Principles of Mathematics 10 PDF

6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page i Series Author and Senior Consultant Marian Small Lead Author...

6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page i Series Author and Senior Consultant Marian Small Lead Author Chris Kirkpatrick Authors Mary Bourassa Crystal Chilvers Santo D’Agostino Ian Macpherson John Rodger Susanne Trew  Principles of Mathematics 10 Series Author and Senior Authors Technology Consultant Consultant Mary Bourassa, Crystal Chilvers, Ian McTavish Marian Small Santo D’Agostino, Ian Macpherson, John Rodger, Susanne Trew Lead Author Chris Kirkpatrick Contributing Authors Dan Charbonneau, Ralph Montesanto, Christine Suurtamm Vice President, Publishing Contributing Editors Senior Production Coordinator Janice Schoening Anthony Arrizza, David Cowan, Kathrine Pummell Beverly Farahani, David Gargaro, General Manager, Mathematics, Elizabeth Pattison Design Director Science, and Technology Ken Phipps Lenore Brooks Editorial Assistants Rachelle Boisjoli Asset Coordinator Publisher, Mathematics Kathryn Chris Suzanne Peden Colin Garnham Executive Director, Content and Interior Design Associate Publisher, Mathematics Media Production Media Services Sandra McTavish Renate McCloy Cover Design Managing Editor, Mathematics Director, Content and Media Courtney Hellam Erynn Marcus Production Cover Image Product Manager Linh Vu © CanStock Images / Alamy Linda Krepinsky Senior Content Production Editor Production Services Program Manager Debbie Davies-Wright Nesbitt Graphics Inc. 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ISBN-10: 0-17-633202-2 be reproduced, transcribed, or Further questions about Printed and bound in Canada used in any form or by any means— permissions can be emailed to 2 3 4 5 12 11 10 09 graphic, electronic, or mechanical, [email protected] including photocopying, recording, For more information contact taping, Web distribution, or Every effort has been made to Nelson Education Ltd., information storage and retrieval trace ownership of all copyrighted 1120 Birchmount Road, Toronto, systems—without the written material and to secure permission ON, M1K 5G4. permission of the publisher. from copyright holders. In the Or you can visit our Internet site at event of any question arising as to http://www.nelson.com the use of any material, we will be pleased to make the necessary corrections in future printings. 6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page iii Reviewers and Advisory Panel Paul Alves Patricia Kehoe Kathy Pilon Department Head of Mathematics Itinerant Teacher, Student Success Program Leader Stephen Lewis Secondary School Department St. John Catholic High School Peel District School Board Ottawa Catholic School Board Catholic District School Board of Eastern Mississauga, ON Ottawa, ON Ontario Perth, ON Anthony Arrizza Michelle Lang Department Head of Mathematics Consultant, Learning Services 7–12 Jennifer Portelli Woodbridge College Waterloo Region District School Board Teacher York Region District School Board Kitchener, ON Holy Cross Catholic Elementary School Woodbridge, ON Dufferin-Peel Catholic District School Board Angelo Lillo Mississauga, ON Terri Blackwell Head of Mathematics Secondary Mathematics Teacher Sir Winston Churchill Secondary School Tamara Porter Burlington Central High School District School Board of Niagara Department Head of Mathematics Halton District School Board St. Catharines, ON Prince Edward Collegiate Institute Burlington, ON Hastings and Prince Edward District School Susan MacRury Board Mark Cassar Senior Mathematics Teacher Picton, ON Principal Lasalle Secondary School Holy Cross Catholic Elementary School Rainbow District School Board Margaret Russo Dufferin-Peel Catholic District School Board Sudbury, ON Mathematics Teacher Mississauga, ON Madonna Catholic Secondary School Frank Maggio Toronto Catholic District School Board Angela Conetta Department Head of Mathematics Toronto, ON Mathematics Teacher Holy Trinity Catholic Secondary School Chaminade College School Halton Catholic District School Board Scott Taylor Toronto Catholic District School Board Oakville, ON Department Head of Mathematics, Computer Toronto, ON Science and Business Peter Matijosaitis Bell High School Tamara Coyle Retired Ottawa-Carleton District School Board Teacher Toronto Catholic District School Board Nepean, ON Mother Teresa Catholic High School Bob McRoberts Ottawa Catholic School Board Joyce Tonner Head of Mathematics Nepean, ON Educator Dr. G W Williams Secondary School Thames Valley District School Board Justin de Weerdt York Region District School Board London, ON Mathematics Department Head Aurora, ON Huntsville High School Salvatore Trabona Cheryl McQueen Trillium Lakelands District School Board Mathematics Department Head Mathematics Learning Coordinator Huntsville, ON Madonna Catholic Secondary School Thames Valley District School Board Toronto Catholic District School Board Sandra Emms Jones London, ON Toronto, ON Math Teacher Kay Minter Forest Heights C.I. James Williamson Teacher Waterloo Region District School Board Teacher Cedarbrae C.I. Kitchener, ON St. Joseph-Scollard Hall C.S.S Toronto District School Board Nipissing-Parry Sound Catholic District School Beverly Farahani Toronto, ON Board Head of Mathematics Reshida Nezirevic