SAT. New. Acing. Math. the PDF. Created for the New SAT Exam!

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1 Created for the New SAT Eam! Full-Length Practice Tests 0 Chapter Tests Definitions, Theorems, and Key Formulas Hundreds of Eamples and Eercises * SAT is a registered trademark of the College Entrance Eamination Board, which is not affiliated with this book Acing the New SAT Math PDF

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3 Acing the New SAT Math by Thomas Hyun GREENHALL PUBLISHING THOUSAND OAKS, CA

4 Copyright 06 by Greenhall Publishing All rights reserved. No part of this book may be used or reproduced in any manner without the written permission of the publisher. This edition published by Greenhall Publishing Greenhall Publishing Thousand Oaks, CA 98 greenhallpublishing.com Written by Thomas Hyun Cover design by Hespenheide Design Printed in the United States of America ISBN-0: ISBN-: SAT is a registered trademark of the College Entrance Eamination Board, which is not affiliated with this book.

5 CONTENTS To the Students Part A Math Review I. Heart of Algebra vi Chapter : The Language and Tools of Algebra - Variables and Epressions - Eponents and Order of Operations 7 - Simplifying Algebraic Epressions 9 - Rational, Irrational, and Decimal Chapter Practice Test Answers and Eplanations Chapter : Solving Linear Equations 9 - Writing Equations 9 - Solving Equations - Solving Equations with Variables on Both Sides - Equations with No Solution and Identity - Solving for a Specific Variable 7 Chapter Practice Test 9 Answers and Eplanations Chapter : Functions and Linear Equations 7 - Relations and Functions 7 - Rate of Change and Slope 0 - Slope-Intercept Form and Point-Slope Form - Parallel and Perpendicular Lines - Solving Systems of Linear Equations 6-6 Absolute Value Equations 0 Chapter Practice Test Answers and Eplanations Chapter : Linear Inequalities and Graphs 6 - Solving Inequalities 6 - Compound and Absolute Value Inequalities 6 - Graphing Inequalities in Two Variables 6 - Graphing Systems of Inequalities 67 Chapter Practice Test 69 Answers and Eplanations 7

6 ii Contents Chapter : Word Problems in Real-Life Situation 77 - Solving Word Problems Using Linear Models 77 - Solving Word Problems with Equations 79 - Solving Word Problems Using Systems of Equations 8 - Solving Word Problems Using Inequalities 8 Chapter Practice Test 8 Answers and Eplanations 87 II. Statistics and Data Analysis Chapter 6: Ratios, Rates, and Proportions 9 6- Ratios and Rates 9 6- Proportions Ratios, Rates, and Proportions Word Problems 99 Chapter 6 Practice Test 0 Answers and Eplanations 0 Chapter 7: Percents Percent of Change Percents and Equations 7- Percent Word Problems Chapter 7 Practice Test Answers and Eplanations 7 Chapter 8: Statistics 8- Mean, Median, Mode, and Range 8- Standard Deviation 8- Graphical Displays 8- Scatter Plots 7 8- Populations, Samples, and Random Selection 9 Chapter 8 Practice Test Answers and Eplanations Chapter 9: Categorical Data and Probability 9 9- Permutations and Combinations 9 9- Rules of Probability 9- Categorical Data and Conditional Probabilities Chapter 9 Practice Test Answers and Eplanations 7

7 Contents iii III. Advanced Math Chapter 0: Law of Eponents and Polynomials 0- Laws of Eponents and Scientific Notation 0- Adding, Subtracting, Multiplying, and Dividing Polynomials 6 0- FOIL Method and Special Products 9 0- Prime Factorization, GCF, and LCM 6 0- Factoring Using the Distributive Property 6 Chapter 0 Practice Test 66 Answers and Eplanations 68 Chapter : Quadratic Functions 7 - Graphs of Quadratic Equations 7 - Factoring Trinomials 76 - Factoring Differences of Squares and Perfect Square Trinomials 79 - Solving Quadratic Equations by Completing the Square 8 - Quadratic Formula and the Discriminant 8-6 Solving Systems Consisting Linear and Quadratic Equations 86 Chapter Practice Test 88 Answers and Eplanations 90 Chapter : Composition, Recursion, and Eponential Functions 97 - Composition of Functions 97 - Recursive Formula 99 - Eponential Functions and Graphs 0 - Eponential Growth and Decay 0 Chapter Practice Test 0 Answers and Eplanations 07 Chapter : Polynomial and Radical Functions - Polynomial Functions and Their Graphs - Remainder Theorem and Factor Theorem - Radical Epressions 7 - Solving Radical Equations 0 - Comple Numbers Chapter Practice Test Answers and Eplanations 7

8 iv Contents Chapter : Rational Epressions - Rational Epressions - Solving Rational Equations 6 - Direct, Inverse, and Joint Variation 8 - Solving Word Problems Using Rational Equations 0 Chapter Practice Test Answers and Eplanations Chapter : Trigonometric Functions - Trigonometric Ratios of Acute Angles - The Radian Measure of an Angle - Trigonometric Functions of General Angles and the Unit Circle Chapter Practice Test 9 Answers and Eplanations 6 IV. Geometry Chapter 6: Lines and Angles Lines, Segments, and Rays Angles Parallel and Perpendicular Lines 7 Chapter 6 Practice Test 7 Answers and Eplanations to Eercises and Practice Test 7 Chapter 7: Triangles Angles of a Triangles Pythagorean Theorem and Special Right Triangles 8 7- Similar Triangles and Proportional Parts 8 7- Area of a Triangle 8 Chapter 7 Practice Test 87 Answers and Eplanations 89 Chapter 8: Polygons 9 8- Parallelograms 9 8- Rectangles, Squares, and Trapezoids Regular Polygons 99 Chapter 8 Practice Test 0 Answers and Eplanations 0

9 Contents v Chapter 9: Circles Arcs, Angles, and Tangents Arc Lengths and Areas of Sectors 9- Inscribed Angles 9- Arcs and Chords 9- Equations of Circles 7 Chapter 9 Practice Test 9 Answers and Eplanations Chapter 0: Surface Areas and Volumes 7 0- Prisms 7 0- Cylinders and Spheres 9 0- Pyramids and Cones Chapter 0 Practice Test Answers and Eplanations Part B Practice Tests PSAT 0 Practice Test Answers and Eplanations 60 SAT Practice Test 67 Answers and Eplanations 88 SAT Practice Test 99 Answers and Eplanations

