Transcription
Elementary Algebra
Elementary AlgebraKatherine YoshiwaraLos Angeles Pierce CollegeSeptember 17, 2020
2019 Katherine YoshiwaraPermission is granted to copy, distribute and/or modify this document underthe terms of the GNU Free Documentation License, Version 1.2 or any laterversion published by the Free Software Foundation; with no Invariant Sections,no Front-Cover Texts, and no Back-Cover Texts. A copy of the license isincluded in the appendix entitled “GNU Free Documentation License.” Alltrademarks are the registered marks of their respective owners.
PrefaceElementary Algebra: Equations and Graphs was written with two goalsin mind: to present the skills of algebra in the context of modeling and problemsolving, and to engage students in an activity-based course.Throughout the book, we use applications to motivate algebraic skills. Wepresent ideas using verbal, numerical, and graphical tools. We use graphsextensively to illustrate algebraic techniques and to help students visualize relationships between variables. In the Homework Problems we break up skillspractice into smaller sets of exercises and combine them with conceptual questions, graphing, and applications of various types.We have included a number of features to encourage student participation. The reading is broken into small segments by using boxes around Examples, definitions, and rules, and flags including Caution, Look Closer,and Look Ahead. We want students to learn how to read a math book, so we have includedReading Questions that mirror the content of the lesson. For example, thefirst Reading Question is ”A numerical quantity that changes over time orin different situations is called a .” Ideally, students would answer thesequestions before class, perhaps through an on-line homework system. Each section ends with a Skills Warm-Up (with answers) for studentsto complete on their own. The Skills Warm-Up problems review anarithmetic or algebraic technique needed for the Lesson that follows. Because choosing appropriate scales for the axes is a time-consumingtask for beginning students, the text includes labeled grids for most ofthe graphing exercises. Ready-made grids allow students to considera wider range of examples (with ”harder” numbers) and to focus on theproperties of the graphs, such as intercepts and slope, and on interpretingthe information given by the graph. If students are using technology tocreate graphs, the grids can help them choose an appropriate window.The textbook is accompanied by an Activities Workbook that provides aLesson for each section in the book. These Lessons consist of Activities forstudents to complete in groups or with guidance from the instructor; or theycan be used as support for a lecture format. Each Lesson ends with a Wrap-Upand a set of Homework Preview exercises.Chapter Reviews include a Summary, Review Questions for writing or discussion, and a set of Review Problems. Answers to odd-numbered Homeworkproblems are provided, as well as a Glossary of mathematical terms. A Reviewof Arithmetic Skills, with Exercises, appears in Appendix A.The content of the text follows our ”guiding principles:”iv
v Develop the concepts and skills of algebra in context. Use applicationsto motivate and illustrate techniques. Don’t frontload with ”theory” orgeneral principles. Keep reading sections short and keep terminology to a minimum. A solid understanding of basic algebra is more important than facilitywith complicated algorithms and procedures. Include graphing from the beginning. The ability to read and analyzegraphs is just as important as mastering algebraic skills. Emphasize theconnection between equations and their graphs. Present material by increasing levels of sophistication, not by taxonomy.In other words, don’t have one chapter on simplifying expressions, andthen a separate chapter on solving all kinds of equations, and so on.Instead, develop algebraic skills (writing expressions, solving equations,modeling) simultaneously