Chapter 3 Resource Masters

Transcription

1 Chapter 3 Resource Masters

2 Consumable Workbooks Man of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook X Skills Practice Workbook Practice Workbook ANSWERS FR WRKBKS The answers for Chapter 3 of these workbooks can be found in the back of this Chapter Resource Masters booklet. Glencoe/McGraw-Hill Copright b The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced onl for classroom use; be provided to students, teacher, and families without charge; and be used solel in conjunction with Glencoe s Algebra 2. An other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill Companies 8787 rion Place Columbus, H ISBN: Algebra 2 Chapter 3 Resource Masters

3 Contents Vocabular Builder vii Lesson 3- Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 3-2 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 3-3 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 3-4 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 3-5 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Chapter 3 Assessment Chapter 3 Test, Form Chapter 3 Test, Form 2A Chapter 3 Test, Form 2B Chapter 3 Test, Form 2C Chapter 3 Test, Form 2D Chapter 3 Test, Form Chapter 3 pen-ended Assessment Chapter 3 Vocabular Test/Review Chapter 3 Quizzes & Chapter 3 Quizzes 3 & Chapter 3 Mid-Chapter Test Chapter 3 Cumulative Review Chapter 3 Standardized Test Practice Standardized Test Practice Student Recording Sheet A ANSWERS A2 A26 Glencoe/McGraw-Hill iii Glencoe Algebra 2

4 Teacher s Guide to Using the Chapter 3 Resource Masters The Fast File Chapter Resource sstem allows ou to convenientl file the resources ou use most often. The Chapter 3 Resource Masters includes the core materials needed for Chapter 3. These materials include worksheets, etensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Algebra 2 TeacherWorks CD-RM. Vocabular Builder Pages vii viii include a student stud tool that presents up to twent of the ke vocabular terms from the chapter. Students are to record definitions and/or eamples for each term. You ma suggest that students highlight or star the terms with which the are not familiar. WHEN T USE Give these pages to students before beginning Lesson 3-. Encourage them to add these pages to their Algebra 2 Stud Notebook. Remind them to add definitions and eamples as the complete each lesson. Stud Guide and Intervention Each lesson in Algebra 2 addresses two objectives. There is one Stud Guide and Intervention master for each objective. WHEN T USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN T USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closel follow the structure of the Practice and Appl section of the Student Edition eercises. These eercises are of average difficult. WHEN T USE These provide additional practice options or ma be used as homework for second da teaching of the lesson. Reading to Learn Mathematics ne master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. Additional questions ask students to interpret the contet of and relationships among terms in the lesson. Finall, students are asked to summarize what the have learned using various representation techniques. WHEN T USE This master can be used as a stud tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Enrichment There is one etension master for each lesson. These activities ma etend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics the are learning. These are not written eclusivel for honors students, but are accessible for use with all levels of students. WHEN T USE These ma be used as etra credit, short-term projects, or as activities for das when class periods are shortened. Glencoe/McGraw-Hill iv Glencoe Algebra 2

5 Assessment ptions The assessment masters in the Chapter 3 Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. Chapter Assessment CHAPTER TESTS Form contains multiple-choice questions and is intended for use with basic level students. Forms 2A and 2B contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms 2C and 2D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with aes are provided for questions assessing graphing skills. Form 3 is an advanced level test with free-response questions. Grids without aes are provided for questions assessing graphing skills. All of the above tests include a freeresponse Bonus question. The pen-ended Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. A Vocabular Test, suitable for all students, includes a list of the vocabular words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate Assessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. Continuing Assessment The Cumulative Review provides students an opportunit to reinforce and retain skills as the proceed through their stud of Algebra 2. It can also be used as a test. This master includes free-response questions. The Standardized Test Practice offers continuing review of algebra concepts in various formats, which ma appear on the standardized tests that the ma encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. Bubble-in and grid-in answer sections are provided on the master. Answers Page A is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages This improves students familiarit with the answer formats the ma encounter in test taking. The answers for the lesson-b-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer kes are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe Algebra 2

6 NAME DATE PERID 3 Reading to Learn Mathematics Vocabular Builder This is an alphabetical list of the ke vocabular terms ou will learn in Chapter 3. As ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. Add these pages to our Algebra Stud Notebook to review vocabular at the end of the chapter. Vocabular Term bounded region Found on Page Definition/Description/Eample Vocabular Builder consistent sstem constraints kuhn STRAYNTS dependent sstem elimination method feasible region FEE zuh buhl inconsistent sstem ihn kuhn SIHS tuhnt independent sstem (continued on the net page) Glencoe/McGraw-Hill vii Glencoe Algebra 2

7 3 NAME DATE PERID Reading to Learn Mathematics Vocabular Builder (continued) Vocabular Term linear programming Found on Page Definition/Description/Eample ordered triple substitution method sstem of equations sstem of inequalities unbounded region vertices Glencoe/McGraw-Hill viii Glencoe Algebra 2

8 3- NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations b Graphing Graph Sstems of Equations A sstem of equations is a set of two or more equations containing the same variables. You can solve a sstem of linear equations b graphing the equations on the same coordinate plane. If the lines intersect, the solution is that intersection point. Eample Solve the sstem of equations b graphing Write each equation in slope-intercept form The graphs appear to intersect at (0, 2). CHECK Substitute the coordinates into each equation ( 2) 4 0 ( 2) The solution of the sstem is (0, 2). (0, 2) Lesson 3- Eercises Solve each sstem of equations b graphing (6, ) 4 (2, 2) 2 4 (4, ) (, 3) ( 4, 3) 2 ( 2, 3) Glencoe/McGraw-Hill 9 Glencoe Algebra 2