North Bay, ON Kingston Collegiate and Vocational Institute Head of Mathematics Limestone District School Board Charles Wyszkowski Blessed Mother Teresa C.S.S Kingston, ON Instructor Toronto Catholic District School Board School of Education, Trent University Richard Gallant Scarborough, ON Peterborough, ON Secondary Curriculum Consultant Elizabeth Pattison Simcoe Muskoka Catholic District School Krista Zupan Mathematics Department Head Board Math Consultant (Numeracy K–12) Grimsby Secondary School Barrie, ON Durham Catholic District School Board District School Board of Niagara Oshawa, ON Jacqueline Hill Grimsby, ON K–12 Mathematics Facilitator Durham District School Board Whitby, ON NEL iii 6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page iv Table of Contents Chapter 1: Systems of Linear Equations 2 Chapter 3: Graphs of Quadratic Relations 128 Getting Started 4 1.1 Representing Linear Relations 8 Getting Started 130 1.2 Solving Linear Equations 15 3.1 Exploring Quadratic Relations 134 1.3 Graphically Solving Linear Systems 21 3.2 Properties of Graphs of Quadratic Relations 138 Curious Math 29 Curious Math 149 3.3 Factored Form of a Quadratic Relation 150 Mid-Chapter Review 30 Mid-Chapter Review 159 1.4 Solving Linear Systems: Substitution 33 1.5 Equivalent Linear Systems 41 3.4 Expanding Quadratic Expressions 161 1.6 Solving Linear Systems: Elimination 49 3.5 Quadratic Models Using Factored Form 169 1.7 Exploring Linear Systems 57 3.6 Exploring Quadratic and Exponential Graphs 179 Chapter Review 60 Chapter Review 183 Chapter Self-Test 64 Chapter Self-Test 187 Chapter Task 65 Chapter Task 188 Chapter 2: Analytic Geometry: Chapters 1–3 Cumulative Review 189 Line Segments and Circles 66 Getting Started 68 Chapter 4: Factoring Algebraic Expressions 192 2.1 Midpoint of a Line Segment 72 2.2 Length of a Line Segment 81 Getting Started 194 2.3 Equation of a Circle 88 4.1 Common Factors in Polynomials 198 Mid-Chapter Review 94 4.2 Exploring the Factorization of Trinomials 205 2.4 Classifying Figures on a Coordinate Grid 96 4.3 Factoring Quadratics: x 2 + bx + c 207 2.5 Verifying Properties of Geometric Figures 104 Mid-Chapter Review 214 2.6 Exploring Properties of Geometric Figures 111 4.4 Factoring Quadratics: ax 2 + bx + c 217 Curious Math 114 4.5 Factoring Quadratics: Special Cases 225 2.7 Using Coordinates to Solve Problems 115 Curious Math 232 Chapter Review 122 4.6 Reasoning about Factoring Polynomials 233 Chapter Self-Test 126 Chapter Review 238 Chapter Task 127 Chapter Self-Test 242 Chapter Task 243 iv Table of Contents NEL 6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page v Chapter 5: Applying Quadratic 7.2 Solving Similar Triangle Problems 382 Models 244 Mid-Chapter Review 389 Getting Started 246 7.3 Exploring Similar Right Triangles 391 5.1 Stretching/Reflecting Quadratic Relations 250 7.4 The Primary Trigonometric Ratios 394 5.2 Exploring Translations of Quadratic Relations 259 7.5 Solving Right Triangles 400 5.3 Graphing Quadratics in Vertex Form 263 Curious Math 407 7.6 Solving Right Triangle Problems 408 Mid-Chapter Review 273 Chapter Review 415 5.4 Quadratic Models Using Vertex Form 275 5.5 Solving Problems Using Quadratic Relations 285 Chapter Self-Test 418 Curious Math 296 Chapter Task 419 5.6 Connecting Standard and Vertex Forms 297 Chapter Review 303 Chapter 8: Acute Triangle Trigonometry 420 Chapter Self-Test 306 Getting Started 422 Chapter Task 307 8.1 Exploring the Sine Law 426 Chapter 6: Quadratic Equations 308 8.2 Applying the Sine Law 428 Getting Started 310 Mid-Chapter Review 435 6.1 Solving Quadratic Equations 314 8.3 Exploring the Cosine Law 437 6.2 Exploring the Creation of Perfect Squares 322 Curious Math 439 Curious Math 324 8.4 Applying the Cosine Law 440 6.3 Completing the Square 325 8.5 Solving Acute Triangle Problems 446 Mid-Chapter Review 333 Chapter Review 452 6.4 The Quadratic Formula 336 Chapter Self-Test 454 6.5 Interpreting Quadratic Equation Roots 345 Chapter Task 455 6.6 Solving Problems Using Quadratic Models 352 Chapter Review 360 Chapters 7–8 Cumulative Review 456 Chapter Self-Test 363 Appendix A: Chapter Task 364 REVIEW OF ESSENTIAL SKILLS AND KNOWLEDGE 459 Appendix B: Chapters 4–6 Cumulative Review 365 REVIEW OF TECHNICAL SKILLS 486 Chapter 7: Similar Triangles Glossary 528 and Trigonometry 368 Answers 536 Getting Started 370 Index 600 7.1 Congruence and Similarity in Triangles 374 Credits 604 NEL Table of Contents v 6706_PM10SB_C1_CO_pp2-3.qxd 5/7/09 3:40 PM Page 2 2 NEL 6706_PM10SB_C1_CO_pp2-3.qxd 5/7/09 3:40 PM Page 3 Chapter 1 Systems of Linear Equations GOALS You will be able to Solve a system of linear equations using a variety of strategies Solve problems that are modelled by linear equations or systems of linear equations Describe the relationship between the number of solutions to a system of linear equations and the coefficients of the equations Comparing Light Bulb Costs y 40 Cost of bulb and electricity ($) 30 20 ? Why does it make sense to buy energy-efficient compact 10 fluorescent light bulbs, even x 0 though they often cost more 5000 10000 than incandescent light bulbs? Hours of use incandescent light bulb compact fluorescent light bulb NEL 3 6706_PM10SB_C1_GS_pp4-7.qxd 5/7/09 3:41 PM Page 4 1 Getting Started WORDS YOU NEED to Know 1. Complete each sentence using one or more of the