Chapter 6 Resource Masters

Transcription

1 Chapter 6 Resource Masters Bothell, WA Chicago, IL Columbus, H New York, NY

2 CNSUMABLE WRKBKS Man of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks in both English and Spanish. ISBN10 ISBN13 Stud Guide and Intervention Workbook Homework Practice Workbook Spanish Version Homework Practice Workbook X Answers For Workbooks The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet. ConnectED All of the materials found in this booklet are included for viewing, printing, and editing at connected.mcgraw-hill.com. Spanish Assessment Masters (MHID: , ISBN: ) These masters contain a Spanish version of Chapter 6 Test Form A and Form C. connected.mcgraw-hill.com Copright b The McGraw-Hill Companies, Inc. All rights reserved. The contents, or parts thereof, ma be reproduced in print form for non-profit educational use with Glencoe Algebra 1, provided such reproductions bear copright notice, but ma not be reproduced in an form for an other purpose without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, network storage or transmission, or broadcast for distance learning. Send all inquiries to: McGraw-Hill Education 8787 rion Place Columbus, H 4340 ISBN: MHID: X Printed in the United States of America DH

3 Contents Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Teacher s Guide to Using the Chapter 6 Resource Masters...iv Chapter Resources Chapter 6 Student-Built Glossar... 1 Chapter 6 Anticipation Guide (English)... 3 Chapter 6 Anticipation Guide (Spanish)... 4 Lesson 6-1 Graphing Sstems of Equations Stud Guide and Intervention... 5 Skills Practice... 7 Practice... 8 Word Problem Practice... 9 Enrichment Graphing Calculator Activit...11 Lesson 6- Substitution Stud Guide and Intervention... 1 Skills Practice Practice Word Problem Practice Enrichment Lesson 6-3 Elimination Using Addition and Subtraction Stud Guide and Intervention Skills Practice... 0 Practice... 1 Word Problem Practice... Enrichment... 3 Lesson 6-4 Elimination Using Multipcation Stud Guide and Intervention... 4 Skills Practice... 6 Practice... 7 Word Problem Practice... 8 Enrichment... 9 Lesson 6-5 Appling Sstems of Linear Equations Stud Guide and Intervention Skills Practice... 3 Practice Word Problem Practice Enrichment Lesson 6-6 Sstems of Inequalities Stud Guide and Intervention Skills Practice Practice Word Problem Practice Enrichment Graphing Calculator... 4 Spreadsheet Student Recording Sheet Rubric for Scoring Etended Response Chapter 6 Quizzes 1 and Chapter 6 Quizzes 3 and Chapter 6 Mid-Chapter Test Chapter 6 Vocabular Test Chapter 6 Test, Form Chapter 6 Test, Form A Chapter 6 Test, Form B Chapter 6 Test, Form C Chapter 6 Test, Form D Chapter 6 Test, Form Chapter 6 Etended-Response Test Standardized Test Practice Unit Test Answers... A1 A3 iii

4 Teacher s Guide to Using the Chapter 6 Resource Masters The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. 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, printing, and editing at connected.mcgraw-hill.com. Chapter Resources Student-Built Glossar (pages 1 ) These masters are 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. Give this to students before beginning Lesson 6-1. Encourage them to add these pages to their mathematics stud notebooks. Remind them to complete the appropriate words as the stud each lesson. Anticipation Guide (pages 3 4) This master, presented in both English and Spanish, is a surve used before beginning the chapter to pinpoint what students ma or ma not know about the concepts in the chapter. Students will revisit this surve after the complete the chapter to see if their perceptions have changed. Lesson Resources Stud Guide and Intervention These masters provide vocabular, ke concepts, additional worked-out eamples and Guided Practice eercises to use as a reteaching activit. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-da teaching of the lesson. Practice This master closel follows the tpes of problems found in the Eercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for secondda teaching of the lesson. Word Problem Practice This master includes additional practice in solving word problems that appl the concepts of the lesson. Use as an additional practice or as homework for second-da teaching of the lesson. Enrichment These activities ma etend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students perspectives on the mathematics the are learning. The are written for use with all levels of students. Graphing Calculator, TI-Nspire, or Spreadsheet Activities These activities present was in which technolog can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of our lesson presentation. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. iv