9 3- NAME DATE PERID Stud Guide and Intervention (continued) Solving Sstems of Equations b Graphing Classif Sstems of Equations The following chart summarizes the possibilities for graphs of two linear equations in two variables. Graphs of Equations Slopes of Lines Classification of Sstem Number of Solutions Lines intersect Different slopes Consistent and independent ne Lines coincide (same line) Same slope, same -intercept Consistent and dependent Infinitel man Lines are parallel Same slope, different -intercepts Inconsistent None Eample Graph the sstem of equations 3 6 and describe it as consistent and independent, 2 3 consistent and dependent, or inconsistent. Write each equation in slope-intercept form The graphs intersect at ( 3, 3). Since there is one solution, the sstem is consistent and independent. ( 3, 3) Eercises Graph the sstem of equations and describe it as consistent and independent, consistent and dependent, or inconsistent consistent inconsistent and dependent inconsistent consistent 2 2 consistent 6 consistent and independent and dependent and independent Glencoe/McGraw-Hill 20 Glencoe Algebra 2

10 3- NAME DATE PERID Skills Practice Solving Sstems of Equations B Graphing Solve each sstem of equations b graphing (2, 0) 2 4 (2, 0) 2 (2, 2) (3, ) 5 3 (3, 4) 3 4 (, ) Lesson (2, ) (4, 0) 2 ( 2, 5) Graph each sstem of equations and describe it as consistent and independent, consistent and dependent, or inconsistent inconsistent consistent and consistent and dependent independent Glencoe/McGraw-Hill 2 Glencoe Algebra 2

11 3- NAME DATE PERID Practice (Average) Solving Sstems of Equations B Graphing Solve each sstem of equations b graphing (2, ) 2 3 (2, ) (3, 3) ( 2, ) 2 2 (2, 2) (, ) Graph each sstem of equations and describe it as consistent and independent, consistent and dependent, or inconsistent consistent and indep. inconsistent consistent and dep. SFTWARE For Eercises 0 2, use the following information. Location Mapping needs new software. Software A costs $3,000 plus $500 per additional site license. Software B costs $2500 plus $200 per additional site license. 0. Write two equations that represent the cost of each software. A: 3, , B: Graph the equations. Estimate the break-even point of the software costs. 5 additional licenses 2. If Location Mapping plans to bu 0 additional site licenses, which software will cost less? B Software Costs Additional Licenses Glencoe/McGraw-Hill 22 Glencoe Algebra 2 Total Cost ($) 24,000 20,000 6,000 2,000 8,000 4,000

12 3- NAME DATE PERID Reading to Learn Mathematics Solving Sstems of Equations b Graphing Pre-Activit How can a sstem of equations be used to predict sales? Read the introduction to Lesson 3- at the top of page 0 in our tetbook. Which are growing faster, in-store sales or online sales? online sales In what ear will the in-store and online sales be the same? 2005 Reading the Lesson. The Stud Tip on page 0 of our tetbook sas that when ou solve a sstem of equations b graphing and find a point of intersection of the two lines, ou must alwas check the ordered pair in both of the original equations. Wh is it not good enough to check the ordered pair in just one of the equations? Sample answer: To be a solution of the sstem, the ordered pair must make both of the equations true. Lesson 3-2. Under each sstem graphed below, write all of the following words that appl: consistent, inconsistent, dependent, and independent. inconsistent consistent; consistent; dependent independent Helping You Remember 3. Look up the words consistent and inconsistent in a dictionar. How can the meaning of these words help ou distinguish between consistent and inconsistent sstems of equations? Sample answer: ne meaning of consistent is being in agreement, or compatible, while one meaning of inconsistent is not being in agreement or incompatible. When a sstem is consistent, the equations are compatible because both can be true at the same time (for the same values of and ). When a sstem is inconsistent, the equations are incompatible because the can never be true at the same time. Glencoe/McGraw-Hill 23 Glencoe Algebra 2

13 3- NAME DATE PERID Enrichment Investments The following graph shows the value of two different investments over time. Line A represents an initial investment of $30,000 with a bank paing passbook savings interest. Line B represents an initial investment of $5,000 in a profitable mutual fund with dividends reinvested and capital gains accepted in shares. B deriving the linear equation m b for A and B, ou can predict the value of these investments for ears to come. Amount Invested (thousands) A B Years Invested. The -intercept, b, is the initial investment.find b for each of the following. a. line A 30,000 b. line B The slope of the line, m, is the rate of return.find m for each of the following. a. line A b. line B What are the equations of each of the following lines? a. line A b. line B What will be the value of the mutual fund after ears of investment? $35, What will be the value of the bank account after ears of investment? $36, When will the mutual fund and the bank account have equal value? after.67 ears of investment 7. Which investment has the greatest paoff after ears of investment? the mutual fund Glencoe/McGraw-Hill 24 Glencoe Algebra 2

14 3-2 NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations Algebraicall Substitution To solve a sstem of linear equations b substitution, first solve for one variable in terms of the other in one of the equations. Then substitute this epression into the other equation and simplif. Eample Use substitution to solve the sstem of equations Solve the first equation for in terms of. 2 9 First equation Subtract 2 from both sides. Multipl both sides b. Substitute the epression 2 9 for into the second equation and solve for. 3 6 Second equation 3(2 9) 6 Substitute 2 9 for Distributive Propert Simplif. 7 2 Add 27 to each side. 3 Divide each side b 7. Now, substitute the value 3 for in either original equation and solve for. 2 9 First equation 2(3) 9 Replace with Simplif. 3 Subtract 6 from each side. 3 Multipl each side b. The solution of the sstem is (3, 3). Lesson 3-2 Eercises Solve each sstem of linear equations b using substitution (, 0) (2, ) ( 2, 7) no solution ( 2, 4) (3, 9) (4, 2), 3 ( 8, 6) , infinitel man, Glencoe/McGraw-Hill 25 Glencoe Algebra 2