given words. Each word can be used only once. i) x-intercept v) coefficient ii) y-intercept vi) point of intersection iii) equation vii) solution iv) variable a) The place where a graph crosses the x-axis is called the _____. b) In the _____ y  5x  2, 5 is a _____ of the _____ x. c) Let x  0 to determine the _____ of y  4x  7. d) You can determine the _____ to 20  3x  10 by graphing y  3x – 10. e) The ordered pair at which two lines cross is called the _____. SKILLS AND CONCEPTS You Need Graphing a Linear Relation You can use different tools and strategies to graph a linear relation: Study Aid a table of values the x- and y-intercepts For more help and the slope and y-intercept a graphing calculator practice, see Appendix A-6 and A-7. EXAMPLE Graph 3x  2y  9. Solution Using the x- and y-intercepts y 5 Let y  0 to determine the x-intercept. (0, 4.5) 4 3x  2(0)  9 3x  9 3 x3 2 The graph passes through (3, 0). 1 Let x  0 to determine the y-intercept. (3, 0) x 3(0)  2y  9 -1 0 1 2 3 4 -1 2y  9 3x  2y  9 y  4.5 -2 The graph passes through (0, 4.5). Plot the intercepts, and join them with a straight line. 4 Getting Started NEL 6706_PM10SB_C1_GS_pp4-7.qxd 5/7/09 3:41 PM Page 5 Getting Started Using the Slope and y-intercept y 5 (0, 4.5) 3x + 2y = 9 4 2y = - 3x + 9 3 (1, 3) 2y - 3x 9 = + 2 2 2 2 y = - 1.5x + 4.5 1 The slope is 1.5. The y-intercept is 4.5, so the line passes through (0, 4.5). x Plot (0, 4.5). Use the rise and run to locate a second point on the line, 0 -1 1 2 3 4 by going right 1 unit and down 1.5 units to (1, 3). -1 3x  2y  9 2. Graph each relation using the slope and y-intercept. -2 a) y = 4x - 7 b) x + 2y = 3 3. Graph each relation using the x- and y-intercepts. a) 4x - 5y = 10 b) y = 2 - 3x 4. Graph each relation using the strategy of your choice. a) x - 3y = 6 b) y = 5 - 2x Expanding and Simplifying an Algebraic Expression You can use an algebra tile model to visualize and simplify an expression. Study Aid If the expression has brackets, you can use the distributive property For more help and to expand it. You can add or subtract like terms. practice, see EXAMPLE Appendix A-8. Expand and simplify 2(3x  1)  3(x  2). Solution Using an Algebra Tile Model Using Symbols 2(3x  1)  3(x  2) 1 1 2(3x  1)  3(x  2) x x x x 1 1 1 1 1 1  6x  2  3x  6 x x x x x  6x  3x  2  6  9x  4 1 1 1 1 1 1 1 1 x x x x x x x x x  9x  4 5. Expand and simplify as necessary. a) 5x  10  3x  12 d) (3x  6)  (2x  7) b) 4(3x  5) e) 6(2x  4)  3(2x  1) c) 2(5x  2) f ) (8x  14)  (7x  6) NEL Chapter 1 5 6706_PM10SB_C1_GS_pp4-7.qxd 5/7/09 3:41 PM Page 6 Study Aid PRACTICE For help, see the Review of Essential Skills and 6. Rearrange each equation to complete the table. Knowledge Appendix. Ax  By  C  0 Form y  mx  b Form Question Appendix a) 3x  4y  6  0 6, 7 A-7 b) y  2x  5 10 A-10 c) 4x  7y  3  0 11, 12 A-9 2 5 d) y = - x - 3 6 7. State the slope and y-intercept of each relation. Then sketch the graph. a) y = 3x - 5 c) y = 0.5x 2 b) y = - x + 1 d) y = 2.6x - 1.2 3 8. Which relations in question 7 are direct variations? Which are partial variations? Explain how you know. Kyle’s Journey Home 9. The graph at the left shows Kyle’s distance from home as he cycles from School home from school. y 6 a) How far is the school from Kyle’s home? Distance from home (km) b) At what speed does Kyle cycle? 4 10. State whether each relation is linear or nonlinear. Explain how you know. a) y  3x  6 b) x 1 2 3 4 5 6 2 y 7 9 11 13 15 17 x c) y = 5x 2 + 6x - 4 0 5 10 15 20 25 d) x 1 2 3 4 5 6 Time (min) y 3 0 5 12 21 32 11. Solve. a) x + 5 = 12 d) -3x = - 21 b) 13 = 9 - x e) 2x - 5 = 15 c) 2x = 18 f ) 4x - 6 = 8x + 2 12. a) If 3x - 2y = 14 and x  1.5, determine the value of y. b) If 0.36x + 0.54y = 1.1 and y  0.7, determine the value of x. 13. a) Make a concept map that shows different strategies you could use to graph 2x  4y  8. b) Which strategy would you use? Explain why. 6 Getting Started NEL 6706_PM10SB_C1_GS_pp4-7.qxd 5/7/09 3:41 PM Page 7 Getting Started APPLYING What You Know YOU WILL NEED grid paper Making Change ruler Barb is withdrawing $100 from her bank account. She asks for the money in $5 bills and $10 bills. ? Which combinations of $5 bills and $10 bills equal $100? A. If the teller gives Barb four $10 bills, how many $5 bills does he give her? B. List four more combinations of $100. Record the combinations in a table. Number of $5 Bills Number of $10 Bills 4 C. Let x represent the number of $5 bills, and let y represent the number of $10 bills. Write an equation for combinations of these bills with a total value of $100. D. Graph your equation for part C. Should you use a solid or broken line? Explain. E. Describe how the number of $10 bills changes as the number of $5 bills increases. F. Explain what the x-intercept and y-intercept represent on your graph. G. Which points on your graph are not possible combinations? Explain why. H. Determine all the possible combinations of $5 bills and $10 bills that equal $100. NEL Chapter 1 7 6706_PM10SB_C1_L1.1_pp8-14.qxd 5/7/09 3:41 PM Page 8 1.1 Representing Linear Relations YOU WILL NEED GOAL grid paper ruler Use tables, graphs, and equations to represent linear relations. graphing calculator LEARN ABOUT the Math Aiko’s cell-phone plan is shown here. Services Cost Aiko has a budget of $30 each month for her cell phone. calls 20¢/min text messages 