and incrementally. Review arithmetic skills ”just-in-time” as needed (and often). Some more advanced topics can be left for later courses, e.g. factoringsums of cubes.Consequently, our text may differ in some ways from other treatments ofthe subject. We begin with writing algebraic expressions instead of manipulating expressions. We use tables and graphs to motivate expressions. Graphing appears in Chapter 1 and is continued throughout the book.Graphs are not restricted to 10 10 grids. We solve equations both algebraically and graphically. Standard word problems are de-emphasized in favor of modeling. Quadratic expressions and equations are treated before polynomials. Westart with quadratic equations that can be solved by extracting roots. We try to minimize factoring in the study of algebraic fractions.An Instructor’s Manual for the text is also available. The Manual containsobjectives and teaching notes for each section of the text, as well as suggestedconcept questions and topics for writing or discussion. The teaching notesinclude suggestions for using the Activities booklet and how to structure classtime.Katherine YoshiwaraAtascadero, CA 2019
ContentsPrefaceiv1 Variables11.1Variables . . . . . . . . . .What is a Variable?. . . . . . .Using Letters to Represent VariablesConnections Between Variables . .Skills Warm-Up . . . . . . . .Lesson . . . . . . . . . . .Homework Preview . . . . . . .Homework 1.1 . . . . . . . . .1.2 Algebraic Expressions . . . . .Writing an Algebraic Expression . .Sums . . . . . . . . . . . .Products. . . . . . . . . . .Differences . . . . . . . . . .Quotients . . . . . . . . . .Evaluating an Algebraic Expression .Some Useful Algebraic Formulas . .Skills Warm-Up . . . . . . . .Lesson . . . . . . . . . . .Homework Preview . . . . . . .Homework 1.2 . . . . . . . . .1.3 Equations and Graphs . . . . .Anatomy of a Graph . . . . . .Graphing an Equation . . . . . .Choosing Scales for the Axes . . .Solving Equations with Graphs . .Skills Warm-Up . . . . . . . .Lesson . . . . . . . . . . .Homework Preview . . . . . . .Homework 1.3 . . . . . . . . .1.4 Solving Equations. . . . . . .What is an Equation? . . . . . .Trial and Error . . . . . . . .Inverse Operations . . . . . . .Solving Equations . . . . . . .Problem Solving with Equations . 4142474748485051
CONTENTSSkills Warm-Up . . . . . . .Lesson . . . . . . . . . .Homework Preview . . . . . .Homework 1.4 . . . . . . . .1.5 Order of Operations . . . . .Addition and Multiplication . . .Subtraction and Division . . . .Combined Operations . . . . .Parentheses. . . . . . . . .Fraction Bars . . . . . . . .Summary . . . . . . . . .Algebraic Expressions . . . . .Skills Warm-Up . . . . . . .Lesson . . . . . . . . . .Homework Preview . . . . . .Homework 1.5 . . . . . . . .1.6 Chapter 1 Summary and ReviewLesson 1.1 Variables . . . . .Lesson 1.2 Algebraic Expressions .Lesson 1.3 Equations and Graphs .Lesson 1.4 Solving Equations . .Lesson 1.5 Order of Operations .Review Questions . . . . . .Review Problems. . . . . . .vii.2 Linear Equations2.1Signed Numbers . . . . . . . . .Types of Numbers . . . . . . . . .Adding Two Numbers with the Same SignAdding Two Numbers with Opposite SignsSubtracting a Positive Number. . . . .Subtracting a Negative Number . . . .Subtracting Signed Numbers . . . . .Multiplying Signed Numbers . . . . .Dividing Signed Numbers. . . . . . .Quotients Involving Zero . . . . . . .Skills Warm-Up . . . . . . . . . .Lesson . . . . . . . . . . . . .Homework Preview . . . . . . . . .Homework 2.1 . . . . . . . . . . .2.2 Expressions and Equations . . . . .Writing Algebraic Expressions . . . . .Evaluating Algebraic Expressions . . . .Negative Numbers . . . . . . . . .Skills Warm-Up . . . . . . . . . .Lesson . . . . . . . . . . . . .Homework Preview . . . . . . . . .Homework 2.2 . . . . . . . . . . 1. 81. 81. 82. 83. 84. 85. 85. 87. 88. 89. 90. 91. 94. 94. 98. 98. 98. 99.100.101.103.104