15 3-2 NAME DATE PERID Stud Guide and Intervention (continued) Solving Sstems of Equations Algebraicall Elimination To solve a sstem of linear equations b elimination, add or subtract the equations to eliminate one of the variables. You ma first need to multipl one or both of the equations b a constant so that one of the variables has the same (or opposite) coefficient in one equation as it has in the other. Eample Eample 2 Use the elimination method to solve the sstem of equations. Multipl the second equation b 4. Then subtract the equations to eliminate the variable Multipl b Multipl the first equation b 3 and the second equation b 2. Then add the equations to eliminate the variable Multipl b Multipl b Eercises Replace with 7 and solve for ( 7) The solution is ( 7, 3). Use the elimination method to solve the sstem of equations. Solve each sstem of equations b using elimination. Replace with 2 and solve for ( 2) The solution is ( 2, 5) (3, ) (2, 4) ( 2, ) (4, 0) no solution (2, ) infinitel man (, 3) m 2n m 3n 2 5 (6, 3) (4, 2), 2 ( 4, 6) Glencoe/McGraw-Hill 26 Glencoe Algebra 2

16 3-2 NAME DATE PERID Skills Practice Solving Sstems of Equations Algebraicall Solve each sstem of equations b using substitution.. m n w z m n 4 (8, 2) (3, 2) 2w 3z 2 (3, 2) 4. 3r s b 3c r s 5 (2, ) b c 3 (3, 0) (3, 2) Solve each sstem of equations b using elimination. 7. 2p q j k c 2d 2 3p q 5 (2, ) 3j k 2 (, ) 3c 4d 50 (6, 8) 0. 2f 3g f g 2 (3, ) 2 3 (, ) 2 6 no solution Solve each sstem of equations b using either substitution or elimination. Lesson r t r t 4 (, 6) 4 2, 4 2 (3, 5) 6. 2p 3q w 8z 6 8. c d 6 2p 3q 6 (3, 0) 3w 4z 8 c d 0 (3, 3) infinitel man 9. 2u 4v a b u 2v 3 no solution 3a b 5 (, 2) (4, 2) z c 2d r 2s 3 z 6 ( 2, 0) 2c 5d 3 ( 4, ) 2r 3s 9 ( 3, 5) 25. The sum of two numbers is 2. The difference of the same two numbers is 4. Find the numbers. 4, Twice a number minus a second number is. Twice the second number added to three times the first number is 9. Find the two numbers., 3 Glencoe/McGraw-Hill 27 Glencoe Algebra 2

17 3-2 NAME DATE PERID Practice (Average) Solving Sstems of Equations Algebraicall Solve each sstem of equations b using substitution g 3h 8 no 3 2 (7, 0) 2 (3, 2) g h 9 3 solution 4. 2a 4b 6 infinitel 5. 2m n a 2b 3 man 5m 6n (5, 4) 2 7 ( 3, 2) 7. u 2v w 3z u 2v 5 no solution 4 9 (, 5) 3w 5z 4, Solve each sstem of equations b using elimination. 0. 2r s 5. 2m n r s 20 (5, 5) 3m 2n 30 (4, 9) (, 4) 3. 3j k no 5. 2g h 6 4j k 6 (6, 8) solution 3g 2h 6 (4, 2) 6. 2t 4v 6 infinitel t 2v 3 man (6, 3) (4, 3) Solve each sstem of equations b using either substitution or elimination q 5r infinitel , 3 4q 2r 56 (0, 8) man 22. 4b 2d 5 no 23. s m 2p 0 2b d solution s (, ) 3m 9p 5 2, 25. 5g 4k h z 3 no 3g 5k 7 (6, 5) 2 8 ( 2, 3) 3h 3z 6 solution SPRTS For Eercises 28 and 29, use the following information. Last ear the volleball team paid $5 per pair for socks and $7 per pair for shorts on a total purchase of $35. This ear the spent $342 to bu the same number of pairs of socks and shorts because the socks now cost $6 a pair and the shorts cost $ Write a sstem of two equations that represents the number of pairs of socks and shorts bought each ear , How man pairs of socks and shorts did the team bu each ear? socks: 2, shorts: 5 Glencoe/McGraw-Hill 28 Glencoe Algebra 2

18 3-2 NAME DATE PERID Reading to Learn Mathematics Solving Sstems of Equations Algebraicall Pre-Activit How can sstems of equations be used to make consumer decisions? Read the introduction to Lesson 3-2 at the top of page 6 in our tetbook. How man more minutes of long distance time did Yolanda use in Februar than in Januar? 3 minutes How much more were the Februar charges than the Januar charges? $.04 Using our answers for the questions above, how can ou find the rate per minute? Find $ Reading the Lesson. Suppose that ou are asked to solve the sstem of equations at the right b the substitution method. 3 9 The first step is to solve one of the equations for one variable in terms of the other. To make our work as eas as possible, which equation would ou solve for which variable? Eplain. Sample answer: Solve the second equation for because in that equation the variable has a coefficient of. Lesson Suppose that ou are asked to solve the sstem of equations at the right b the elimination method To make our work as eas as possible, which variable would ou eliminate? Describe how ou would do this. Sample answer: Eliminate the variable ; multipl the second equation b 3 and then add the result to the first equation. Helping You Remember 3. The substitution method and elimination method for solving sstems both have several steps, and it ma be difficult to remember them. You ma be able to remember them more easil if ou notice what the methods have in common. What step is the same in both methods? Sample answer: After finding the value of one of the variables, ou find the value of the other variable b substituting the value ou have found in one of the original equations. Glencoe/McGraw-Hill 29 Glencoe Algebra 2

