Solve a formula for a variable. Strategy for Solving Word Problems.

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CHAPTER 6

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  1. CHAPTER 6 Algebra: Equations and Inequalities

  2. 6.3 • Applications of Linear Equations

  3. Objectives Use linear equations to solve problems. Solve a formula for a variable.

  4. 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.

  5. Algebraic Translations of English Phrases

  6. Example: Education Pays Off • The bar graph shows the ten most popular college majors with median, or middlemost, starting salaries for recent college graduates.

  7. 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.

  8. 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.

  9. 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.

  10. Example: continued • starting salary of psychology majors: x = 35 • starting salary of business majors: x + 8 = 35 + 8 = 43 • starting salary of English majors: x + 3 = 35 + 3 = 38. • Step 5: Check the proposed solution in the wording of the problem. The solution checks.

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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. • x − 0.4x = 276 • 0.6x = 276 x = 460 The camera’s price before the reduction was $460.

  17. 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.

  18. 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

  19. 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 T – D = pm m m T – D = p m Example: Solving a Formula for One of its Variables

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