CONTENTSviii2.3Graphs of Linear Equations . . . . . .Graphing Equations . . . . . . . . .Cartesian Coordinate System . . . . . .Skills Warm-Up . . . . . . . . . . .Lesson . . . . . . . . . . . . . .Homework Preview . . . . . . . . . .Homework 2.3 . . . . . . . . . . . .2.4 Linear Equations and Inequalities . . . .Solving Equations . . . . . . . . . .Applied Problems . . . . . . . . . .Solving Inequalities . . . . . . . . . .Compound Inequalities . . . . . . . .Skills Warm-Up . . . . . . . . . . .Lesson . . . . . . . . . . . . . .Homework Preview . . . . . . . . . .Homework 2.4 . . . . . . . . . . . .2.5 Like Terms . . . . . . . . . . . .Equivalent Expressions . . . . . . . .Like Terms . . . . . . . . . . . . .Adding and Subtracting Like Terms . . . .Removing Parentheses . . . . . . . . .Solving Equations and Inequalities . . . .Skills Warm-Up . . . . . . . . . . .Lesson . . . . . . . . . . . . . .Homework Preview . . . . . . . . . .Homework 2.5 . . . . . . . . . . . .2.6 Chapter 2 Summary and Review . . . .Lesson 2.1 Signed Numbers . . . . . . .Section 2.2 Expressions and Equations . . .Section 2.3 Graphs of Linear Equations . .Section 2.4 Linear Equations and InequalitiesSection 2.5 Like Terms. . . . . . . . .Review Questions . . . . . . . . . .Review Problems. . . . . . . . . . .3 Graphs of Linear Equations3.1Intercepts . . . . . . . .Intercepts of a Line . . . . . .The Intercept Method of GraphingInterpreting the Intercepts . . .Skills Warm-Up . . . . . . .Lesson . . . . . . . . . .Homework Preview . . . . . .Homework 3.1 . . . . . . . .3.2 Ratio and Proportion . . . .Ratios. . . . . . . . . . .Rates . . . . . . . . . . .Proportions. . . . . . . . .Proportional Variables . . . . .Skills Warm-Up . . . . . . .Lesson . . . . . . . . . .Homework Preview . . . . . .Homework 3.2 . . . . . . . 172.172
CONTENTS3.3Slope . . . . . . . . . . .Rate of Change . . . . . . . .Slope . . . . . . . . . . . .Meaning of Slope. . . . . . . .Geometrical Meaning of Slope . . .Skills Warm-Up . . . . . . . .Lesson . . . . . . . . . . .Homework Preview . . . . . . .Homework 3.3 . . . . . . . . .3.4 Slope-Intercept Form . . . . .Writing a Linear Equation . . . .Slope-Intercept Method of Graphing.Finding the Slope-Intercept Form . .Skills Warm-Up . . . . . . . .Lesson . . . . . . . . . . .Homework Preview . . . . . . .Homework 3.4 . . . . . . . . .3.5 Properties of Lines . . . . . .Parallel and Perpendicular Lines . .Equations for Horizontal and VerticalDistance Between Points . . . . .Subscript Notation . . . . . . .A New Formula for Slope. . . . .Skills Warm-Up . . . . . . . .Lesson . . . . . . . . . . .Homework Preview . . . . . . .Homework 3.5 . . . . . . . . .3.6 Chapter 3 Summary and Review .Lesson 3.1 Intercepts . . . . . .Lesson 3.2 Ratio and Proportion . .Lesson 3.3 Slope . . . . . . . .Lesson 3.4 Slope-Intercept Form . .Lesson 3.5 Properties of Lines . . .Review Questions . . . . . . .Review Problems. . . . . . . .ix. . . . . . . . . . . . . . . . . . . .Lines. . . . . . . . . . . . . . . .4 Applications of Linear Equations4.1The Distributive Law . . . . . . .Products. . . . . . . . . . . . .The Distributive Law . . . . . . . .Solving Equations and Inequalities . . .Algebraic Expressions . . . . . . . .Skills Warm-Up . . . . . . . . . .Lesson . . . . . . . . . . . . .Homework Preview . . . . . . . . .Homework 4.1 . . . . . . . . . . .4.2 Systems of Linear Equations . . . . .What is a System of Equations? . . . .Solving a System of Equations by GraphingInconsistent and Dependent Systems . .Problem Solving with Systems . . . . .Skills Warm-Up . . . . . . . . . .Lesson . . . . . . . . . . . . 246.247