19 3-2 NAME DATE PERID Enrichment Using Coordinates From one observation point, the line of sight to a downed plane is given b. This equation describes the distance from the observation point to the plane in a straight line. From another observation point, the line of sight is given b 3 2. What are the coordinates of the point at which the crash occurred? Solve the sstem of equations ( ) 2 Substitute for Substitute 6 for The coordinates of the crash are (6, 5). Solve the following.. The lines of sight to a forest fire are as follows. From Ranger Station A: 3 9 From Ranger Station B: Find the coordinates of the fire. 2. An airplane is traveling along the line when it sees another airplane traveling along the line If the continue along the same lines, at what point will their flight paths cross? 3. Two mine shafts are dug along the paths of the following equations If the shafts meet at a depth of 200 feet, what are the coordinates of the point at which the meet? Glencoe/McGraw-Hill 30 Glencoe Algebra 2

20 3-3 NAME DATE PERID Stud Guide and Intervention Solving Sstems of Inequalities b Graphing Graph Sstems of Inequalities To solve a sstem of inequalities, graph the inequalities in the same coordinate plane. The solution set is represented b the intersection of the graphs. Eample Solve the sstem of inequalities b graphing. 2 and 2 3 The solution of 2 is Regions and 2. The solution of 2 is Regions and 3. 3 The intersection of these regions is Region, which is the solution set of the sstem of inequalities. Region 3 Region 2 Region Eercises Solve each sstem of inequalities b graphing Lesson Glencoe/McGraw-Hill 3 Glencoe Algebra 2

21 3-3 NAME DATE PERID Stud Guide and Intervention (continued) Solving Sstems of Inequalities b Graphing Find Vertices of a Polgonal Region Sometimes the graph of a sstem of inequalities forms a bounded region. You can find the vertices of the region b a combination of the methods used earlier in this chapter: graphing, substitution, and/or elimination. Eample Find the coordinates of the vertices of the figure formed b , 2 3, and 3 4. Graph the boundar of each inequalit. The intersections of the boundar lines are the vertices of a triangle. The verte (4, 0) can be determined from the graph. To find the coordinates of the second and third vertices, solve the two sstems of equations and For the first sstem of equations, rewrite the first equation in standard form as 2 3. Then multipl that equation b 4 and add to the second equation. 2 3 Multipl b ( ) Then substitute 3 and solve for in one of the original equations The coordinates of the second verte are,4. Eercises For the second sstem of equations, use substitution. Substitute 2 3 for in the second equation to get 3(2 3) Then substitute in the 5 first equation to solve for The coordinates of the third verte are 3 2, Thus, the coordinates of the three vertices are (4, 0),,4, and 2, 2. Find the coordinates of the vertices of the figure formed b each sstem of inequalities (2, ), ( 4, 2), 3 ( 3, 4), ( 2, 4), (2, 2), (3, 2) ( 3, 4), (3, 2) , Glencoe/McGraw-Hill 32 Glencoe Algebra 2

22 3-3 NAME DATE PERID Skills Practice Solving Sstems of Inequalities b Graphing Solve each sstem of inequalities b graphing no solution Lesson Find the coordinates of the vertices of the figure formed b each sstem of inequalities ( 2, 4), (0, 0), (0, ), (, 0) (0, 3), ( 5, 3), ( 5, 8) ( 2, 4), (2, 0) Glencoe/McGraw-Hill 33 Glencoe Algebra 2

23 3-3 NAME DATE PERID Practice (Average) Solving Sstems of Inequalities b Graphing Solve each sstem of inequalities b graphing Find the coordinates of the vertices of the figure formed b each sstem of inequalities (, 0), (3, 2), (3, 2) ( 2, 4), ( 2, 4), (2, 0) 3 ( 3, 4),,, (3, 4) DRAMA For Eercises 0 and, use the following information. The drama club is selling tickets to its pla. An adult ticket costs $5 and a student ticket costs $. The auditorium will 400 seat 300 ticket-holders. The drama club wants to collect at 350 least $3630 from ticket sales. 0. Write and graph a sstem of four inequalities that describe how man of each tpe of ticket the club must sell to meets its goal. 0, 0, 300, List three different combinations of tickets sold that satisf the inequalities. Sample answer: 250 adult and 50 student, 200 adult and 00 student, 45 adult and 48 student Pla Tickets Adult Tickets Glencoe/McGraw-Hill 34 Glencoe Algebra 2 Student Tickets

24 3-3 NAME DATE PERID Reading to Learn Mathematics Solving Sstems of Inequalities b Graphing Pre-Activit How can ou determine whether our blood pressure is in a normal range? Read the introduction to Lesson 3-3 at the top of page 23 in our tetbook. Satish is 37 ears old. He has a blood pressure reading of 35/99. Is his blood pressure within the normal range? Eplain. Sample answer: No; his sstolic pressure is normal, but his diastolic pressure is too high. It should be between 60 and 90. Reading the Lesson. Without actuall drawing the graph, describe the boundar lines 3 for the sstem of inequalities shown at the right. 5 Two dashed vertical lines ( 3 and 3) and two solid horizontal lines ( 5 and 5) 2. Think about how the graph would look for the sstem given above. What will be the shape of the shaded region? (It is not necessar to draw the graph. See if ou can imagine it without drawing anthing. If this is difficult to do, make a rough sketch to help ou answer the question.) a rectangle 3. Which sstem of inequalities matches the graph shown at the right? B A. 2 B C. 2 D Lesson 3-3 Helping You Remember 4. To graph a sstem of inequalities, ou must graph two or more boundar lines. When ou graph each of these lines, how can the inequalit smbols help ou remember whether to use a dashed or solid line? Use a dashed line if the inequalit smbol is or, because these smbols do not include equalit and the dashed line reminds ou that the line itself is not included in the graph. Use a solid line if the smbol is or, because these smbols include equalit and tell ou that the line itself is included in the graph. Glencoe/McGraw-Hill 35 Glencoe Algebra 2