15¢/message ? How can Aiko show how many messages and calls she can make each month for $30? EXAMPLE 1 Representing a linear relation Show the combinations of messages and calls that are possible each month for $30. Aiko’s Solution: Using a table I made a table to show how many Text Messages Calls messages and calls are possible for $30. Number Number Total I started with 0 messages and let the of Cost of Cost Cost Messages ($) Minutes ($) ($) number of messages increase by 20 each time. I calculated the cost of the 0 0 150 30 30 messages by multiplying the number in 20 3 135 27 30 the first column by $0.15. Then I subtracted the cost of the messages 40 6 120 24 30 from $30 to determine the amount of : : : money that was left for calls. I calculated the number of minutes for calls by 200 30 0 0 30 dividing this amount by $0.20. As the number of text messages increases, the 40 text messages a month is about number of minutes available for calls decreases. 1 per day. 120 min a month for calls is Aiko can make choices based on the numbers in about 4 min per day. the table. For example, if Aiko sends 40 text messages, she can talk for 120 min. 8 1.1 Representing Linear Relations NEL 6706_PM10SB_C1_L1.1_pp8-14.qxd 5/7/09 3:41 PM Page 9 1.1 Malcolm’s Solution: Using an equation and a graph Let x represent the number of text messages per I used letters for the variables. month. Let y represent the number of minutes of calls per month. I wrote an equation based on Aiko’s Aiko has a budget of $30 for text messages and budget of $30. In my equation, x text calls, so 0.15x  0.20y  30. messages cost $0.15x and y minutes of calls cost $0.20y. At the x-intercept, y  0. At the y-intercept, x  0. I used my equation to calculate the 0.15x  0.20(0)  30 0.15(0)  0.20y  30 maximum number of text messages and 30 30 the maximum time for calls. To do this, x = y = I determined the intercepts. 0.15 0.20 x  200 y  150 Number of Minutes of Calls vs. Number of Text Messages y Number of minutes of calls 240 200 I drew a graph by plotting the x-intercept 160 and y-intercept, and joining them. 120 0.15x  0.20y  30 I used a broken line because x represents 80 whole numbers only in this equation. 40 x 0 40 80 120 160 200 240 Number of text messages Aiko’s options for text messages and calls are displayed as points on the graph. Each point on the graph The point (40, 120) shows that if Aiko sends represents an ordered pair (x, y), where 40 text messages in a month, she has a maximum x is the number of text messages per of 120 min for calls to stay within her budget. month and y is the number of minutes of calls per month. Reflecting A. How does the table show that the relationship between the number of text messages and the number of minutes of calls is linear? B. How did Malcolm use his equation to draw a graph of Aiko’s choices? C. Which representation do you think Aiko would find more useful: the table or the graph? Why? NEL Chapter 1 9 6706_PM10SB_C1_L1.1_pp8-14.qxd 5/7/09 3:41 PM Page 10 APPLY the Math EXAMPLE 2 Representing a linear relation using graphing technology Patrick has saved $600 to buy British pounds and euros for a school trip to Europe. On the day that he goes to buy the currency, one pound costs $2 and one euro costs $1.50. a) Create a table, an equation, and a graph to show how many pounds and euros Patrick can buy. b) Explain why the relationship between pounds and euros is linear. c) Describe how Patrick can use each representation to decide how much Career Connection of each currency he can buy. Careers as diverse as sales consultants, software Brittany’s Solution developers, and financial analysts have roles in currency a) Let x represent the pounds that I chose letters for the variables. exchange. x pounds cost $2x and y euros Patrick buys. Let y represent the euros that he buys. cost $1.50y. Patrick has $600. I wrote an equation based on the 2x  1.50y  600 cost of the currency. I rearranged 1.50y  600  2x my equation into the form 1.50y 600 2x y  mx  b so I could enter = - 1.50 1.50 1.50 it into a graphing calculator. y = 400 - a bx 2 1.50 Tech Support I graphed the equation using For help using a TI-83/84 these window settings because graphing calculator to enter I knew that the y-intercept would then graph relations and be at 400 and the x-intercept use the Table Feature, see would be at 300. Appendix B-1, B-2, and B-6. If you are using a TI-nspire, see Appendix B-37, B-38, and B-42. 10 1.1 Representing Linear Relations NEL 6706_PM10SB_C1_L1.1_pp8-14.qxd 5/7/09 3:41 PM Page 11 1.1 I set the decimal setting to two decimal places because x and y represent money. Then I created a table of values. b) Since the degree of the equation In the table, each increase Tech Support is one and the graph is a straight of 1 in the x-values results For help creating a difference line, the relationship is linear. in a decrease of about 1.33 table with a TI-83/84 graphing in the y-values. calculator, see Appendix B-7. The first differences in the table If you are using a TI-nspire, are constant. see Appendix B-43. c) By tracing up and down the line, or by scrolling up and down the table, Patrick can see the combinations of pounds and euros. He can use the equation, in either form, to calculate specific numbers of pounds or euros. EXAMPLE 3 Selecting a representation for a linear relation Judy is considering two sales positions. Sam’s store offers $1600/month plus 2.5% commission on sales. Carol’s store offers $1000/month plus 5% commission on sales. In the past, Judy has had about $15 000 in sales each month. a) Represent Sam’s offer so that Judy can check what her monthly pay would be. b) Represent the two offers so that Judy can compare them. Which offer pays more? Justine’s Solution a) Let x represent her sales in dollars. I chose letters for the variables. Let y represent her earnings I wrote an equation to describe in dollars. what Judy’s monthly pay would An equation will help Judy check be. Her base salary is $1600. Her earnings for her monthly her pay. sales would be $0.025x, y  1600  0.025x since 2.5% 2.5 or 0.025. 