CONTENTSxHomework Preview . . . . . . . .Homework 4.2 . . . . . . . . . .4.3 Algebraic Solution of Systems . . .Introduction . . . . . . . . . .Substitution Method . . . . . . .Elimination Method . . . . . . .Inconsistent and Dependent Systems .Skills Warm-Up . . . . . . . . .Lesson . . . . . . . . . . . .Homework Preview . . . . . . . .Homework 4.3 . . . . . . . . . .4.4 Problem Solving with Systems . . .Introduction . . . . . . . . . .Interest . . . . . . . . . . . .Mixtures . . . . . . . . . . . .Motion . . . . . . . . . . . .Skills Warm-Up . . . . . . . . .Lesson . . . . . . . . . . . .Homework Preview . . . . . . . .Homework 4.4 . . . . . . . . . .4.5 Point-Slope Formula . . . . . . .Finding an Equation for a Line . . .The Line Through Two Points . . . .Applications . . . . . . . . . .Skills Warm-Up . . . . . . . . .Lesson . . . . . . . . . . . .Homework Preview . . . . . . . .Homework 4.5 . . . . . . . . . .4.6 Chapter 4 Summary and Review . .Lesson 4.1 The Distributive Law . . .Lesson 4.2 Systems of Linear EquationsLesson 4.3 Algebraic Solution of SystemsLesson 4.4 Applications of Systems . .Lesson 4.5 Point-Slope Form . . . .Review Questions . . . . . . . .Review Problems. . . . . . . . .5 Exponents and Roots5.1Exponents . . . . . .What is an Exponent? . . .Powers of Negative NumbersUsing a Calculator . . . .Powers of Variables . . . .Order of Operations . . .Like Terms . . . . . . .Skills Warm-Up . . . . .Lesson . . . . . . . .Homework Preview . . . .Homework 5.1 . . . . . 9.299.301.301.302.303.305.306.306.308.309
CONTENTSxi5.2Square Roots and Cube Roots . . . .What is a Square Root? . . . . . . .Radicals . . . . . . . . . . . . .Square Root of a Negative Number . . .Rational and Irrational Numbers . . . .Decimal Form . . . . . . . . . . .Real Numbers . . . . . . . . . . .Order of Operations . . . . . . . .Cube Roots. . . . . . . . . . . .Skills Warm-Up . . . . . . . . . .Lesson . . . . . . . . . . . . .Homework Preview . . . . . . . . .Homework 5.2 . . . . . . . . . . .5.3 Using Formulas . . . . . . . . .Volume and Surface Area. . . . . . .Solving Equations with x2 . . . . . .Pythagorean Theorem . . . . . . . .Solving for One Variable . . . . . . .Skills Warm-Up . . . . . . . . . .Lesson . . . . . . . . . . . . .Homework Preview . . . . . . . . .Homework 5.3 . . . . . . . . . . .5.4 Products of Binomials . . . . . . .Products of Variables . . . . . . . .Using the Distributive Law . . . . . .Some Terminology . . . . . . . . .Multiplying Binomials . . . . . . . .The Four Terms in a Binomial Product .Skills Warm-Up . . . . . . . . . .Lesson . . . . . . . . . . . . .Homework Preview . . . . . . . . .Homework 5.4 . . . . . . . . . . .5.5 Chapter 5 Summary and Review . . .Lesson 5.1 Exponents . . . . . . . .Lesson 5.2 Square Roots and Cube Roots .Lesson 5.3 Using Formulas . . . . . .Lesson 5.4 Products of Binomials . . . .Review Questions . . . . . . . . .Review Problems. . . . . . . . . .6 Quadratic Equations6.1Extracting Roots . . . . .Quadratic Equations . . . .Graphs of Quadratic EquationsSolving Quadratic Equations .Extraction of Roots . . . . .Solving Formulas. . . . . .Skills Warm-Up . . . . . .Lesson . . . . . . . . .Homework Preview . . . . .Homework 6.1 . . . . . . 47.347353.353.353.354.354.356.358.358.359.361.362
CONTENTSxii6.2Some Quadratic Models . . . . . . . . . .Revenue . . . . . . . . . . . . . . . . .The Zero-Factor Principle . . . . . . . . . .Factoring . . . . . . . . . . . . . . . .Solving Quadratic Equations by Factoring . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 6.2 . . . . . . . . . . . . . . .6.3 Solving Quadratic Equations by Factoring . . . .The Zero-Factor Principle . . . . . . . . . .Factoring Quadratic Trinomials . . . . . . . .Solving a Quadratic Equation . . . . . . . . .Application . . . . . . . . . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 6.3 . . . . . . . . . . . . . . .6.4 Graphing Quadratic Equations . . . . . . . .x-Intercepts of a Parabola . . . . . . . . . .The Vertex . . . . . . . . . . . . . . . .Graphing Parabolas. . . . . . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 6.4 . . . . . . . . . . . . . . .6.5 The Quadratic Formula . . . . . . . . . .Using the Quadratic Formula . . . . . . . . .Non-Real Solutions . . . . . . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 6.5 . . . . . . . . . . . . . . .6.6 Chapter 6 Summary and Review . . . . . . .Lesson 6.1 Extracting Roots. . . . . . . . . .Lesson 6.2 Some Quadratic Models . . . . . . .Lesson 6.3 Solving Quadratic Equations by FactoringLesson 6.4 Graphing Quadratic Equations. . . . .Lesson 6.5 The Quadratic Formula . . . . . . .Review Questions . . . . . . . . . . . . .Review Problems. . . . . . . . . . . . . .7 Polynomials7.1Polynomials . . . .What is a Polynomial?.Degree . . . . . .Evaluating PolynomialsLike Terms . . . . .Adding Polynomials. .Subtracting PolynomialsApplication . . . . .Skills Warm-Up . . .Lesson . . . . . 421.422