25 3-3 NAME DATE PERID Enrichment Tracing Strateg Tr to trace over each of the figures below without tracing the same segment twice. A B K J P Q L D C M The figure at the left cannot be traced, but the one at the right can. The rule is that a figure is traceable if it has no more than two points where an odd number of segments meet. The figure at the left has three segments meeting at each of the four corners. However, the figure at the right has onl two points, L and Q, where an odd number of segments meet. Determine if each figure can be traced without tracing the same segment twice. If it can, then name the starting point and name the segments in the order the should be traced.. E F 2. A B C W X Y D E F K H es; X; X Y, Y F, F X, X G, G F, C K, F E, E H, H X, X W, W H, H G G G H J es; E; E D, D A, A E, E B, B F, F C, K F, F J, J H, H F, F E, E H, H G, G E 3. T U P V S R Glencoe/McGraw-Hill 36 Glencoe Algebra 2

26 3-4 NAME DATE PERID Stud Guide and Intervention Linear Programming Maimum and Minimum Values When a sstem of linear inequalities produces a bounded polgonal region, the maimum or minimum value of a related function will occur at a verte of the region. Eample Graph the sstem of inequalities. Name the coordinates of the vertices of the feasible region. Find the maimum and minimum values of the function f(, ) 3 2 for this polgonal region First find the vertices of the bounded region. Graph the inequalities. The polgon formed is a quadrilateral with vertices at (0, 4), (2, 4), (5, ), and (, 2). Use the table to find the maimum and minimum values of f(, ) 3 2. (, ) 3 2 f(, ) (0, 4) 3(0) 2(4) 8 (2, 4) 3(2) 2(4) 4 (5, ) 3(5) 2() 7 (, 2) 3( ) 2( 2) 7 The maimum value is 7 at (5, ). The minimum value is 7 at (, 2). Eercises Graph each sstem of inequalities. Name the coordinates of the vertices of the feasible region. Find the maimum and minimum values of the given function for this region f(, ) 3 2 f(, ) 4 f(, ) 4 3 Lesson 3-4 vertices: (, 2), (, 4), vertices: ( 5, 2), vertices (0, 2), (4, 3), (5, 8), (5, 2); ma: ; (3, 2), (, 2); 7, ; ma: 25; min: 6 min; 5 ma: 0; min: 8 Glencoe/McGraw-Hill 37 Glencoe Algebra 2

27 3-4 NAME Real-World Problems When solving linear programming problems, use the following procedure.. Define variables. 2. Write a sstem of inequalities. 3. Graph the sstem of inequalities. 4. Find the coordinates of the vertices of the feasible region. 5. Write an epression to be maimized or minimized. 6. Substitute the coordinates of the vertices in the epression. 7. Select the greatest or least result to answer the problem. Eample A painter has eactl 32 units of ellow de and 54 units of green de. He plans to mi as man gallons as possible of color A and color B. Each gallon of color A requires 4 units of ellow de and unit of green de. Each gallon of color B requires unit of ellow de and 6 units of green de. Find the maimum number of gallons he can mi. Step Define the variables. the number of gallons of color A made the number of gallons of color B made Step 2 Write a sstem of inequalities. Since the number of gallons made cannot be negative, 0 and 0. There are 32 units of ellow de; each gallon of color A requires 4 units, and each gallon of color B requires unit. So Similarl for the green de, Steps 3 and 4 Graph the sstem of inequalities (0, 9) and find the coordinates of the vertices of the feasible region. The vertices of the feasible region are (0, 0), (0, 9), (6, 8), and (8, 0). Steps 5 7 Find the maimum number of gallons,, that he can make. The maimum number of gallons the painter can make is 4, 6 gallons of color A and 8 gallons of color B. Eercises DATE PERID Stud Guide and Intervention (continued) Linear Programming Color A (gallons) (, ) f(, ) (0, 0) (0, 9) (6, 8) (8, 0) FD A delicatessen has 8 pounds of plain sausage and 0 pounds of garlic-flavored sausage. The deli wants to make as much bratwurst as possible. Each pound of 3 bratwurst requires pound of plain sausage and pound of garlic-flavored sausage. 4 4 Find the maimum number of pounds of bratwurst that can be made lb 2. MANUFACTURING Machine A can produce 30 steering wheels per hour at a cost of $6 per hour. Machine B can produce 40 steering wheels per hour at a cost of $22 per hour. At least 360 steering wheels must be made in each 8-hour shift. What is the least cost involved in making 360 steering wheels in one shift? $94 Glencoe/McGraw-Hill 38 Glencoe Algebra 2 Color B (gallons) (6, 8) (8, 0)

28 3-4 NAME DATE PERID Skills Practice Linear Programming Graph each sstem of inequalities. Name the coordinates of the vertices of the feasible region. Find the maimum and minimum values of the given function for this region f(, ) f(, ) 3 f(, ) ma.: 9, min.: 3 ma.: 2, min.: 5 ma.: 2, min.: f(, ) f(, ) 4 3 f(, ) 3 5 ma.: 3, no min. no ma., min.: 20 ma.: 22, min.: 2 7. MANUFACTURING A backpack manufacturer produces an internal frame pack and an eternal frame pack. Let represent the number of internal frame packs produced in one hour and let represent the number of eternal frame packs produced in one hour. Then the inequalities 3 8, 2 6, 0, and 0 describe the constraints for manufacturing both packs. Use the profit function f() and the constraints given to determine the maimum profit for manufacturing both backpacks for the given constraints. $620 Lesson 3-4 Glencoe/McGraw-Hill 39 Glencoe Algebra 2