10 vi To the Students To the Students If you re reading this, you are on your way to a better score on the New SAT math! Below are guidelines for getting the most out of this book, as well as information about changes to the math section of the New SAT. Using this Book This book helps students review and master mathematical concepts in the most concise and straightforward manner possible. Each of the twenty chapters in this book teaches a particular group of mathematical concepts you need to know for the New SAT. Each chapter contains lessons that convey key concepts along with illustrations and diagrams. Important vocabularies are printed in boldface. Net to the key terms and illustrations, you will find eamples, with complete solutions, that apply the application of the concepts you have just learned. Keep a pencil and sheet of paper handy. Follow along, working out the sample questions yourself this will help you later in solving the practice problems. This book is not only helpful for SAT Math, but is also a very useful supplement for high school math courses. Eercise Sets and Practice Tests Each lesson includes a set of practice problems for the lesson. Each chapter includes a practice test followed by answers and eplanations, to ensure that you master the material. PSAT and SAT Practice Tests At the end of the book, you will find one PSAT and two SAT practice tests. For maimum benefit, these should be taken under realistic testing-center conditions timed and free from outside distractions. About the New SAT Math Content Starting in spring of 06, the math portion of the SAT will cover Algebra I and II, Geometry, parts of Trigonometry, and parts of Statistics. For the new eam students must also be familiar with materials typically taught in Algebra II, Trigonometry, and Statistics. The math portion of the SAT will consist of two sections one -minute no-calculator section and one -minute calculator section., for a total of 80 minutes in math testing. The new PSAT/NMSQT will also include the new topics, although the SAT will test for these concepts at a higher level. The math portion of the PSAT/NMSQT will consist of two sections one -minute no-calculator section and one -minute calculator section, for a total of 70 minutes in math testing. Math questions on the SAT and PSAT fall into two main categories: multiple choice (with four answer choices given) and student produced response questions. Student produced response questions, commonly referred to as grid-ins, require students to formulate their own answers, then enter the numeric values into a special grid. Multiple choice questions have no penalty for guessing!

11 vii To the Students No-Calculator Section Multiple Choice (MC) Number of Questions SAT Time Allocated Number of Questions PSAT Time Allocated Student-Produced Response (SPR - Grid-In) minutes minutes Sub Total 0 7 Calculator Section Number of Questions Time Allocated Number of Questions Time Allocated Multiple Choice (MC) 0 7 Student-Produced Response (SPR - Grid-In) Sub Total 8 8 minutes Total 8 80 minutes 8 70 minutes minutes Study Timetable As with any test, students will see the best results by studying consistently over at least several weeks before the eam date, rather than trying to cram and learn test tricks in a day or two. Students studying an hour a day over two months should be more than able to finish all the eercises in this book. The clear subject organization by chapter means you can focus your efforts and spend more time on topics you are struggling with. While chapters can be broken up over a couple sessions, students should finish each practice test in one uninterrupted sitting. Calculators Students may use a four-function, scientific, or graphing calculator during the SAT eam (calculators with QWERTY keyboard are not allowed). Use of a calculator is highly recommended, though most of the problems can be solved without one. Becoming comfortable with your calculator during test preparation will help you use this tool to solve problems more quickly and efficiently during the actual eam. Scoring Scores on the math portion of the SAT range from 00 to 800. An average score on the math section is about 00, but this can vary with each individual test. Consult the College Board Web site ( for further specifics. Signing Up for the SAT Students can register online for the SAT at the College Board Web site. Students can also pick up registration packets at their high schools and sign up via snail mail. Generally the registration deadlines are a little over a month before the actual testing date. Late registrations incur additional fees. This book was written to be a straightforward study guide for the New SAT Math. There are no shortcuts or gimmicks, but taking the time to work through this book should leave you feeling confident and well prepared for the test. Good luck!

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13 PART A Math Review I. Heart of Algebra II. Statistics and Data Analysis III. Advanced Math IV. Geometry

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15 I. Arithmetic and Algebra Chapter : The language and Tools of Algebra Chapter : Solving Linear Equations Chapter : Functions and Linear Equations Chapter : Linear Inequalities and Graphs Chapter : Word Problems in Real-Life Situation

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17 CHAPTER The Language and Tools of Algebra -. Variables and Epressions In algebra, variables are symbols used to represent unspecified numbers or values. An algebraic epression is a collection of numbers, variables, operations, and grouping symbols. Verbal Phrase Addition The sum of twenty and a number n 0 + n Nine more than twice a number a a + 9 A number m increased by m + Subtraction The difference between a number and 7 7 Three less than a number b b Three times a number k decreased by five k Nine minus a number n 9 n Multiplication Eleven times z to the third power The product of and a number n Two thirds of a number Algebraic Epressions z n Division n divided by 8 n /8 The quotient of fifteen and a number d d Eample Write an algebraic epression for each verbal epression. a. Ten less than one-fourth the cube of p. b. Twice the difference between and siteen. c. Four times the sum of a number and three. d. Four times a number increased by three. Solution a. p 0 b. ( 6) c. ( n + ) d. n + Eample Mr. and Mrs. Sawyer are taking their three children to an amusement park. The admission is a dollars per adult, c dollars per child and the cost of each ride is r dollars per person. a. Write an epression for the cost of admission plus 0 rides for the family. b. Find the cost of admission plus 0 rides if the admission per adult is 0 dollars, the admission per child is dollars, and the cost of each ride is 8 dollars. Solution a. a+ c+ 0r b Substitute 0 for a, for c, and 8 for r. = 6

18 6 Chapter Eercise - Variables and Epressions Twice the product of m and n decreased by the square of the sum of m and n. Which of the following is an epression for the statement above? Mario received y tet messages each minute for 0 minutes yesterday and received t tet messages each minute for 0 minutes today. What is the total number of tet messages he received for two days in terms of y and t? A) B) C) D) mn ( m + n ) mn ( m + n) ( m + n) mn ( m + n ) mn A) 0yt B) 00yt C) 0y+ 0t D) 0y+ 0t The product of a number and four decreased by twelve. Which of the following is an epression for the statement above? A) + B) ( + ) C) ( ) D) The quotient of 9 and a number d increased by seven. Which of the following is an epression for the statement above? A) 9 7 d + d B) C) 9 + d 7 6 Which of the following epressions represents the product of k and the sum of m and one third of n? A) B) C) D) km + n k ( m+ n) km ( + n) km ( + n+ ) The difference between two numbers is eight. If the smaller number is n to the third power what is the greater number? A) B) C) D) n 8 n n 8n D) d + 7 9