CONTENTSxiiiHomework Preview . . . . . . . . . . . . .Homework 7.1 . . . . . . . . . . . . . . .7.2 Products of Polynomials . . . . . . . . . .Products of Powers . . . . . . . . . . . . .Products of Monomials . . . . . . . . . . .Multiplying by a Monomial . . . . . . . . . .Products of Polynomials . . . . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 7.2 . . . . . . . . . . . . . . .7.3 More About Factoring . . . . . . . . . . .Quotients of Powers. . . . . . . . . . . . .Quotients of Monomials . . . . . . . . . . .Greatest Common Factors . . . . . . . . . .Factoring Quadratic Trinomials by Guess-and-Check .Quadratic Trinomials in Two Variables . . . . . .Combining Factoring Techniques . . . . . . . .Optional Extension: The Box Method . . . . . .Factoring Quadratic Trinomials by the Box Method .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 7.3 . . . . . . . . . . . . . . .7.4 Special Products and Factors . . . . . . . .Squares of Monomials . . . . . . . . . . . .Squares of Binomials . . . . . . . . . . . .Difference of Two Squares . . . . . . . . . .Factoring Special Products . . . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 7.4 . . . . . . . . . . . . . . .7.5 Chapter 7 Summary and Review . . . . . . .Lesson 7.1 Polynomials . . . . . . . . . . .Lesson 7.2 Products of Polynomials . . . . . . .Lesson 7.3 More About Factoring . . . . . . . .Lesson 7.4 Special Products and Factors . . . . .Review Questions . . . . . . . . . . . . .Review Problems. . . . . . . . . . . . . .8 Algebraic Fractions8.1Algebraic Fractions . . . . .What is an Algebraic Fraction? .Applications of Algebraic FractionsReducing Fractions . . . . . .Negative of a Binomial . . . .The Opposite of a Binomial . . .Skills Warm-Up . . . . . . .Lesson . . . . . . . . . .Homework Preview . . . . . .Homework 8.1 . . . . . . . 469
CONTENTSxiv8.2Operations on Fractions . . . . . . .Products of Fractions . . . . . . . . .Quotients of Fractions . . . . . . . . .Adding Like addition . . . . . . . . .Subtracting Like Fractions . . . . . . .Polynomial Division. . . . . . . . . .Skills Warm-Up . . . . . . . . . . .Lesson . . . . . . . . . . . . . .Homework Preview . . . . . . . . . .Homework 8.2 . . . . . . . . . . . .8.3 Lowest Common Denominator . . . . .Building Fractions . . . . . . . . . .Adding and Subtracting Unlike Fractions . .Finding the Lowest Common Denominator .Skills Warm-Up . . . . . . . . . . .Lesson . . . . . . . . . . . . . .Homework Preview . . . . . . . . . .Homework 8.3 . . . . . . . . . . . .8.4 Equations with Fractions . . . . . . .”Clearing” Fractions . . . . . . . . .Using an LCD to Clear Fractions . . . . .Variables in the Denominator . . . . . .Extraneous Solutions . . . . . . . . .Formulas. . . . . . . . . . . . . .Applications . . . . . . . . . . . .Skills Warm-Up . . . . . . . . . . .Lesson . . . . . . . . . . . . . .Homework Preview . . . . . . . . . .Homework 8.4 . . . . . . . . . . . .8.5 Chapter 8 Summary and Review . . . .Lesson 8.1 Algebraic Fractions . . . . . .Lesson 8.2 Operations on Algebraic FractionsLesson 8.3 Lowest Common Denominator . .Lesson 8.4 Equations with Fractions . . . .Review Questions . . . . . . . . . .Review Problems. . . . . . . . . . .9 More About Exponents and Roots9.1Laws of Exponents . . .Products and Quotients . .Power of a Power. . . . .Power of a Product . . . .Power of a Quotient . . .Using the Laws of ExponentsSkills Warm-Up . . . . .Lesson . . . . . . . .Homework Preview . . . .Homework 9.1 . . . . . 0.510.511.512.512.513.515.515.516.517