5 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Assessment ptions The assessment masters in the Chapter 6 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment. Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter. Etended Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions. Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter. Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions. Vocabular Test This test is suitable for all students. It includes a list of vocabular words and 9 questions to assess students knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests. Leveled Chapter Tests Form 1 contains multiple-choice questions and is intended for use with below grade level students. Forms A and B contain multiplechoice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations. Forms C and D contain freeresponse questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations. Form 3 is a free-response test for use with above grade level students. All of the above mentioned tests include a free-response Bonus question. Etended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation. Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and shortanswer free-response questions. Answers The answers for the Anticipation Guide and Lesson Resources are provided as reduced pages. Full-size answer kes are provided for the assessment masters. v

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7 NAME DATE PERID 6 Student-Built Glossar This is an alphabetical list of the ke vocabular terms ou will learn in Chapter 6. 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. consistent kuhn SIHS tuhnt Vocabular Term Found on Page Definition/Description/Eample Chapter Resources dependent elimination ih LIH muh NAY shuhn inconsistent Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. independent substitution SUHB stuh T shuhn sstem of equations sstem of inequalities Chapter 6 1 Glencoe Algebra 1

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9 NAME DATE PERID 6 Anticipation Guide Solving Sstems of Linear Equations Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 1 Before ou begin Chapter 6 Read each statement. Decide whether ou Agree (A) or Disagree (D) with the statement. Write A or D in the first column R if ou are not sure whether ou agree or disagree, write NS (Not Sure). STEP 1 A, D, or NS Step Statement 1. A solution of a sstem of equations is an ordered pair that satisfies one of the equations. A sstem of equations of parallel lines will have no solutions. 3. A sstem of equations of two perpendicular lines will have infinitel man solutions. 4. It is not possible to have eactl two solutions to a sstem of linear equations 5. The most accurate wa to solve a sstem of equations is to graph the equations to see where the intersect. 6. To solve a sstem of equations, such as - = 1 and 3 = - 6, b substitution, solve one of the equations for one variable and substitute the result into the other equation. 7. When solving a sstem of equations, a result that is a true statement, such as -5 = -5, means the equations do not share a common solution. 8. Adding the equations 3-4 = 8 and + 4 = 7 results in a 0 coefficient for. 9. The equation 7 - = 1 can be multiplied b so that the coefficient of is The result of multipling -7-3 = 11 b -3 is = 11. After ou complete Chapter 6 Reread each statement and complete the last column b entering an A or a D. Did an of our opinions about the statements change from the first column? For those statements that ou mark with a D, use a piece of paper to write an eample of wh ou disagree. STEP A or D Chapter Resources Chapter 6 3 Glencoe Algebra 1

10 NMBRE FECHA PERÍD 6 Ejercicios preparatorios Resuelve sistemas de ecuaciones lineales Paso 1 Antes de comenzar el Capítulo 6 Lee cada enunciado. Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado. Escribe A o D en la primera columna si no estás seguro(a) de la respuesta, escribe NS (No esto seguro(a)). PAS 1 A, D o NS Paso Enunciado 1. Una solución de un sistema de ecuaciones es cualquier par ordenado que satisface una de las ecuaciones. Un sistema de ecuaciones de rectas paralelas no tendrá soluciones. 3. Un sistema de ecuaciones de dos rectas perpendiculares tendrá un número infinito de soluciones. 4. No es posible tener eactamente dos soluciones para un sistema de ecuaciones lineales. 5. La forma más precisa de resolver un sistema de ecuaciones es graficar las ecuaciones ver dónde se intersecan. 6. Para resolver un sistema de ecuaciones, como - = 1 3 = - 6, por sustitución, despeja una variable en una de la ecuaciones reemplaza el resultado en la otra ecuación. 7. Cuando se resuelve un sistema de ecuaciones, un resultado que es un enunciado verdadero, como -5 = -5, significa que las ecuaciones no comparten una solución. 8. El sumar las ecuaciones 3-4 = = 7 resulta en un coeficiente de 0 para. 9. La ecuación 7 - = 1 se puede multiplicar por, de modo que el coeficiente de es El resultado de multiplicar -7-3 = 11 por -3 es = 11. Después de completar el Capítulo 6 PAS A o D Vuelve a leer cada enunciado completa la última columna con una A o una D. Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna? En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Capítulo 6 4 Álgebra 1 de Glencoe