19 The Language and Tools of Algebra 7 -. Eponents and Order of Operations An epression like = factors of is called a power. The number is the base, and the number is the eponent. To evaluate an epression involving more than one operation, we agree to perform operations in the following order. Order of Operations. Simplify the epressions inside grouping symbols, such as parentheses, brackets, and fraction bars.. Evaluate all powers.. Do all multiplications and divisions in order from left to right.. Do all additions and subtractions in order from left to right. Eample Evaluate ( 0 + 8). Solution ( 0 + 8) = ( 0 + 8) Evaluate power inside grouping symbols. = ( 0 + 8) Evaluate epression inside grouping symbols. = ( 0 + 8) Evaluate epression inside grouping symbols. = ( + 8) Divide 0 by. = (6 + 8) Subtract from. = () Evaluate epression inside grouping symbols. = 8 Multiply. Eample Evaluate b + c a + ( ab + c) a if a =, b =, and c = 7. Solution b + c a + ( ab + c) a ( ) + 7 = + ( ( ) + 7) Replace a with, b with, and c with = 6 + ( + 7) Evaluate, ( ), and ( ). 6 = 6 + ( ) Evaluate epression inside grouping symbols. = 6 + ( ) Divide 6 by. = Subtract and add.

20 8 Chapter Eercise - Eponents and Order of Operations [(7 9) 8] = c ( b+ a) What is the value of ( ) 0 a c if a =, b =, and c =? [0 8] = What is the value of 9 ( z y) if =, y =, and z =? = What is the value of if p =, q =, and r =? 7 ( q) p+ 6q r p q 8 ( + ) = What is the value of b =, and c =? c ( a+ b) ( c a) if a =,

21 The Language and Tools of Algebra 9 -. Simplifying Algebraic Epressions A term is a number, a variable, or a product or quotient of numbers and variables. For eample,, 7a, b, and m n are all terms. Like terms contain identical variables. For eample, in +, the terms and are like terms because the variable part of each term is identical. The coefficient of a term is a number that multiplies a variable. For eample, in 8 y, the coefficient is 8, and in m, the coefficient is. An epression is in simplest form when it is replaced by an equivalent epression having no like terms or parentheses. Simplifying means rewriting in simpler form. Distributive Property Symbols For any real numbers a, b, and c a( b + c) = ab + ac a( b c) = ab ac Eamples (7 + ) = 7 + (7 ) = 7 Commutative Property Symbols For any real numbers a and b, a+ b = b+ a ab = ba Eamples + = + = Associative Property Symbols For any real numbers a, b, and c ( a+ b) + c = a+ ( b+ c ) ( ab) c = a( bc ) Eamples ( + ) + 7 = + ( + 7) ( ) 7 = ( 7) Eample Simplify each epression. a. m n+ m b. ( + ) c. a( b) ( a b) Solution a. m n+ m = ( + ) m n Combine like terms. = m n b. ( + ) = ( ) ( ) Distributive property = 6 Multiply. = 6 Combine like terms. c. a( b) ( a b) = a ab a+ b Distributive property = a a ab + b Commutative property = ab + b

22 0 Chapter Eercise - Simplifying Algebraic Epressions Which of the following epressions is equivalent to ( a a ) + ( a + a+ 6)? A) B) C) a a a + a a Which of the following epressions is equivalent to a( b c) b( a+ c) c( a b)? A) bc B) ac C) bc D) ac D) a Which of the following epressions is equivalent to.( y).7( y)? Which of the following epressions is NOT equivalent to [6a ( a) ( a+ )]? A).7( + y) B).7( y) C).7+.6y D).7.6y A) a B) ( a ) C) ( a ) D) ( ) a Which of the following epressions is equivalent to (a+ b+ ) c ( b+ ) c? 6 Which of the following epressions is NOT equivalent to p (p q) ( p+ q)? A) a c B) a+ c C) a+ c D) a c A) B) C) D) ( ) p q p+ q ( ) p+ q ( p q )

23 The Language and Tools of Algebra -. Rational, Irrational, and Decimal Numbers can be pictured as points on a horizontal line called a number line. The point for 0 is the origin. Points to the left of 0 represent negative numbers, and points to the right of 0 represent positive numbers. Numbers increase in value from left to right. The point that corresponds to a number is called the graph of the number. Each number in a pair such as and is called the opposite of the other number. The opposite of a is written a. Negative Numbers.7 Origin Positive Numbers π. 0 Increasing natural numbers: {,,,...} whole numbers: {0,,,,...} integers: {...,,,, 0,,,,...} rational numbers: A rational number is one that can be epressed as a ratio a, where a and b are b integers and b is not zero, such as.7,, 0,, and.. The decimal form of a rational number is either a terminating or repeating decimal. irrational numbers Any real number that is not rational is irrational.,, and π are irrational. Rounding Decimals To round a decimal to the desired place, underline the digit in the place to be rounded. ) If the digit to the right of the underlined digit is or more, increase the underlined digit by one (round up). ) If the digit to the right of the underlined digit is less than, leave the underlined digit as it is (round down). ) Drop all digits to the right of the underlined digit. Eample On the number line below, the marks are equally spaced. What is the coordinate of P? P 0 0. Solution Since the distance between the marks is 0., the coordinate of P is 7 0., or.7. Eample Round.8 to the nearest a) integer and b) hundredth. Solution a) Underline, the digit in the ones place..8 The digit to the right of the underlined digit is less than, keep the underlined digit. Therefore,.8 rounded to the nearest integer is. b) Underline, the digit in the hundredths place..8 The digit to the right of the underlined digit is more than, round up. Therefore,.8 rounded to the nearest hundredths place is..