29 3-4 NAME DATE PERID Practice (Average) Linear Programming Graph each sstem of inequalities. Name the coordinates of the vertices of the feasible region. Find the maimum and minimum values of the given function for this region f(, ) 2 f(, ) f(, ) 3 ma.: 8, min.: 4 ma.: 2, min.: 6 ma.: 5, min.: f(, ) 4 f(, ) 3 0 f(, ) ma.: 28, min.: 0 ma.:, no min. no ma., min.: 7 PRDUCTIN For Eercises 7 9, use the following information. A glass blower can form 8 simple vases or 2 elaborate vases in an hour. In a work shift of no more than 8 hours, the worker must form at least 40 vases. 7. Let s represent the hours forming simple vases and e the hours forming elaborate vases. Write a sstem of inequalities involving the time spent on each tpe of vase. s 0, e 0, s e 8, 8s 2e If the glass blower makes a profit of $30 per hour worked on the simple vases and $35 per hour worked on the elaborate vases, write a function for the total profit on the vases. f(s, e) 30s 35e 9. Find the number of hours the worker should spend on each tpe of vase to maimize profit. What is that profit? 4 h on each; $260 Glencoe/McGraw-Hill 40 Glencoe Algebra 2

30 3-4 NAME DATE PERID Reading to Learn Mathematics Linear Programming Pre-Activit How is linear programming used in scheduling work? Reading the Lesson. Complete each sentence. Read the introduction to Lesson 3-4 at the top of page 29 in our tetbook. Name two or more facts that indicate that ou will need to use inequalities to model this situation. Sample answer: The buo tender can carr up to 8 new buos. There seems to be a limit of 24 hours on the time the crew has at sea. The crew will want to repair or replace the maimum number of buos possible. a. When ou find the feasible region for a linear programming problem, ou are solving a sstem of linear inequalities called constraints. The points in the feasible region are solutions of the sstem. b. The corner points of a polgonal region are the vertices of the feasible region. 2. A polgonal region alwas takes up onl a limited part of the coordinate plane. ne wa to think of this is to imagine a circle or rectangle that the region would fit inside. In the case of a polgonal region, ou can alwas find a circle or rectangle that is large enough to contain all the points of the polgonal region. What word is used to describe a region that can be enclosed in this wa? What word is used to describe a region that is too large to be enclosed in this wa? bounded; unbounded 3. How do ou find the corner points of the polgonal region in a linear programming problem? You solve a sstem of two linear equations. 4. What are some everda meanings of the word feasible that remind ou of the mathematical meaning of the term feasible region? Sample answer: possible or achievable Helping You Remember 5. Look up the word constraint in a dictionar. If more than one definition is given, choose the one that seems closest to the idea of a constraint in a linear programming problem. How can this definition help ou to remember the meaning of constraint as it is used in this lesson? Sample answer: A constraint is a restriction or limitation. The constraints in a linear programming problem are restrictions on the variables that translate into inequalit statements. Lesson 3-4 Glencoe/McGraw-Hill 4 Glencoe Algebra 2

31 3-4 NAME DATE PERID Enrichment Computer Circuits and Logic Computers operate according to the laws of logic. The circuits of a computer can be described using logic.. With switch A open, no current flows. The value 0 is assigned A to an open switch. 2. With switch A closed, current flows. The value is assigned A to a closed switch. 3. With switches A and B open, no current flows. This circuit A B can be described b the conjunction, A B. A 4. In this circuit, current flows if either A or B is closed. This circuit can be described b the disjunction, A B. B A B A B Truth tables are used to describe the flow of current in a circuit. The table at the left describes the circuit in diagram 4. According to the table, the onl time current does not flow through the circuit is when both switches A and B are open. Draw a circuit diagram for each of the following.. (A B) C 2. (A B) C 3. (A B) (C D) 4. (A B) (C D) 5. Construct a truth table for the following circuit. A B C Glencoe/McGraw-Hill 42 Glencoe Algebra 2

32 3-5 NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations in Three Variables Sstems in Three Variables Use the methods used for solving sstems of linear equations in two variables to solve sstems of equations in three variables. A sstem of three equations in three variables can have a unique solution, infinitel man solutions, or no solution. A solution is an ordered triple. Eample Solve this sstem of equations. 3 z 6 2 2z 8 4 3z 2 Step Use elimination to make a sstem of two equations in two variables. 3 z 6 First equation 2 2z 8 Second equation ( ) 2 2z 8 Second equation ( ) 4 3z 2 Third equation 5 z 2 Add to eliminate. 6 z 3 Add to eliminate. Step 2 Solve the sstem of two equations. 5 z 2 ( ) 6 z 3 Add to eliminate z. Divide both sides b. Substitute for in one of the equations with two variables and solve for z. 5 z 2 5( ) z 2 5 z 2 z 7 Equation with two variables Replace with. Multipl. Add 5 to both sides. The result so far is and z 7. Step 3 Substitute for and 7 for z in one of the original equations with three variables. 3 z 6 3( ) 7 6 Replace with and z with Multipl. 4 Simplif. The solution is (, 4, 7). Eercises riginal equation with three variables Solve each sstem of equations z z 3. 2 z 8 2 4z 4 2 6z 2 z 0 3 8z z 3 6 3z 24 (4, 3, ) 2, 5, infinitel man solutions 4. 3 z z z z 4 8 2z z z 2 3 z , 2, 5 no solution 6,, Lesson 3-5 Glencoe/McGraw-Hill 43 Glencoe Algebra 2