CONTENTS9.2Negative Exponents and Scientific Notation . . .Zero as an Exponent . . . . . . . . . . . .Negative Exponents. . . . . . . . . . . . .Laws of Exponents . . . . . . . . . . . . .Powers of Ten . . . . . . . . . . . . . . .Scientific Notation . . . . . . . . . . . . .Calculators and Scientific Notation . . . . . . .Mental Calculation with Scientific Notation . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 9.2 . . . . . . . . . . . . . . .9.3 Properties of Radicals . . . . . . . . . . .Product and Quotient Rules. . . . . . . . . .Simplifying Square Roots. . . . . . . . . . .Square Root of a Variable Expression . . . . . .Sums and Differences . . . . . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 9.3 . . . . . . . . . . . . . . .9.4 Operations on Radicals. . . . . . . . . . .Products. . . . . . . . . . . . . . . . .Quotients . . . . . . . . . . . . . . . .Fractions . . . . . . . . . . . . . . . .Rationalizing the Denominator. . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 9.4 . . . . . . . . . . . . . . .9.5 Equations with Radicals . . . . . . . . . .Radical Equations . . . . . . . . . . . . .Extraneous Solutions . . . . . . . . . . . .Equations with Cube Roots . . . . . . . . . .Extraction of Roots . . . . . . . . . . . . .Skills Warm-Up . . . . . . . . . . . . . .Lesson . . . . . . . . . . . . . . . . .Homework Preview . . . . . . . . . . . . .Homework 9.5 . . . . . . . . . . . . . . .9.6 Chapter 9 Summary and Review . . . . . . .Lesson 9.1 Laws of Exponents . . . . . . . . .Lesson 9.2 Negative Exponents and Scientific NotationLesson 9.3 Properties of Radicals . . . . . . . .Lesson 9.4 Operations on Radicals . . . . . . .Lesson 9.5 Equations with Radicals . . . . . . .Review Questions . . . . . . . . . . . . .Review Problems. . . . . . . . . . . . . 63Index646
Chapter 1VariablesVariablesWhat is a Variable?Definition 1.1 Variable.A variable is a numerical quantity that changes over time or in differentsituations.We can show the values of a variable in a table or a graph.Example 1.2Life expectancy is the average age to which people will live. The tablebelow shows that your life expectancy depends on the year of your birth.Thus, life expectancy is a variable.Year Born Life 70197071198074199075a To what age could people born in 1940 expect to live?b In what year did people’s life expectancy reach 70 years of age?Solution.a People born in 1940 lived to 63 years of age on average.b Life expectancy reached 70 years of age in 1960.1
CHAPTER 1. VARIABLES2Example 1.3Average salary (in thousands of dollars)The graph below shows the average annual salaries of NFL footballplayers, starting in 1940. The average salary of NFL football players isa variable.40035030025020015010050019401960 1970 1980 1990Yeara Use the graph to estimate the average annual salary of NFL football players in 1975.b By how much did salaries increase from 1975 to 1980?Solution.a In 1975, the average NFL player’s salary was 40,000.b In 1980, the average salary was 80,000, so salaries had increasedfrom 40,000 to 80,000, or by 80, 000 40, 000 40, 000Reading QuestionsRQ 1.4 A numerical quantity that changes over time or in different situationsis called a.Answer. variableRQ 1.5 We can show the values of a variable in aor a.Answer. table, graphLook Closer.By displaying the values of a variable in a table or a graph, we sometimessee trends or patterns in those values. For example, we see that lifeexpectancy has been increasing slowly over time, and that salaries ofNFL football players have been increasin
Elementary Algebra: Equations and Graphs was written with two goals in mind: to present the skills of algebra in the context of modeling and problem solving, and to engage students in an activity-based course. Throughout the book, we use applications to motivate algebraic skills. We present ideas using verbal, numerical, and graphical tools.