24 Chapter Eercise Rational, Irrational, and Decimal Which of the following shows the numbers arranged in increasing order? A),,,, 0 Which of the following is an irrational number? A). B) B) C),,, 0,,,, 0, C) 6 D) 0 D),, 0,, Which of the following is a rational number? A).6 ( 0.8 ) + ( ) = A).68 B).8 C).78 D).8 6 B) 9 6 C) 0.9 D) 0 In three plays, a football team loses yards and then gains yards by completing a pass. Then a penalty was called and the team lost 0 yards. How many yards did the team actually gain? To the nearest cents, what is the value of 8 8 $00( + 0.0) $00( + 0.0)? A) $. B) $6.00 C) $6.77 D) $8. 7 A B. 0. On the number line above, if BC = AB what is the value of? C

25 The Language and Tools of Algebra Chapter Practice Test Two less than the quotient of three and a number n Which of the following is an epression for the statement above? A) n n B) C) D) n n What number is halfway between a number line? A) B) C) D) and 6 on How much greater than n is n +? A) 8 B) 0 C) D) What is.98 rounded to the nearest hundredth? A).9 B).98 C).99 D).0 Johnny received m tet messages on Friday, three less than twice as many tet messages on Saturday than on Friday, and five more tet messages on Sunday than on Saturday. What is the total number of tet messages he received over the three days? A) m + B) m C) m D) m + 6 Which of the following epressions is equivalent to a+ ( b ) c (a+ ) b? A) a b c B) a+ b c C) a b c D) a+ b c

26 Chapter 7 How many minutes are there in h hours and 6m minutes? A) 60h+ m B) 0h+ 6m C) 60h+ 6m D) 0h+ 60m 0 If = 0, what is the value of + +? Add to a number n.. Divide by 8.. Subtract by.. Multiply by 8. If and y are positive integers and + y = 8, what is the value of? When the sequence of operations above has been completed in order, which of the following is an epression for the statement above? A) n B) n C) n D) n 9 Which of the following epressions is equivalent to ( y ) ( y z) ( + z)? A) B) C) + 8z D) 8z If a =, b =, and c =, what is the value a ( b) of 7? c +

27 Answers and Eplanations Answer Key Section -. B. D. A. D. C 6. B Section / Section -. A. B. D. D. B 6. C Section -. C. B. C. D. B Chapter Practice Test. C. D. B. A. D 6. C 7. B 8. C 9. B Note: Throughout the book, the symbol is used to indicate that one step of an equation implies the net step of the equation. Answers and Eplanations Section -. B. D. A mn ( m + n) twice the product of m and n decreased by the product of a number and 9 d the quotient of 9 and a number d decreased by + increased by the square of the sum of m and n twelve 7 seven. D The number of tet messages Mario received yesterday = 0y. The number of tet messages Mario received today = 0t The total number of tet messages he received for two days = 0y+ 0t.. C k ( m+ n ) times k 6. B sum of m and one third of n km ( + n) stands for k ( m+ n). If the smaller number is n to the third power, which is n, the greater number is 8 more than n, which is n + 8. Section -. 0 [(7 9) 8] = [(9 9) 8] = [(0) 8] = [] = [0 8] 6 7 = 9 [0 8] 9 7 = 9 [0 8] = 9 [0 ] = 9 [0 8] = 9 [] = 9 6 = = = = = ( + ) = ( + ) = (6) = 8 = 7

28 6 Chapter c ( b+ a) ( + ) ( ) 0 = ( ) 0 a c = ( ) 0 ( ) = = 9 ( z y) = 9 () ( ( )) = 9 8 ( ) = 9 8 ( 8) = 9 ( ) = 0. D (a+ b+ ) c ( b+ ) c = a+ b+ c b c = a c. D a( b c) b( a+ c) c( a b) = ab ac ab bc ac + bc = ac 8. 7 ( q) p+ 6q r p q 7 ( ) () + 6( ) = () 7 8 = = 6 = 7 6 = ( + ) () c ( a+ b) = = ( c a) ( ( )) () = = 9 8. B 6. C [6a ( a) ( a+ )] = [6a + a a ] = [a 8] = a All of the answer choices ecept B are equivalent to a. p (p q) ( p+ q) = p p+ q p q = p+ q All of the answer choices ecept C are equivalent to p+ q. Section -. A ( a a ) + ( a + a+ 6) = a a + a + a+ = a. B.( y).7( y) =. 0.8y.7+ 8.y =.7.7y =.7( y) Section -. C, (.7),, 0(.), Answer choice C shows the numbers arranged in increasing order.. B ( 0.8 ) + ( ) = (.6) + ( 6). = (.6) 6 =.8

29 Answers and Eplanations 7. C. D. B 6. 7 Use calculator ( + 0.0) 00( + 0.0) 8 8 = 00[(.0) (.0) ] = 00(0.0) rounded to the nearest hundredth is is an irrational number. 6 = is not an irrational number = = is a rational number = 7 A B. 0. BC = AB ( 0.) = [ 0. (.)] + 0. = (0.8) + 0. =.7 =.7 0. =. Chapter Practice Test. C. D. B The phrase two less than the quotient of three and a number n is translated n, not. n ( n+ ) ( n ) = n+ n+ = Number of tet messages he received on Friday is m, on Saturday is m, and on Sunday is m + The total number of tet messages he received over the three days is m+ (m ) + (m + ) = m C. A To find a number which is halfway between two numbers, find the average of the two numbers. + ( + ) = = = = ()6. D 6. C Underline 9, the digit in the hundredths place..98 The digit to the right of the underlined digit is more than, round up. Therefore,.98 rounded to the nearest hundredths place is.0. a+ ( b ) c (a+ ) b = a+ b c a b = a b c 7. B There are h 60 minutes in h hours. There are (0h+ 6 m) minutes in h hours and 6m minutes. 8. C 9. B. Add to a number n. n +. Divide by 8. ( n + ) 8. Subtract by. ( n + ) 8. Multiply by 8. ( n + ) [ ] 8 8 ( n + ) [ ] 8 = ( n+ ) 8 = n 8 ( y ) ( y z) ( + z) = y y+ z z = = = =.