33 3-5 Real-World Problems Eample NAME DATE PERID The Laredo Sports Shop sold 0 balls, 3 bats, and 2 bases for $99 on Monda. n Tuesda the sold 4 balls, 8 bats, and 2 bases for $78. n Wednesda the sold 2 balls, 3 bats, and base for $ What are the prices of ball, bat, and base? First define the variables. price of ball price of bat z price of base Translate the information in the problem into three equations z z z Stud Guide and Intervention (continued) Solving Sstems of Equations in Three Variables Subtract the second equation from the first equation to eliminate z z 99 ( ) 4 8 2z Multipl the third equation b 2 and subtract from the second equation z 78 ( ) 4 6 2z So a ball costs $8, a bat $5.40, and a base $.40. Substitute 5.40 for in the equation (5.40) Substitute 8 for and 5.40 for in one of the original equations to solve for z z 99 0(8) 3(5.40) 2z z 99 2z 2.80 z.40 Eercises. FITNESS TRAINING Carl is training for a triathlon. In her training routine each week, she runs 7 times as far as she swims, and she bikes 3 times as far as she runs. ne week she trained a total of 232 miles. How far did she run that week? 56 miles 2. ENTERTAINMENT At the arcade, Ran, Sara, and Tim plaed video racing games, pinball, and air hocke. Ran spent $6 for 6 racing games, 2 pinball games, and game of air hocke. Sara spent $2 for 3 racing games, 4 pinball games, and 5 games of air hocke. Tim spent $2.25 for 2 racing games, 7 pinball games, and 4 games of air hocke. How much did each of the games cost? Racing game: $0.50; pinball: $0.75; air hocke: $ FD A natural food store makes its own brand of trail mi out of dried apples, raisins, and peanuts. ne pound of the miture costs $3.8. It contains twice as much peanuts b weight as apples. ne pound of dried apples costs $4.48, a pound of raisins $2.40, and a pound of peanuts $3.44. How man ounces of each ingredient are contained in pound of the trail mi? 3 oz of apples, 7 oz of raisins, 6 oz of peanuts Glencoe/McGraw-Hill 44 Glencoe Algebra 2

34 3-5 NAME DATE PERID Skills Practice Solving Sstems of Equations in Three Variables Solve each sstem of equations.. 2a c 0 (5, 5, 20) 2. z 3 (0, 2, ) b c 5 3 2z 2 a 2b c 5 5z z 6 ( 3, 2, ) 4. 4 z no solution z 0 z 4 z z 6 (2,, 3) z 2 ( 2,, 3) 2 3 z 2 6 2z 2 2 3z z 5 (0, 0, ) 8. 3r 2t (, 6, 2) 3 0 4r s 2t 6 3 2z 2 r s 4t z 3 no solution 0. 5m 3n p 4 ( 2, 3, 5) 2 2 6z 6 3m 2n 0 5z 3 2m n 3p z 2 infinitel man 2. 2 z 4 (, 2, ) 2 3 2z 4 3 2z 3 z 3 z z (5, 7, 0) z 2 infinitel man z 2 4 2z z z z 2 (, 3, ) z 0 (3, 0, 2) z 3 2 5z z z z 2 (, 2, 3) z infinitel man 3 3 z 0 2 z 6 z z 3 9. The sum of three numbers is 8. The sum of the first and second numbers is 5, and the first number is 3 times the third number. Find the numbers. 9, 6, 3 Lesson 3-5 Glencoe/McGraw-Hill 45 Glencoe Algebra 2

35 3-5 NAME DATE PERID Practice (Average) Solving Sstems of Equations in Three Variables Solve each sstem of equations.. 2 2z z a b 3 z z 22 b c 3 3 2z 8 4z 6 a 2c 0 (3,, 5) (, 4, 4) (2,, 4) 4. 3m 2n 4p g 3h 8j z 8 m n p 3 g 4h 4 2z 3 m 4n 5p 0 2g 3h 8j 5 3 2z 5 (3, 3, 3) no solution ( 2, 3, ) z z 2 9. p 4r z z 0 p 3q z 7 5 2z 4 q r (5,, 0) (2, 2, 2) (, 3, 2) z 8. d 3e f z z 0 d 2e f 4 2z z 4 4d e f 2 3 2z 2 infinitel man (,, 2) (2, 3, 4) z z z 0 2 z z z z z z 9 ( 7, 6, ) infinitel man (, 3, 2) 6. 2u v w z z 3u 2v 3w z 22 2 z 6 u v 2w z z (0,, 3) (, 2, 3) no solution z z z z z 4 0 2z z z z 9 (4, 5, 0) (2, 2, 3) (, 2, 5) 22. The sum of three numbers is 6. The third number is the sum of the first and second numbers. The first number is one more than the third number. Find the numbers. 4,, The sum of three numbers is 4. The second number decreased b the third is equal to the first. The sum of the first and second numbers is 5. Find the numbers. 3, 2, 24. SPRTS Aleandria High School scored 37 points in a football game. Si points are awarded for each touchdown. After each touchdown, the team can earn one point for the etra kick or two points for a 2-point conversion. The team scored one fewer 2-point conversions than etra kicks. The team scored 0 times during the game. How man touchdowns were made during the game? 5 Glencoe/McGraw-Hill 46 Glencoe Algebra 2

