Precalculus Essentials Fifth Edition Chapter P Prerequisites: Fundamental Concepts of Algebra 1 If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed: 1) MathType Plugin 2) Math Player (free versions available) 3) NVDA Reader (free versions available) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

P.8 Modeling with Equations

Learning Objectives Use equations to solve problems.

Problem Solving with Equations A model is a mathematical representation of a real-world situation. We obtain models by translating from the ordinary language of English into the language of algebraic equations.

Strategy for Solving Word Problems (1 of 2) 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 unknown 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.

Strategy for Solving Word Problems (2 of 2) 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.

Example 1: Application (1 of 5) 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 1: Application (2 of 5) 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 1: Application (3 of 5) 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 1: Application (4 of 5) Step 4 Solve the equation and answer the question. 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 1: Application (5 of 5) 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 2: Application (1 of 3) 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 2: Application (2 of 3) 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 2: Application (3 of 3) Step 5 Check the proposed solution in the original wording of the problem. This verifies that the computer’s price before the reduction was $1200.

Example 3: Application (1 of 3) 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 3: Application (2 of 3) 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 3: Application (3 of 3) 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.

The Pythagorean Theorem The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse. If the legs have lengths a and b, and the hypotenuse has length c, then

Example 4: Using the Pythagorean Theorem (1 of 5) A radio tower is supported by two wires that are each 130 yards long and attached to the ground 50 yards from the base of the tower. How tall is the tower? Step 1 Let x represent one of the unknown quantities. x = the height of the tower Step 2 Represent other unknown quantities in terms of x. There are no other unknown quantities, so we skip this step.

Example 4: Using the Pythagorean Theorem (2 of 5) A radio tower is supported by two wires that are each 130 yards long and attached to the ground 50 yards from the base of the tower. How tall is the tower? Step 3 Write an equation in x that models the conditions.

Example 4: Using the Pythagorean Theorem (3 of 5) A radio tower is supported by two wires that are each 130 yards long and attached to the ground 50 yards from the base of the tower. How tall is the tower? Step 4 Solve the equation and answer the question. x represents the height of the tower, the measurement must be positive. We reject −120. Thus, the height of the tower is 120 yards.

Example 4: Using the Pythagorean Theorem (4 of 5) A radio tower is supported by two wires that are each 130 yards long and attached to the ground 50 yards from the base of the tower. How tall is the tower? Step 5 Check the proposed solution in the original wording of the problem. This can be checked using the converse of the Pythagorean Theorem: If a triangle has sides of lengths a, b, and c, where c is the length of the longest side, and if then the triangle is a right triangle.

Example 4: Using the Pythagorean Theorem (5 of 5) The height of the tower is 120 yards.