30 8 Chapter. Choose the first few positive integers for and make substitutions for the given equation. Construct a table of values. y not an integer not an integer not an integer. 8 Both and y are positive integers when equals and y equals. Therefore the value of is. a ( b) ( ( )) 7 = 7 c + + () = 7 = 7 = 7 = 7+ = 8

31 CHAPTER Solving Linear Equations -. Writing Equations An equation is a mathematical sentence with an equal sign. To translate a word sentence into an equation, choose a variable to represent one of the unspecified numbers or measures in the sentence. This is called defining a variable. Then use the variable to write equations for the unspecified numbers. Eample Translate each sentence into an equation. a. Twice a number increased by fourteen is identical to fifty. b. Half the sum of seven and a number is the same as the number decreased by two. c. The quotient of m and n equals four more than one-third the sum of m and n. d. The cube of plus the square of y is equal to fifty two. Solution a. Let c be the number. Define a variable. c + = 0 Twice a number c increased by fourteen is identical to fifty. b. Let n be the number. Define a variable. (7 + n) = n is the same as the number decreased by two. Half the sum of 7 and a number c. m n = + ( m+ n) The quotient equals four more than one-third the sum of mand n. of m and n d. + y = The cube of plus the square of y is equal to fifty two. Consecutive Numbers Consecutive Integers...,,,, 0,,,,... n, n +, n + are three consecutive integers if n is an integer. Consecutive Even Integers..., 6,,, 0,,, 6,... n, n +, n + are three consecutive even integers if n is an even integer. Consecutive Odd Integers...,,,,,,,... n, n +, n + are three consecutive odd integers if n is an odd integer. Eample Write an equation to represent the given relationship between integers. a. The sum of four consecutive integers is. b. The product of three consecutive odd integers is 69. Solution a. Let n be the first integer. Define a variable. n+ ( n+ ) + ( n+ ) + ( n+ ) = The sum of four consecutive integers is b. Let n be the first odd integer. Define a variable. nn ( + )( n+ ) = 69 The product of three consecutive odd integers is 69

32 0 Chapter Eercise - Writing Equations Eighteen more than the number n is. What is the value of n? 6 The sum of three fourths of the number a and is negative 9. What is the value of a? Twenty is 7 less than twice the number w. What is the value of w? Nine less than twice is three more than. What is the value of? 7 A) B) 0 C) 0 D) A number g is decreased by and then multiplied by. The result is 8 more than twice the number g. A) B) C) 9 D) 8 Eight less than four times the number c is twenty. What is the value of c? 8 The quotient of p and q is twelve less than three times the sum of p and q. Which of the following equations represents the statement above? The sum of four consecutive odd integers is 96. What is the greatest of the four consecutive odd integers? A) p ( p q) q = + B) p ( p q) q = + C) p ( p q) q = + D) p ( p q) q = +

33 Solving Linear Equations -. Solving Equations To solve an equation means to find all values of the variable that make the equation a true statement. One way to do this is to isolate the variable that has a coefficient of onto one side of the equation. You can do this using the rules of algebra called properties of equality. Properties of Equality Symbols Eamples. Addition Property If a = b, then a+ c = b+ c. If =, then ( ) + = () +.. Subtraction Property If a = b, then a c = b c. If + = 6, then ( + ) = (6).. Multiplication Property If a = b, then ca = cb. If =, then. Division Property If a = b and c 0, then a = b. If =, then =. c c Eample Solve each equation. a. a + ( ) = b. = 8y Solution a. a + ( ) = a + ( ) + = + Add to each side. a = + = 0 and + = b. = 8y 8y = Divide each side by = y = and 8 y = y 8 8 Many equations require more than one operation to solve. Such equations are called multi-step equations. To solve multi-step equations, first simplify each side of the equation, if needed, and then use inverse operations to isolate the variable. Eample Solve ( ) ( 0) =. Solution ( ) ( 0) = 9 + = 9 Distributive Property 9 = + = 9+ Add to each side. = ( ) = () Multiply each side by. =

34 Chapter Eercise - Solving Equations + = 9 Given the above equation, what is the value of 0 ( )? 6 Two and three fifths of a number equals 6. What is the number? A) B) 0 C) D) 0 If a = a+ 7 a, what is the value of + a? 7 There are one hundred forty-two students in a high school band. These students represent two ninth of the total students in the high school. How many students attend the school? If + =, what is the value of? A) 87 B) 6 C) 69 D) 66 If ( ) + ( ) = 7, what is the value of? If three quarters of a number decreased by twenty is equal to eighty two, what is that number? 8 80c+ 80r =, 60 The above equation models the amount of calories in a snack of c cups of cashews and r cups of raisins. The amount of calories per cup of cashews is 80 and the amount of calories per cup of raisins is 80. According to the equation, how many cups of raisins are used, if cups of cashews are used to make the snack? A) B) C) D) 6

35 Solving Linear Equations -. Solving Equations with Variables on Both Sides Some equations have variables on both sides. To solve such equations, first use the Addition or Subtraction Property of Equality to write an equivalent equation that has all of the variables on one side. Then use the Multiplication or Division Property of Equality to simplify the equation if necessary. When solving equations that contain grouping symbols, use the Distributive Property to remove the grouping symbols. Eample Solve each equation. a. 7 8= 6+ b. ( k+ ) = k Solution a. 7 8= = 6+ Subtract from each side. 8= = 6+ 8 Add 8 to each side. = = Divide each side by. = 7 b. ( k+ ) = k k = k Distributive Property k = k k+ k = k+ k Add k to each side. k = 0 k = 0 Subtract from each side. k = k = Divide each side by. k = Eample Four times the sum of three and a number equals nine less than the number. a. Write an equation for the problem. b. Then solve the equation. Solution a. ( + n) = n 9 b. + n = n 9 Distributive Property + n n = n 9 n Subtract n from each side. + n = 9 + n = 9 Subtract from each side. n = n = Divide each side by. n = 7

36 Chapter Eercise - Solving Equations with Variables on Both Sides If 7n+ = n, what is the value of n +? A $8 shirts costs $ more than one half the cost of a pair of pants. How much does the pair of pants cost? If 7( h ) h = h, what is the value of 7 h? 6 Twice a number n, increased by is the same as si times the number decreased by 9. What is the value of n? r s 9 + = Given the above equation, if s =, what is the value of r? 7 One half of a number increased by is five less than two thirds of the number. If 9 k = k, what is the value of k? 8 Four times the greatest of three consecutive odd integers eceeds three times the least by. What is the greatest of the three consecutive odd integers?

37 Solving Linear Equations -. Equation with No Solution and Identity It is possible that an equation may have no solution. That is, there is no value of the variable that will result in a true equation. It is also possible that an equation may be true for all values of the variable. Such an equation is called an identity. Eample Solve each equation. If the equation has no solution or it is an identity, write no solution or identity. a. ( ) + = ( + ) b. w ( w) = ( w) c. (8y 6) = y ( y+ ) Solution a. ( ) + = ( + ) + = + Distributive property + = + + = + Subtract from each side. = The given equation is equivalent to the false statement =. Therefore the equation has no solution. b. w ( w) = ( w) w + w= 6+ w Distributive property 8w = 6+ w 8w w= 6+ w w Subtract w from each side. 6w = 6 6w + = 6+ Add to each side. 6w = 6w = 6 6 Divide each side by 6. w = c. (8y 6) = y ( y+ ) y = y y Distributive property y = y The given equation is equivalent to y = y, which is true for all values of y. This equation is an identity.