11 NAME DATE PERID 6-1 Stud Guide and Intervention Graphing Sstems of Equations Possible Number of Solutions Two or more linear equations involving the same variables form a sstem of equations. A solution of the sstem of equations is an ordered pair of numbers that satisfies both equations. The table below summarizes information about sstems of linear equations. Graph of a Sstem intersecting lines same line parallel lines Lesson 6-1 Number of Solutions eactl one solution infi nitel man solutions no solution Terminolog consistent and independent consistent and dependent inconsistent Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eample Use the graph at the right to determine whether each sstem is consistent or inconsistent and if it is independent or dependent. a. = - + = = + 1 Since the graphs of = - + and = + 1 intersect, there is one solution. Therefore, the sstem is consistent and independent. b. = = -3 Since the graphs of = - + and = -3 are parallel, there are no solutions. Therefore, the sstem is inconsistent. c = -3 = Since the graphs of = -3 and = coincide, = -3 there are infinitel man solutions. Therefore, the sstem is consistent and dependent. Eercises Use the graph at the right to determine whether each sstem is consistent or inconsistent and if it is independent or dependent. 1. = = -6 = - 1 = = 4 + = -6 = - 1 = = + 1 = = 3 3. = = -6 + = = 3 Chapter 6 5 Glencoe Algebra 1

12 NAME DATE PERID 6-1 Stud Guide and Intervention (continued) Graphing Sstems of Equations Solve b Graphing ne method of solving a sstem of equations is to graph the equations on the same coordinate plane. Eample Graph each sstem and determine the number of solutions that it has. If it has one solution, name it. a. + = - = 4 The graphs intersect. Therefore, there is one solution. The point (3, -1) seems to lie on both lines. Check this estimate b replacing with 3 and with -1 in each equation. + = 3 + (-1) = - = (-1) = or 4 The solution is (3, -1). b. = = ( 3, 1) - = 4 = 4 + = + 1 = 4 + The graphs coincide. Therefore there are infinitel man solutions. Eercises Graph each sstem and determine the number of solutions it has. If it has one solution, name it. 1. = -. = 3. = = -1 + = 1 + = = = 6 6. = = = -4 = - + Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 6 Glencoe Algebra 1

13 NAME DATE PERID 6-1 Skills Practice Graphing Sstems of Equations Use the graph at the right to determine whether each sstem is consistent or inconsistent and if it is independent or dependent. 1. = = -4 = = = = = - = - = -4 = + 4 = - 1 = = = Lesson 6-1 Graph each sstem and determine the number of solutions that it has. If it has one solution, name it = 1 6. = = -3 = -3 + = = 3 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 8. = = = -1 - = = -3 + = = = = = 3 = = 6-6 Chapter 6 7 Glencoe Algebra 1

14 NAME DATE PERID 6-1 Practice Graphing Sstems of Equations Use the graph at the right to determine whether each sstem is consistent or inconsistent and if it is independent or dependent = 3. - = -3 + = = = = -6 - = -3 + = 3 + = = = 3 + = -3 - = -3 Graph each sstem and determine the number of solutions that it has. If it has one solution, name it = - 6. = = = 0 4 = = BUSINESS Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $0 in operating costs per week plus $0.50 to produce each treat. He plans to sell each treat for $1.50. a. Graph the sstem of equations = and = 1.5 to represent the situation. b. How man treats does Nick need to sell per week to break even? 9. SALES A used book store also started selling used CDs and videos. In the first week, the store sold 40 used CDs and videos, at $4.00 per CD and $6.00 per video. The sales for both CDs and videos totaled $ a. Write a sstem of equations to represent the situation. b. Graph the sstem of equations. Video Sales ($) Cost ($) Dog Treats Sales ($) CD and Video Sales Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. c. How man CDs and videos did the store sell in the first week? CD Sales ($) Chapter 6 8 Glencoe Algebra 1

15 NAME DATE PERID 6-1 Word Problem Practice Graphing Sstems of Equations 1. BUSINESS The widget factor will sell a total of widgets after das according to the equation = The gadget factor will sell gadgets after das according to the equation = Look at the graph of the sstem of equations and determine whether it has no solution, one solution, or infinitel man solutions. 4. AVIATIN Two planes are in flight near a local airport. ne plane is at an altitude of 1000 meters and is ascending at a rate of 400 meters per minute. The second plane is at an altitude of 5900 meters and is descending at a rate of 300 meters per minute. a. Write a sstem of equations that represents the progress of each plane Lesson = Widgets Items sold = Gadgets b. Make a graph that represents the progress of each plane Das 5 6 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. ARCHITECTURE An office building has two elevators. ne elevator starts out on the 4th floor, 35 feet above the ground, and is descending at a rate of. feet per second. The other elevator starts out at ground level and is rising at a rate of 1.7 feet per second. Write a sstem of equations to represent the situation. 3. FITNESS livia and her brother William had a biccle race. livia rode at a speed of 0 feet per second while William rode at a speed of 15 feet per second. To be fair, livia decided to give William a 150-foot head start. The race ended in a tie. How far awa was the finish line from where livia started? Altitude (m) 0 Time (min.) Chapter 6 9 Glencoe Algebra 1