100 NEL Chapter 1 11 6706_PM10SB_C1_L1.1_pp8-14.qxd 5/7/09 3:41 PM Page 12 Tech Support b) The equation for Carol’s offer is I wrote an equation for Carol’s For help changing the y  1000  0.05x. offer and graphed both relations window settings and tracing Judy can use a graph to compare using a graphing calculator. on a graph using a TI-83/84 graphing calculator, see her pay for a typical month. Appendix B-4 and B-2. If you are using a TI-nspire, I adjusted the settings, as see Appendix B-40 and B-38. shown, so I could see the point where the graphs crossed. I used Trace to compare the two offers. Sam’s offer pays more. In Summary Key Idea Three useful ways to represent a linear relation are a table of values a graph an equation Need to Know A linear relation has the following characteristics: The first differences in a table of values are constant. The graph is a straight line. The equation has a degree of 1. The equation of a linear relation can be written in a variety of equivalent forms, such as standard form: Ax  By  C  0 slope y-intercept form: y  mx  b A graph and a table of values display some of the ordered pairs for a relation. You can use the equation of a relation to calculate ordered pairs. CHECK Your Understanding 1. Which of these ordered pairs are not points on the graph of 2x  4y  20? Justify your decision. a) (10, 0) b) (3, 7) c) (6, 2) d) (0, 5) e) (12, 1) 12 1.1 Representing Linear Relations NEL 6706_PM10SB_C1_L1.1_pp8-14.qxd 5/7/09 3:41 PM Page 13 1.1 2. Jacob has $15 to buy muffins and doughnuts at the school bake sale, as a treat for the Camera Club. Muffins are 75¢ each and doughnuts are 25¢ each. How many muffins and doughnuts can he buy? a) Create a table to show the possible combinations of muffins and doughnuts. b) What is the maximum number of muffins that Jacob can buy? c) What is the maximum number of doughnuts that he can buy? d) Write an equation that describes Jacob’s options. e) Graph the possible combinations. 3. Refer to question 2. Which representation do you think is more useful for Jacob? Justify your choice. PRACTISING 4. State two ordered pairs that satisfy each linear relation and one ordered K pair that does not. a) y  5x  1 c) y  25x  10 b) 3x - 4y = 24 d) 5x = 30 - 2y 5. Define suitable variables for each situation, and write an equation. a) Caroline has a day job and an evening job. She works a total of 40 h/week. b) Caroline earns $15/h at her day job and $11/h at her evening job. Last week, she earned $540. c) Justin earns $500/week plus 6% commission selling cars. d) Justin is offered a new job that would pay $800/week plus 4% commission. e) A piggy bank contains $5.25 in nickels and dimes. 6. Graph the relations in question 5, parts a) and b). 7. Refer to question 5, parts c) and d). Justin usually has about $18 000 in weekly sales. Should he take the new job? Justify your decision. 8. Deb pays 10¢/min for cell-phone calls and 6¢/text message. She has a budget of $25/month for both calls and text messages. a) Create a table to show the ways that Deb can spend up to $25 each month on calls and text messages. b) Create a graph to show the information in the table. 9. Leah earns $1200/month plus 3.5% commission. C a) Create an equation that she can use to check her paycheque each month. b) Last month, Leah had $96 174 in sales. Her pay before deductions was $4566.09. Is this amount correct? Explain your answer. NEL Chapter 1 13 6706_PM10SB_C1_L1.1_pp8-14.qxd 5/7/09 3:41 PM Page 14 10. Ben’s Bikes rents racing bikes for $25/day and mountain bikes A for $30/day. Yesterday’s rental charges were $3450. a) Determine the greatest number of racing bikes that could have been rented. b) Determine the greatest number of mountain bikes that could have been rented. c) Write an equation and draw a graph to show the possible combinations of racing and mountain bikes rented yesterday. 11. Abigail is planning to fly to Paris and then travel through Switzerland and Austria to Italy by train. On the day that she goes to buy the foreign currencies she needs, one euro costs $1.40 and one Swiss franc costs $0.90. What combinations of these currencies can Abigail buy for $630? Use two different strategies to show the possible combinations. 12. A student council invested some of the money from a fundraiser in a savings account that pays 3%/year and the rest of the money in a government bond that pays 4%/year. The investments earned $150 in the first year. a) Define two variables for the information, and write an equation. b) Graph the information. 