38 6 Chapter Eercise - Equation with No Solution and Identity If (9 6 ) =, what is the value of? A) B) C) D) The equation has no solution. If ( ) = ( ), which of the following must be true? A) is. B) is. C) is. D) The equation is true for all values of. 6 What is the value of n 7 if (n ) + n = ( + n)? What is the value of k if 7( k + ) = k? ( 6 ) = a If the linear equation above is an identity, what is the value of a? 7 What is the value of if [ ( )] + =? A) B) C) D) + = 7( ) + b If the linear equation above has no solution, which of the following could be the value of b? 8 What is the value of m if 0.(m 9) = m (0. m)? A) B) C) D)

39 Solving Linear Equations 7 -. Solving for a Specific Variable A formula is an equation that states the relationship between two or more variables. Formulas and some equations contain more than one variable. It is often useful to solve formulas for one of the variables. Eample Solve each equation or formula for the specified variable. a. a = k + b, for b. A= ( a+ bh ), for h c. C = ( F ), for F 9 Solution a. a = k + b a k = k + b k Subtract k from each side. a k = b a k+ a = b+ a Add a to each side. k = b + a ( k) = b+ a Distributive property ( k) b+ a = k k Divide each side by k. b+ a = k b. A= ( a+ bh ) A= ( a+ bh ) Multiply each side by. A= ( a+ bh ) A ( a+ b) = h a+ b a+ b Divide each side by a+ b. A h a b c. C = ( F ) C = ( F ) 9 Multiply each side by 9. 9 C F 9 C + = F + Add to each side. 9 F = C+

40 8 Chapter Eercise - Solving for a Specific Variable If + y = 8, which of the following gives y in terms of? ab If = c, which of the following gives b in terms of the other variables? A) B) C) D) y = 6 + y = 6 y = 6 + y = 6 A) B) C) D) c + b = a c b = a c b = + a c b = a If P = l+ w, which of the following gives w in terms of P and l? If gh f = g h, which of the following gives g in terms of the other variables? A) w= P l B) w= P l A) g = f + h h C) P w= l B) g = f h h + D) w= P l C) g = f + h h + D) g = f h h a If c = a + b, which of the following gives a in terms of b and c? bc A) a = c bc B) a = + c bc C) a = b c bc D) a = b + c 6 If n = a+ ( k ) d, which of the following gives k in terms of the other variables? A) B) C) D) n a+ k = d n+ a k = d n a d k = d n a+ d k = d

41 Solving Linear Equations 9 Chapter Practice Test If =, what is the value of? 6 A) B) C) D) When one half of the number n is decreased by, the result is 6. What is three times n added to 7? P = F( v + ) The above equation gives pressure P, which is eerted by a fluid that is forced to stop moving. The pressure depends on the initial force, F, and the speed of the fluid, v. Which of the following epresses the square of the velocity in terms of the pressure and the force? A) B) C) D) v = ( P F) v = ( P F ) v v P = ( ) F P F = ( ) F A) 7 B) C) D) One half of the number n increased by 0 is the same as four less than twice the number. Which of the following equations represents the statement above? If 7 is less than, what is the value of? A) B) 9 C) 6 D) A) ( 0) ( ) n+ = n B) 0 ( ) n+ = n C) 0 n+ = n D) ( 0) n+ = n

42 0 Chapter 6 If a is b less than one-half of c, what is b in terms of a and c? 9 a If =, what is the value of a b b? A) c a 7 B) a c C) a c D) c a If = y and = 8 y, what is the value of? A) B) C) D) 0 When an object is thrown from the ground into the air with an initial upward speed of v 0 meters per second, the speed v, in meters per second, is given by the equation v = v0 9.8t, where t is the time in seconds. The speed of an object becomes 0 when the object reaches its maimum height. If an object is thrown upward with an initial speed of m/sec, how many seconds does it taken an to reach its maimum height? (Round your answer to the nearest hundredth of a second.) 8 The quotient of a number and five equals nine less than one half of the number. What is the number? When an object is dropped from a height of s feet above the ground, the height h of the object is given by the equation h = 6t + s, where t is the time in seconds after the object has dropped. If an object is dropped from a height of feet above the ground, how many seconds will it take to hit the ground? A) 0 B) 0 C) 0 D) 0

43 Answers and Eplanations Answer Key Section A 7. A 8. C Section B 7. C 8. C Section Section -. D. D. A. C. 6. Section B. C. A. A.D 6. D Chapter Practice Test. C. B.C. D. C 6. A 7. B 8. D Answers and Eplanations Section n + 8 = 8 more than n n = 8 = = w 7 7 less than twice w = w Add 7 to each side. 7 = w 7 w = Divide each side by.. = w A 7. A 9 = + 9 less than twice more than 9 = + Subtract from each side. 9= = c 8 = 0 8 less than times c c = Add 8 to each side. c = 8 c = 7 Let n = the smallest of four consecutive odd integers. Then, n+ ( n+ ) + ( n+ ) + ( n+ 6) = 96. n + = 96 n = 8 n = 7 The greatest of the four consecutive odd integers is n + 6. Therefore, n + 6 = 7+ 6 = 77 a + = 9 the sum of three fourths of a and a + = 9 Subtract from each side. a = a = ( ) Multiply each side by. a = g ( g ) = g+ 8 is decrease by and then multiplied by. 8 more than twice g ( g ) = (g+ 8) Multiply each side by. g = g+ 6 g + = g+ 6 + Add to each side. g = g+ 9 g g = g+ 9 g Subtract g. g = 9 g =

44 Chapter. 8. C p q the quotient of p and q Section = ( p+ q) three times the sum of p and q twelve less than Given + = 9. 0 ( ) = 0 + = 0 + ( + ) = = 9 a = a+ 7 a a = 7 a a+ a = 7 a+ a Add a to each side. + a = 7 = Multiply by on both sides of the equation to simplify the given equation. ( ) = ( ) = Distributive Property + = + Add to each side. 6 = 6 + = + Add to each side. 6 = = = 6 ( ) + ( ) = = 7 + = 7 + = 7 Subtract. = 8 8 = = = decreased by twenty equals three quarters of a number eighty two 6. B 7. C 8. C = Add 0 to each side. 0 = = 0 = 6 6 = equals two and three fifth of a number negative twenty si 6 = = = 6 Multiply each side by. = 0 Let = the total students in the high school. Then 9 =. 9 = = 69 80c+ 80r =, 60 Substitute for c in the equation above since c represents the number of cups of cashews. 80() + 80r =, 60, r =, 60 80r =,900 r = Section n+ = n n = n = Therefore, n + = ( ) + = + = 6.. 7( h ) h = h 7h h = h h = h h h = h = h = = Therefore, h = () =. 7 7

