Algebra: Equations and Inequalities CHAPTER 6 Algebra: Equations and Inequalities

Applications of Linear Equations 6.3 Applications of Linear Equations

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

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.

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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