36 3-5 NAME DATE PERID Reading to Learn Mathematics Solving Sstems of Equations in Three Variables Pre-Activit How can ou determine the number and tpe of medals U.S. lmpians won? Reading the Lesson Read the introduction to Lesson 3-5 at the top of page 38 in our tetbook. At the 996 Summer lmpics in Atlanta, Georgia, the United States won 0 medals. The U.S. team won 2 more gold medals than silver and 7 fewer bronze medals than silver. Using the same variables as those in the introduction, write a sstem of equations that describes the medals won for the 996 Summer lmpics. g s b 0; g s 2; b s 7. The planes for the equations in a sstem of three linear equations in three variables determine the number of solutions. Match each graph description below with the description of the number of solutions of the sstem. (Some of the items on the right ma be used more than once, and not all possible tpes of graphs are listed.) a. three parallel planes II I. one solution b. three planes that intersect in a line III II. no solutions c. three planes that intersect in one point I III. infinite solutions d. one plane that represents all three equations III 2. Suppose that three classmates, Monique, Josh, and Lill, are studing for a quiz on this lesson. The work together on solving a sstem of equations in three variables,,, and z, following the algebraic method shown in our tetbook. The first find that z 3, then that 2, and finall that. The students agree on these values, but disagree on how to write the solution. Here are their answers: Monique: (3, 2, ) Josh: ( 2,, 3) Lill: (, 2, 3) a. How do ou think each student decided on the order of the numbers in the ordered triple? Sample answer: Monique arranged the values in the order in which she found them. Josh arranged them from smallest to largest. Lill arranged them in alphabetical order of the variables. b. Which student is correct? Lill Helping You Remember 3. How can ou remember that obtaining the equation 0 0 indicates a sstem with infinitel man solutions, while obtaining an equation such as 0 8 indicates a sstem with no solutions? 0 0 is alwas true, while 0 8 is never true. Lesson 3-5 Glencoe/McGraw-Hill 47 Glencoe Algebra 2

37 3-5 NAME DATE PERID Enrichment Billiards The figure at the right shows a billiard table. The object is to use a cue stick to strike the ball at point C so that the ball will hit the sides (or cushions) of the table at least once before hitting the ball located at point A. In plaing the game, ou need to locate point P. Step Find point B so that BC ST and BH CH. B is called the reflected image of C in ST. K T C H B P A R S Step 2 Draw AB. Step 3 AB intersects ST at the desired point P. For each billiards problem, the cue ball at point C must strike the indicated cushion(s) and then strike the ball at point A. Draw and label the correct path for the cue ball using the process described above.. cushion KR 2. cushion RS K A R K R C A C T S T S 3. cushion TS, then cushion RS 4. cushion KT, then cushion RS K R K R A C C T S A T S Glencoe/McGraw-Hill 48 Glencoe Algebra 2

38 Glencoe/McGraw-Hill A2 Glencoe Algebra 2 3- NAME DATE PERID Stud Guide and Intervention Solving Sstems of Equations b Graphing Graph Sstems of Equations A sstem of equations is a set of two or more equations containing the same variables. You can solve a sstem of linear equations b graphing the equations on the same coordinate plane. If the lines intersect, the solution is that intersection point. Eample Solve the sstem of equations b graphing Write each equation in slope-intercept form The graphs appear to intersect at (0, 2). CHECK Substitute the coordinates into each equation ( 2) 4 0 ( 2) The solution of the sstem is (0, 2). Eercises Solve each sstem of equations b graphing (6, ) 4 (2, 2) 2 4 (4, ) (, 3) ( 4, 3) 2 2 ( 2, 3) (6, ) (, 3) ( 4, 3) ( 2, 3) Glencoe/McGraw-Hill 9 Glencoe Algebra 2 (2, 2) (4, ) (0, 2) Lesson 3-3- Classif Sstems of Equations The following chart summarizes the possibilities for graphs of two linear equations in two variables. Graphs of Equations Slopes of Lines Classification of Sstem Number of Solutions Lines intersect Different slopes Consistent and independent ne Lines coincide (same line) Lines are parallel Same slope, same -intercept Same slope, different -intercepts Consistent and dependent Inconsistent Infinitel man Glencoe/McGraw-Hill 20 Glencoe Algebra 2 None Eample Graph the sstem of equations 3 6 and describe it as consistent and independent, 2 3 consistent and dependent, or inconsistent. Write each equation in slope-intercept form The graphs intersect at ( 3, 3). Since there is one solution, the sstem is consistent and independent. Eercises Graph the sstem of equations and describe it as consistent and independent, consistent and dependent, or inconsistent consistent inconsistent and dependent inconsistent consistent 2 2 consistent 6 consistent and independent and dependent and independent NAME DATE PERID Stud Guide and Intervention (continued) Solving Sstems of Equations b Graphing ( 3, 3) Answers (Lesson 3-)

39 Glencoe/McGraw-Hill A3 Glencoe Algebra 2 3- NAME DATE PERID Skills Practice Solving Sstems of Equations B Graphing Solve each sstem of equations b graphing (2, 0) 2 4 (2, 0) 2 (2, 2) (3, ) 5 3 (3, 4) 3 4 (, ) (3, ) (2, ) (4, 0) 2 ( 2, 5) Graph each sstem of equations and describe it as consistent and independent, consistent and dependent, or inconsistent (2, 0) (2, ) (3, 4) inconsistent consistent and consistent and dependent independent Glencoe/McGraw-Hill 2 Glencoe Algebra 2 (4, 0) (2, 0) ( 2, 5) 2 (, ) (2, 2) 2 (5, 0) Lesson 3-3- Solve each sstem of equations b graphing (2, ) 2 3 (2, ) (3, 3) ( 2, ) 2 2 (2, 2) (, ) ( 2, ) NAME DATE PERID Practice (Average) Solving Sstems of Equations B Graphing Graph each sstem of equations and describe it as consistent and independent, consistent and dependent, or inconsistent (2, ) (2, 0) consistent and indep. inconsistent consistent and dep. SFTWARE For Eercises 0 2, use the following information. Location Mapping needs new software. Software A costs $3,000 plus $500 per additional site license. Software B costs $2500 plus $200 per additional site license. 0. Write two equations that represent the cost of each software. A: 3, , B: Graph the equations. Estimate the break-even point of the software costs. 5 additional licenses 2. If Location Mapping plans to bu 0 additional site licenses, which software will cost less? B Software Costs (5, 20,500) Glencoe/McGraw-Hill 22 Glencoe Algebra 2 (2, 2) (2, ) Total Cost ($) 24,000 20,000 6,000 2,000 8,000 4,000 A (3, 3) B (, ) Additional Licenses Answers (Lesson 3-) Answers