45 Answers and Eplanations r s 9 r 9 + = and s =. + = To simplify the equation, multiply both sides of the equation by, which is the LCD of and. r 9 ( + ) = r + 6 = 9 r = r = 9 k = k To simplify the equation, multiply both sides of the equation by. 9 k ( ) = ( k ) 9 k = k 6 k k = 6 9 k = k = Let p = the cost of a pair of pants. Since a $8 shirts costs $ more than one half the cost of a pair of pants, you can set up the following equation. 8 = p + 6 = p = p n+ = 6 n 9 twice a number increased by 7. 8 si times the number decreased by 9 n+ = 6n 9 0 = n n = n+ = n one half of anumber increased by three five less than two thirds of the number To simplify the equation, multiply both sides of the equation by 6, which is the LCD of and. 6( n+ ) = 6( n ) n+ 8 = n 0 Solving for n yields n = Let n be the first of the three consecutive odd integers, so n, n +, and n + are the three consecutive odd integers. ( n+ ) = n + times the greatest of consecutive odd integers ( n+ ) = n+ n+ 6 = n+ n = times the least of eceeds consecutive odd integers by The greatest of the three consecutive odd integers is n + = + = 9. Section -. D. D. A. C. (9 6 ) = = Distributive Property + = + Add to each side. = The given equation is equivalent to the false statement =. Therefore the equation has no solution. ( ) = ( 0) 0 = 0 Distributive Property 0 = 0 The given equation is equivalent to 0 = 0, which is true for all values of. ( 6 ) = a = a Distributive Property If the linear equation is an identity, the value of a is. + = 7( ) + b + = 7 + b + = (7 + b ) If = 7+ b, the linear equation has no solution. Solving for b yields b =. 7 (n ) + n = ( + n)

46 Chapter 6. To simplify the equation, multiply both sides of the equation by. 7 [ (n ) + n] = [ ( + n)] 7(n ) + 8n = ( + n) Distributive Property n+ + 8n = + 6n 6n+ = + 6n 6n+ + 6n = + 6n+ 6n Add 6n to each side. = + n = + n Subtract. 6 = n or n = 6 6 n = = 7( k + ) = k To simplify the equation, multiply both sides of the equation by. 7( k + ) [ ] = [k ] 7( k+ ) = 9k 6 7k 7 = 9k 6 Distributive Property 6 7k = 9k 6 6 7k 6= 9k 6 6 Subtract 6. 7k = 9k 7k 9k = 9k 9k Subtract 9k. 6k = k = = 6 Section -. B. C. A + y = 8 + y = 8 Subtract from each side. y = 8 y 8 = Divide each side by. y = 6 P = l+ w P l = l+ w l Subtract l from each side. P l = w P l w = Divide each side by. P l = w a c = a + b a ( a+ b) c = ( a+ b) Multiply each side by a a + + b. b ac + bc = a ac + bc ac = a ac Subtract ac from each side. bc = a ac bc = a( c) Factor. bc c = a Divide each side by c. 7. [ ( )] + = [ + ] + = [7 ] + = + + = + 7 = 8 = 6 = (m 9) = m (0. m) m.6 = m. + m m.6 = m. m =. m = 0.8. A ab = c ab [ ] = c Multiply each side by. ab = c ab + = c + Add to each side. ab = c + ab c + = a a Divide each side by a. c + b = a

47 Answers and Eplanations. D 6. D gh f = g h gh f + f = g h + f Add f to each side. gh = g h + f gh g = g h + f g Subtract g from each side. gh g = f h gh ( ) = f h Factor. g = f h h Divide each side by h. n = a+ ( k ) d n = a + kd d Distributive Property n a+ d = a+ kd d a+ d Add a+ d to each side. n a + d = kd n a+ d = k Divide each side by d. d Chapter Practice Test. C. B = = Multiply each side by 6 6. = n = 6 of a number n decreased by negative 6 n + = 6+ Add to each side. n = n = Multiply each side by. n = Three times n added to 7 is n + 7. n + 7 = ( ) + 7 Substitute for n. =. C. D. C 6. A 7 = 7 is less than 7 = 8 7 = = = ( ) = 6 P = F( v + ) P F = ( v + ) Divide each side by F. F F P v F = + P v F = + Subtract from each side. P v F = P ( ) = v Multiply each side by. F P ( ) v F = P F ( ) v F F = F F = P F ( ) = v The common denominator is F. F Combine the numerators. of the number increased by 0 n+ 0 = n n a = c b a is b less than of c four less than twice the number a c = c b c Add c to each side. a c = b ( )[ a c] = ( )( b) Multiply each side by. a+ c = b or c a = b

48 6 Chapter 7. B = y First equation = 8 y Second equation Solving the first equation for y yields y =. Substitute for y in the second equation. = 8 ( ) Substitution = 8 + Distributive property = + = + Subtract from each side. = = Divide each side by. =. When the object hits the ground, the height is 0. Substitute 0 for h and for s in the equation 0 = 6t +. Solving the equation for t gives t = = 9. 6 Therefore, t = 9 =. 8. D 9. 0 = 9 the quotient of of a number and nine less than one half of the number 0( ) = 0( 9) Multiply each side by 0. = 90 Distributive Property = 90 Subtract from each side. = = Divide each side by. = 0 a b = a b( ) = b() b Multiply each side by b. a = b a b = b b Subtract b from each side. a b = As the object moves upward, its speed decreases continuously and becomes 0 as it reaches its maimum height. v = v0 9.8t is the given equation. Substituting for v 0 and 0 for v gives 0 = 9.8t. Solving the equation for t gives t = = seconds, which is. to the nearest hundredth of a second.