13. Maureen pays a $350 registration fee and an $85 monthly fee to belong T to a fitness club. Lia’s club has a higher registration fee but a lower monthly fee. After five months, both Maureen and Lia have paid $775. Determine the possible fees at Lia’s club. 14. a) Use the chart to show what you know about linear relations. Characteristics: Methods of Representation: Linear Relation Examples: Non-examples: b) List the advantages and disadvantages of each of the three ways to represent a linear relation. Describe situations in which each representation might be preferred. Extending 15. Create a situation that can be represented by each equation. a) 0.10x + 0.25y = 4.65 b) y = 900 + 0.025x 16. Allan plans to create a new coffee blend using Brazilian beans that cost $12/kg and Ethiopian beans that cost $17/kg. He is going to make 150 kg of the blend and sell it for $14/kg. Write and graph two equations for this situation. 14 1.1 Representing Linear Relations NEL 6706_PM10SB_C1_L1.2_pp15-20.qxd 5/7/09 3:42 PM Page 15 1.2 Solving Linear Equations GOAL YOU WILL NEED grid paper Connect the solution to a linear equation and the graph ruler of the corresponding relation. graphing calculator LEARN ABOUT the Math Joe downloads music to his MP3 player from a site that charges $9.95 per month plus $0.55 for each song. Joe has budgeted $40 per month to spend on music downloads. ? How can Joe determine the greatest number of songs that he can download each month? EXAMPLE 1 Selecting a strategy to solve the problem Determine the maximum number of songs that Joe can download each month. William’s Solution: Solving a problem by reasoning $40.00  $9.95  $30.05 I calculated how much of Joe’s budget he can spend on the songs he downloads, by subtracting the $9.95 monthly fee from $40. $30.05 , $0.55  54.63 Each song costs $0.55, so I divided this into the amount he would have left to spend on songs. Joe can download a maximum I rounded down to 54, since of 54 songs. 55 songs would cost more than he can spend. Tony’s Solution: Solving a problem by using an equation Let n represent the number of songs and let C represent the cost. C  9.95  0.55n I created an equation and 40  9.95  0.55n substituted the $40 Joe has budgeted for C. NEL Chapter 1 15 6706_PM10SB_C1_L1.2_pp15-20.qxd 5/7/09 3:42 PM Page 16 40  9.95  9.95  0.55n  9.95 I solved for n using inverse 30.05  0.55n operations. 30.05 = n 0.55 54.6  n Joe can download a maximum Since n has to be a whole of 54 songs. number, I used the nearest whole number less than 54.6 for my answer. Lucy’s Solution: Solving a problem using graphing technology Let X represent the number of songs and Y1 the cost. Y 1  9.95  0.55X I entered the equation for the cost of music downloads into a graphing calculator. The number of songs downloaded has to be a whole number, so X represents a whole number. I graphed using Zoom Integer, so the x-values would go up by 1 when I traced the graph. I used Trace to determine which Tech Support point on the graph is closest For help graphing and to y  40 (but less than $40). tracing along relations using a TI-83/84 graphing calculator, This point is (54, 39.65). see Appendix B-2. If you are using a TI-nspire, see Appendix B-38. Joe can download 54 songs in a month for $39.65. Reflecting A. How are William’s and Tony’s solutions similar? How are they different? B. How did a single point on Lucy’s graph represent a solution to the problem? C. Which strategy do you prefer? Explain why. 16 1.2 Solving Linear Equations NEL 6706_PM10SB_C1_L1.2_pp15-20.qxd 5/7/09 3:42 PM Page 17 1.2 APPLY the Math EXAMPLE 2 Representing and solving a problem that involves a linear equation At 9:20 a.m., Adrian left Windsor with 64 L of gas in his car. He drove east at 100 km/h. The low fuel warning light came on when 10 L of gas were left. Adrian’s car uses gas at the rate of 8.8 L/100 km. When did the warning light come on? Stefani’s Solution: Solving an equation algebraically Adrian’s car uses 8.8 L of gas I calculated how much gas every 100 km. Since he drove the car used each hour. at 100 km/h, he used 8.8 L/h. I wrote an equation for the amount of gas used. I let t represent the time in hours, and G  64  8.8t I let G represent the amount of gas in litres. 10  64  8.8t 10 10  64  8.8t  10 The warning light came on 0  54  8.8t when G  10, so I let G  10 8.8t  54 and solved for t using inverse 54 operations. t = 8.8 t  6.14 The warning light came on after Adrian had been driving about 6.14 h. 0.14  60  8.4 I wrote the time in hours and minutes by multiplying the part The warning light came on about of the number to the right of 6 h 8 min after 9:20 a.m., which the decimal point by 60. is about 3:28 p.m. Henri’s Solution: Solving a problem by using a graph y = 64 - 8.8x I wrote an equation for the amount of gas in the tank at any time. I let x represent the time in hours, and I let y represent the amount of gas in litres. NEL Chapter 1 17 6706_PM10SB_C1_L1.2_pp15-20.qxd 5/7/09 3:42 PM Page 18 Graph Y 1 = 64 - 8.8X. I graphed the equation on a graphing calculator. I knew that the y-intercept was 64, and I estimated that the x-intercept was about 7, so I used the window settings shown. After about 6.17 h, there was about 9.7 L of gas in the tank. I used Trace to locate the point with a y-value closest to 10. Tech Support To get an exact solution, For help determining the point of intersection