Equation Solving Principles. Equations and Solutions. Zeros of Linear Functions. August 30, 2012

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1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling 1.6 Solving Linear Inequalities Aug 25 10:04 PM Aug 30 9:30 AM Equations and Solutions An equation is a statement that two expressions are equal. To solve an equation in one variable is to find all the values of the variable that make the equation true. Each of these numbers is a solution of the equation. The set of all solutions of an equation is its solution set. A linear equation in one variable is an equation that can be expressed in the form mx + b = 0, where m and b are real numbers and m 0. Equation Solving Principles The Addition Principle: If a = b is true, then a + c = b + c is true. The Multiplication Principle: If a = b is true, then ac = bc is true. Equations that have the same solution set are equivalent equations. Aug 25 10:09 PM Aug 25 10:14 PM Distance Formula The distance d traveled by an object moving at rate r in time t is given by d = rt. Simple Interest I = Prt I = the simple interest ($) P = the principal ($) r = the interest rate (%) t = time (years) Zeros of Linear Functions An input c of a function f is called a zero of the function, if the output for the function is 0 when the input is c. That is, c is a zero of f if f (c) = 0. A linear function f (x) = mx + b, with m 0, has exactly one zero. Aug 25 10:16 PM Aug 25 10:20 PM 1

2 A formula is an equation that can be used to model a situation. 134/4. Solve (6/5)y + 3 = 3/10 We have used the motion formula, d = r t. We have used the simple interest formula, I = Prt. We can use the equation solving principles to solve a formula for a given variable. 134/10. Solve 10x 3 = x Aug 25 10:22 PM 134/22. Solve 4x + 3 = 2x 7 134/37. Ocean Depth. The average depth of the Pacific Ocean is 14,040 ft. This is 8890 ft less than the sum of the average depths of the Atlantic Ocean and the Indian Ocean. The average depth of the Indian Ocean is 272 ft less than four fifths of the average depth of the Atlantic Ocean. (Source: Time Almanac, 2010) Find the average depth of the Indian Ocean. 134/30. Solve 4(3x + 2) 7 = 3(x 2) 135/38. Where the Textbook Dollar Goes. Of each dollar spent on textbooks at college bookstores, 22.3 cents goes to the college store for profit, store operations, and personnel. On average, a college student at a four year college spends $ 667 per year for textbooks. (Source: National Association of College Stores) How much of this expenditure goes to the college store? 135/40. Nutrition. A slice of carrot cake from the popular restaurant The Cheesecake Factory contains 1560 calories. This is three fourths of the average daily calorie requirement for many adults. ( Source: The Center for Science in the Public Interest) Find the average daily calorie requirement for these adults. 2

3 135/46. Commission vs. Salary. Juliet has a choice between receiving an $ 1800 monthly salary from Furniture by Design or a base salary of $ 1600 and a 4% commission on the amount of furniture she sells during the month. For what amount of sales will the two choices be equal? 136/52. Garden Dimensions. The children at Tiny Tots Day Care plant a rectangular vegetable garden with a perimeter of 39 m. The length is twice the width. Find the dimensions of the garden. Use this to graph, calculate perimeter and area of rectangle given the width and the relationship between length and width 136/56. Water Weight. Water accounts for 60% of a man's weight ( Source: ga.water.usgs.gov/edu). Jake weighs 186 lb. How much of his body weight is water? 136/60. Traveling Downstream. Angelos kayak travels 14 km/hr in still water. If the rivers current flows at a rate of 2 km/hr, how long will it take him to travel 20 km downstream? Aug 26 7:11 AM 136/64. Student Loans. Dimitri's two student loans total $ One loan is at 5% simple interest and the other is at 6% simple interest. At the end of 1 yr, Dimitri owes $ 492 in interest. What is the amount of each loan? 137/76. Find the zero of the linear function: f(x) = 2x + 7. Solve algebraically and graphically. 3

4 137/82. Find the zero of the linear function: f(x) = 3x Solve algebraically and graphically. In Exercises 87 92, use the given graph to find each of the following: (a) the x intercept and (b) the zero of the function. 138/90. Aug 30 2:59 PM Aug 30 3:08 PM In Exercises 87 92, use the given graph to find each of the following: (a) the x intercept and (b) the zero of the function. 138/92. Aug 30 9:07 AM 4

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