Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.3 Models and Applications Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

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

Problem Solving with Linear Equations A model is a mathematical representation of a real-world situation. We will use the following general steps in solving word problems: Step 1 Let x represent one of the unknown quantities. Step 2 Represent other unknown quantities in terms of x. Step 3 Write an equation in x that models the conditions. Step 4 Solve the equation and answer the question. Step 5 Check the proposed solution in the original wording of the problem.

Example: Using Linear Equations to Solve Problems The median starting salary of a computer science major exceeds that of an education major by $21 thousand. The median starting salary of an economics major exceeds that of an education major by $14 thousand. Combined, their median starting salaries are $140 thousand. Determine the median starting salaries of education majors, computer science majors, and economics majors with bachelor’s degrees.

Example: Using Linear Equations to Solve Problems (continued) Step 1 Let x represent one of the unknown quantities. x = median starting salary of an education major Step 2 Represent other unknown quantities in terms of x. x + 21 = median starting salary of a computer science major x + 14 = median starting salary of an economics major Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. starting salary of an education major is x = 35 starting salary of a computer science major is x + 21 = 56 starting salary of an economics major is x + 14 = 49

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. (continued) The median starting salary of an education major is $35 thousand, the median starting salary of a computer science major is $56 thousand, and the median starting salary of an economics major is $49 thousand.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem states that combined, the median starting salaries are $140 thousand. Using the median salaries we determined in Step 4, the sum is $35 thousand + $56 thousand + $49 thousand, or $140 thousand, which verifies the problem’s conditions.

Example: Using Linear Equations to Solve Problems You are choosing between two texting plans. Plan A has a monthly fee of $15 with a charge of $0.08 per text. Plan B has a monthly fee of $3 with a charge of $0.12 per text. 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. x = charge per text Step 2 Represent other unknown quantities in terms of x. Plan A = 0.08x +15 Plan B = 0.12x + 3

Example: Using Linear Equations to Solve Problems (continued) Step 3 Write an equation in x that models the conditions. Step 4 Solve the equation and answer the question. The cost of the two plans will be the same for 300 text messages.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem asks for how many text messages will the two plans be the same? The cost of Plan A for 300 text messages is 0.08(300) + 15 = 24 + 15 = 39 The cost of Plan B for 300 text messages is 0.12(300) + 3 = 36 + 3 = 39 For 300 text messages, the plan costs are equal, which verifies the problem’s conditions.

Example: Using Linear Equations to Solve Problems After a 30% price reduction, you purchase a new computer for $840. What was the computer’s price before the reduction? Step 1 Let x represent one of the unknown quantities. x = price before reduction Step 2 Represent other unknown quantities in terms of x. price of new computer = x – 0.3x Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. The price of the computer before the reduction was $1200. Step 5 Check the proposed solution in the original wording of the problem. The price before the reduction, $1200, minus the 30% reduction should equal the reduced price given in the original wording, $840.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. (continued) This verifies that the computer’s price before the reduction was $1200.

Example: Using Linear Equations to Solve Problems You inherited $5000 with the stipulation that for the first year the money had to be invested in two funds paying 9% and 11% annual interest. How much did you invest at each rate if the total interest earned for the year was $487? Step 1 Let x represent one of the unknown quantities. x = the amount invested at 9% Step 2 Represent other unknown quantities in terms of x. 5000 – x = the amount invested at 11% Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. the amount invested at 9% = x = 3150 the amount invested at 11% = 5000 – x = 1850 You should invest $3150 at 9% and $1850 at 11%.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem states that the total interest from the dual investments should be $487. The interest earned at 9% is 0.09(3150) = 283.50 The interest earned at 11% is 0.11(1850) = 203.50 The total interest is 283.50 + 203.50 = 487. This verifies that the amount invested at 9% should be $3150 and that the amount invested at 11% should be $1850.

Example: Using Linear Equations to Solve Problems The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions? Step 1 Let x represent one of the unknown quantities. x = the width of the basketball court Step 2 Represent other unknown quantities in terms of x. x + 44 = the length of the basketball court Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. the width of the basketball court is x = 50 ft the length of the basketball court is x + 44 = 94 ft The dimensions of the basketball court are 50 ft by 94 ft

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem states that the perimeter of the basketball court is 288 feet. If the dimensions are 50 ft by 94 ft, then the perimeter is 2(50) + 2(94) = 100 + 188 = 288. This verifies the conditions of the problem.

Solving a Formula For A Variable Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation.

Example: Solving a Formula for a Variable

Example: Solving a Formula for a Variable That Occurs Twice