between I entered the line Y2  10. two relations on a TI-83/84 The x-coordinate of the point graphing calculator, see of intersection between the Appendix B-11. If you are two lines tells the time when using a TI-nspire, see 10 L of gas is left in the tank. Appendix B-47. Based on the graph, the warning light came on about 6.14 h after Adrian started, at about 3:28 p.m. In Summary Key Idea You can solve a problem that involves a linear relation by solving the associated linear equation. Cost of Car Rental vs. Distance Need to Know y You can solve a linear equation in one variable by graphing the associated linear relation and using the appropriate coordinate of an ordered pair on the line. For example, to solve 3x  2  89, graph 100 y  3x  2 and look for the value of x at the point where y  89 Cost ($) on the line. 50 CHECK Your Understanding x 1. Estimate solutions to the following questions using the graph at the left. 0 200 400 600 800 a) What is the rental cost to drive 500 km? Distance (km) b) How far can you drive for $80, $100, and $75? 18 1.2 Solving Linear Equations NEL 6706_PM10SB_C1_L1.2_pp15-20.qxd 5/7/09 3:42 PM Page 19 1.2 2. a) Write an equation for the linear relation in question 1. b) Use your equation to answer question 1. c) Compare your answers for question 1 with your answers for part b) above. Which strategy gave the more accurate answers? 3. Apple juice is leaking from a carton at the rate of 5 mL/min. There are 1890 mL of juice in the container at 10:00 a.m. a) Write an equation for this situation, and draw a graph. b) When will 1 L of juice be left in the carton? PRACTISING 4. The graph at the right shows how the charge for a banquet hall Cost of Banquet K relates to the number of people attending a banquet. vs. Number of People a) Locate the point (160, 5700) on the graph. What do these y coordinates tell you about the charge for the banquet hall? 9000 b) What is the charge for the banquet hall if 200 people attend? 8000 c) Write an equation for this linear relation. 7000 d) Use your equation to determine how many people can attend for $3100, $4400, and $5000. 6000 Cost ($) e) Why is a broken line used for this graph? 5000 5. Max read on the Internet that 1 U.S. gallon is approximately 4000 equal to 3.785 L. 3000 a) Draw a graph that you can use to convert U.S. gallons 2000 into litres. b) Use your graph to estimate the number of litres in 6 gallons. 1000 x c) Use your graph to estimate the number of gallons in 14 L. 0 50 100 150 200 250 300 6. Melanie drove at 100 km/h from Ajax to Ottawa. She left Ajax Number of people at 2:15 p.m., with 35 L of gas in the tank. The low fuel warning light came on when 9 L was left in the tank. If Melanie’s SUV uses gas at the rate of 9.5 L/100 km, estimate when the warning light came on. 7. Hank sells furniture and earns $280/week plus 4% commission. a) Determine the sales that Hank needs to make to meet his weekly budget requirement of $900. b) Write an equation for this situation, and use it to verify your answer for part a). 8. The Perfect Paving Company charges $10 per square foot to install A interlocking paving stones, as well as a $40 delivery fee. a) Determine the greatest area that Andrew can pave for $3500. b) Andrew needs to include 5 cubic yards of sand, costing $15 per cubic yard, to the total cost of the project. How much will this added cost reduce the area that he can pave with his $3500 budget? NEL Chapter 1 19 6706_PM10SB_C1_L1.2_pp15-20.qxd 5/7/09 3:42 PM Page 20 9. A student athletic council raised $4000 for new sports equipment and uniforms, which will be purchased 3 years from now. Until then, the money will be invested in a simple interest savings account that pays 3.5%/year. a) Write an equation and draw a graph to represent the relationship between time (in years) and the total value of their investment. b) Use the graph to determine the value of their investment after 2 years. c) Use the equation to determine when their investment is worth $4385. 10. Maria has budgeted $90 to take her grandmother for a drive. Katey’s Kars rents cars for $65 per day plus $0.12/km. Determine how far Maria and her grandmother can travel, including the return trip. 11. Cam earns $400/week plus 2.5% commission. He has been offered C another job that pays $700/week but no commission. a) Describe three strategies that you could use to compare Cam’s earnings for the two jobs. b) Which job should Cam take? Justify your decision. 12. At 9:00 a.m., Chantelle starts jogging north at 6 km/h from the south T end of a 21 km trail. At the same time, Amit begins cycling south at 15 km/h from the north end of the same trail. Use a graph to determine when they will meet. 13. Explain how to determine the value of x, both graphically and algebraically, in the linear relation 2x - 3y = 6 when y = 5. Extending 14. The owner of a dart-throwing stand at a carnival pays 75¢ every time the bull’s-eye is hit, but charges 25¢ every time it is missed. After 25 tries, Luke paid $5.25. How many times did he hit the bull’s-eye? Health Connection 15. Adriana earns 5% commission on her sales up to $25 000, 5.5% on Jogging is an exercise that any sales between $25 000 and $35 000, 6% on any sales between keeps you healthy and can burn about 650 calories per hour. $35 000 and $45 000, and 7% for any sales over $45 000. Draw a graph to represent how Adriana’s earnings depend on her sales. What sales volume does she need to earn $2000? Number of Cost per 16. A fabric store sells fancy buttons for the prices in the table at the left. Buttons Button ($) a) Make a table of values and draw a graph to show the cost 1 to 25 1.00 of 0 to 125 buttons. b) Compare the cost of 100 buttons with the cost of 101 buttons. 