Chapter P Prerequisites: Fundamental Concepts of Algebra 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 P.8 Modeling with Equations
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Use equations to solve problems. Objectives:
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Strategy for Solving Word Problems
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Application 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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Application (continued) 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. x = median starting salary of an education major x + 21 = median starting salary of a computer science major x + 14 = median starting salary of an economics major
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Application (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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Application (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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Application (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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Application 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. Step 2 Represent other unknown quantities in terms of x. Step 3 Write an equation in x that models the conditions. x = price before reduction price of new computer = x – 0.3x
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Application (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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Application (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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Application 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. Step 2 Represent other unknown quantities in terms of x. Step 3 Write an equation in x that models the conditions. x = the width of the basketball court x + 44 = the length of the basketball court
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Application (continued) Step 4 Solve the equation and answer the question. The dimensions of the basketball court are 50 ft by 94 ft. The width of the basketball court is x = 50 ft. The length of the basketball court is x + 44 = 94 ft.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 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) = = 288. This verifies the conditions of the problem.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 The Pythagorean Theorem
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Using the Pythagorean Theorem 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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Using the Pythagorean Theorem (continued) 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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Using the Pythagorean Theorem (continued) 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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Using the Pythagorean Theorem (continued) 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 a 2 + b 2 = c 2, then the triangle is a right triangle. The height of the tower is 120 yards.