49 CHAPTER Functions and Linear Equations -. Relations and Functions A coordinate plane is formed by the horizontal line called the - ais and the vertical line called the y- ais, which meet at the origin (0,0). The aes divide the plane into four parts called quadrants. An ordered pair gives the coordinates and location of a point. The ordered pairs (,), (, ), (, ), and (, ) are located in Quadrant I, Quadrant II, Quadrant III, and Quadrant IV respectively. A relation is a set of ordered pairs. A relation can be represented by a graph, a table, or a mapping. The domain of a relation is the set of all - coordinates and the range of a relation is the set of all y- coordinates from the ordered pairs. A function is a special type of relation in which each element of the domain is paired with eactly one element of the range. Quadrant II (,) O Quadrant III (, ) Table y y Quadrant I (,) (, ) Quadrant IV Mapping y Eample Epress each relation below as a set of ordered pairs and determine whether it is a function. a. y b. y c. y 7 0 O 9 Solution a. {(,7), (, ), (0, ), (, 9)} The mapping represents a function. b. {(,), (,), (, ), (, )} The element in the domain is paired with both and in the range. This relation does not represent a function. c. {(, ), (,), (,), (, )} The table represents a function.

50 8 Chapter You can use the vertical line test to see if a graph represents a function. A relation is a function if and only if no vertical line intersects its graph more than once. Function Not a Function Function y y y O O O No vertical line intersects the graph more than once. A vertical line intersects the graph at two points. No vertical line intersects the graph more than once. Function Values Equations that are functions can be written in a form called function notation. In function notation, the equation y = + is written as f( ) = +. The function value of f at = c is denoted as f() c. For instance, if f( ) =, f () is the value of f at = and f () = () = 7. Eample If f( ) = +, find each value. a. f ( ) b. f( c ) c. [ f( )] + f( ) d. f( + ) Solution a. f ( ) = ( ) + Substitute for. = 6+ = Multiply and simplify. b. f( c ) = ( c ) + Substitute c for. = c 6+ Multiply. = c c. [ f( )] + f( ) = [( ) + ] + [( ) + ] Substitute for and for. = [ + ] + [ 6 + ] Multiply. = [ ] + [ ] = = d. f( + ) = ( + ) + Substitute = Multiply. = + for.

51 Functions and Linear Equations 9 Eercises - Relations and Functions What is the domain of the function that contains points at (, ), (,), (0, ), and (, )? If f( ) = + 7, what is f( + ) equal to? A) {,, } B) {,, 0} C) {,,, } A) + B) + C) + D) + 0 D) {,, 0, } 7 y 8 Which of the following relation is a correct representation of the mapping shown above? g( ) = k + For the function g defined above, k is a constant and g( ) =. What is the value of g ()? A) B) C) D) A) {(,7), (, ), (, ), (,8)} B) {(,8), (,7), (, ), (,8)} C) {(7, ), (, ), (, ), (8,)} 6 If f( + ) = + 6, what is the value of f ( )? D) {(8, ), (7, ), (, ), (8,)} If point (7, b ) is in Quadrant I and point ( a, ) is in Quadrant III, in which Quadrant is the point ( ab, )? A) Quadrant I B) Quadrant II C) Quadrant III D) Quadrant IV 7 f( ) = b In the function above, b is a constant. If f ( ) = 7, what is the value of f( b )?

52 0 Chapter -. Rate of Change and Slope The average rate of change is a ratio that describes, on average, change in one quantity with respect to change in another quantity. If is the independent variable and y is the dependent variable, then change in y the average rate of change =. change in Geometrically, the rate of change is the slope of the line through the points (, y ) and (, y ). y y change in y rise The slope m of a line through (, y ) and (, y ) is m = = =. change in run positive slope negative slope zero slope undefined slope y y y y O line slopes up from left to right O line slopes down from left to right O horizontal line O vertical line The standard form of a linear equation of a line is A + By = C, in which A, B, and C are integers. The - coordinate of the point at which the graph of an equation crosses the - ais is called the - intercept. To find the - intercept, let y = 0 and solve the equation for. The y- coordinate of the point at which the graph of an equation crosses the y- ais is called the y- intercept. To find the y- intercept, let = 0 and solve the equation for y. The intercepts of a line provide a quick way to sketch the line. Values of for which f( ) = 0 are called zeros of the function f. A function s zero is its - intercept. Eample The table at the right shows Evan s height from age to 8. Find the average rate of change in Evan s height from age to 8. Age (years) 6 8 Height (inches) Solution change in height Average rate of change = change in years = = =. inches per year 8 6 Eample Find the slope of the line that passes through (, ) and (, ). y y Solution m = ( ) 6 = = = 8 Eample Find the - intercept and y- intercept of + y = 6. Solution To find the - intercept, let y = 0. + (0) = 6 = The - intercept is. To find the y- intercept, let = 0. (0) + y = 6 y = The y- intercept is.

53 Functions and Linear Equations Eercises - Rate of Change and Slope (,) (0, ) y O What is the rate of change shown in the graph of the line above? The graph of the linear function f passes through the points ( a,) and (, b ) in the y- plane. If the slope of the graph of f is, which of the following is true? A) a b = B) a+ b = C) a b = D) a+ b = A) B) What is the slope of the line that passes through (, ) and (, 8)? C) D) What is the value of r if the line that passes through (,) and (, r) has a slope of? 0 6 y What is the average rate of change for the relation shown in the table above? A) 6 What is the value of a if the line that passes through ( a,7) and (, a ) has a slope of? 9 B) C) 7 + y = 6 D) 6 What is the slope of the line in the equation above?.

54 Chapter -. Slope-Intercept Form and Point-Slope Form The slope-intercept form of the equation of a line is y = m + b, in which m is the slope and b is the y- intercept. The point-slope form of the equation of a line is y y = m ( ), in which (, y ) are the coordinates of a point on the line and m is the slope of the line. Eample The graph of a linear equation is shown on the diagram at the right. y a. Find the slope of the line on the graph. b. Write the equation of the line in point-slope form. c. Write the equation of the line in slope-intercept form and find the y- intercept. d. Write the equation of the line in standard form. (, ) O (,) e. Find the - intercept on the graph. y y Solution a. m = ( ) = (, y ) = (, ), (, y ) = (,) ( ) 6 = = 9 b. y y m = ( ) Point-slope form y = ( ) m =. Choose either point for (, y ). 0 c. y = Simplify point-slope form. y = Add to each side. The y- intercept is. d. y = Slope-intercept form y = ( ) Multiply each side by. y = + y = or y = Subtract from each side. e. + (0) = To find the - intercept, let y = 0. = The - intercept is.