26 to 50 0.80 What advice would you give someone who needed 100 buttons? 51 to 100 0.60 Comment on this pricing structure. 101 or more 0.20 c) Write equations to describe the relationship between the cost and the number of buttons purchased. 20 1.2 Solving Linear Equations NEL 6706_PM10SB_C1_L1.3_pp21-29.qxd 5/7/09 3:43 PM Page 21 1.3 Graphically Solving Linear Systems GOAL YOU WILL NEED grid paper Use graphs to solve a pair of linear equations simultaneously. ruler graphing calculator INVESTIGATE the Math Matt’s health-food store sells roasted almonds for $15/kg and dried cranberries for $10/kg. ? How can he mix the almonds and the cranberries to create 100 kg of a mixture that he can sell for $12/kg? A. Let x represent the mass of the almonds. Let y represent the mass of the cranberries. i) Write an equation for the total mass of the mixture. ii) Write an equation for the total value of the mixture. B. Graph your equation of the total mass for part A. What do the points on the line represent? C. Graph your equation of the total value for part A on the same axes. What do the points on this line represent? D. Identify the coordinates of the point where the two lines intersect. State what each value represents. How accurately can you estimate these values from your graph? E. The equations for part A form a system of linear equations. system of linear equations Explain why the coordinates for part D give the solution to a set of two or more linear a system of linear equations. equations with two or more variables F. Substitute the coordinates into each equation to verify your solution. For example, x  y  10 4x  2y  22 Reflecting solution to a system of linear equations G. Explain why you needed two linear relations to describe the problem. the values of the variables in the system that satisfy all the H. Explain how graphing both relations on the same axes helped you equations solve the problem. For example, (7, 3) is the solution to I. Explain why the coordinates of the point of intersection provide x  y  10 4x  2y  22 an ordered pair that satisfies both relations. NEL Chapter 1 21 6706_PM10SB_C1_L1.3_pp21-29.qxd 5/7/09 3:43 PM Page 22 APPLY the Math EXAMPLE 1 Selecting a graphing strategy to solve a linear system Solve the system y = 2x + 1 and x + 2y = - 8 using a graph. Leslie’s Solution y = 2x + 1 The slope of At the y-intercept, I determined the slope and the the line is 2. x  0. y-intercept of the first equation. 2 rise y  2(0)  1 = 1 run y1 x + 2y = -8 At the x-intercept, At the y-intercept, I determined the x- and y = 0. x = 0. y-intercepts of the second x + 2(0) = - 8 0 + 2y = - 8 equation. x = -8 2y 8 = - 2 2 y = -4 y 6 I graphed the first line (blue) by 1 4 plotting the y-intercept and 2x using the rise and run to plot y 2 x another point on the line. 0 I graphed the second line (red) -8 -6 -4 -2 2 4 -2 by plotting points (8, 0) and x -4 2y (0,4) and joining them with a  8 straight line. -6 At the point of intersection, x = - 2 I located the point of intersection and y = - 3. and read its coordinates using The solution is (2, 3). the axes of my graph. y  2x  1 x  2y  8 I checked my solution by Left Side Right Side Left Side Right Side substituting the x- and y-values into each equation. y 2x  1 x  2y 8  3  2(2) 1  2  2(3)  3  8 22 1.3 Graphically Solving Linear Systems NEL 6706_PM10SB_C1_L1.3_pp21-29.qxd 5/7/09 3:43 PM Page 23 1.3 EXAMPLE 2 Solving a problem using a graphing strategy Ellen drives 450 km from her university in Kitchener-Waterloo to her home in Smiths Falls. She travels along one highway to Kingston at 100 km/h and then along another highway to Smiths Falls at 80 km/h. The journey takes her 4 h 45 min. What is the distance from Kingston to Smiths Falls? Bob’s Solution Let x represent the distance that I used letters to identify the variables Ellen travels at 100 km/h. Let y in this situation. represent the distance that she travels at 80 km/h. The total trip is 450 km, so I wrote an equation for the total x  y  450. distance. distance Since speed = , then x y 3 time + = 4 distance 100 80 4 time =. speed I wrote an equation to describe the total x time (in hours) for her trip, where 100 is the time spent driving at 100 km/h y and is the time spent driving 80 at 80 km/h. x + y = 450 At the x-intercept, y  0. At the y-intercept, x  0. I determined the x- and y-intercepts x  0  450 0  y  450 of the first equation. x  450 y  450 x y 3 + = 4 100 80 4 At the x-intercept, y  0. At the y-intercept, x  0. I determined the x- and y-intercepts x 3 y 3 + 0 = 4 0 + = 4 of the second equation. 100 4 80 4 x = 100 a 4 b 3 y = 80 a4 b 3 4 4 x = 100 a b 19 y = 80a b 19 4 4 x = 475 y = 380 NEL Chapter 1 23 6706_PM10SB_C1_L1.3_pp21-29.qxd 5/7/09 3:43 PM Page 24 Distance at 80 km/h vs. Distance at 100 km/h I graphed each equation by plotting the y Distance at 80 km/h (km) x- and y-intercepts and joining them 500 with a straight line. 400 300 200 100 x 0 100 200 300 400 500 Distance at 10