16 NAME DATE PERID 6-1 Enrichment Graphing a Trip The distance formula, d = rt, is used to solve man tpes of problems. If ou graph an equation such as d = 50t, the graph is a model for a car going at 50 mph. The time the car travels is t; the distance in miles the car covers is d. The slope of the line is the speed. Suppose ou drive to a nearb town and return. You average 50 mph on the trip out but onl 5 mph on the trip home. The round trip takes 5 hours. How far awa is the town? The graph at the right represents our trip. Notice that the return trip is shown with a negative slope because ou are driving in the opposite direction. 50 d slope is 50 slope is 5 t Solve each problem. 1. Estimate the answer to the problem in the above eample. About how far awa is the town? Graph each trip and solve the problem.. An airplane has enough fuel for 3 hours of safe fling. n the trip out the pilot averages 00 mph fling against a headwind. n the trip back, the pilot averages 50 mph. How long a trip out can the pilot make? 3. You drive to a town 100 miles awa. 4. You drive at an average speed of 50 mph n the trip out ou average 5 mph. to a discount shopping plaza, spend n the trip back ou average 50 mph. hours shopping, and then return at an How man hours do ou spend driving? average speed of 5 mph. The entire trip takes 8 hours. How far awa is the shopping plaza? 50 d 50 d 100 d 1 t Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. t t Chapter 6 10 Glencoe Algebra 1

17 NAME DATE PERID 6-1 Graphing Calculator Activit Solution to a Sstem of Linear Equations A graphing calculator can be used to solve a sstem of linear equations graphicall. The solution of a sstem of linear equations can be found b using the TRACE feature or b using the intersect command under the CALC menu. Eample Solve each sstem of linear equations. Lesson 6-1 a. + = 0 - = -4 Using TRACE: Solve each equation for and enter each equation into Y=. Then graph using Zoom 8: ZInteger. Use TRACE to find the solution. Kestrokes: Y= ( ) ENTER + 4 ZM 6 ZM 8 ENTER TRACE. The solution is (-, ). b. + = = 3 [-47, 47] scl:10 b [-31, 31] scl:10 Using CALC: Solve each equation for, enter each into the calculator, and graph. Use CALC to determine the solution. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Kestrokes: Y= ( ) + 4 ENTER ( ( ) 4 3 ) + 1 ZM 6 nd [CALC] 5 ENTER ENTER ENTER. To change the -value to a fraction, press nd [QUIT] MATH ENTER ENTER. The solution is (4.5, -5) or ( 9, -5 ). Eercises Solve each sstem of linear equations. 1. =. = = = 18 = = = = = = = 0-5 = 5 [-10, 10] scl:1 b [-10, 10] scl: = = = = = = 10 Chapter 6 11 Glencoe Algebra 1

18 NAME DATE PERID 6- Stud Guide and Intervention Substitution Solve b Substitution ne method of solving sstems of equations is substitution. Eample 1 Use substitution to Eample Solve for one variable, solve the sstem of equations. = 4 - = -4 then substitute. + 3 = 7-4 = -6 Substitute for in the second equation. 4 - = -4 Second equation 4 - = -4 = -4 = Combine like terms. = - Divide each side b and simplif. Use = to find the value of. = First equation = (-) = - = -4 Simplif. The solution is (-, -4). Solve the first equation for since the coefficient of is = 7 First equation = 7-3 = 7-3 Subtract 3 from each side. Simplif. Find the value of b substituting 7-3 for in the second equation. - 4 = -6 Second equation (7-3) - 4 = -6 = = -6 Distributive Propert = -6 Combine like terms = Subtract 14 from each side. -10 = -0 Simplif. Eercises Use substitution to solve each sstem of equations. = Divide each side b -10 Use = to find the value of. = 7-3 = 7-3() = 1 The solution is (1, ). and simplif. 1. = 4. = 3. = = 1 = - = = = = 0 3 = = = 4 7. b = 6a = = a - b = 7 = = 4 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 10. = = = = 10 + = = 1.1 Chapter 6 1 Glencoe Algebra 1

19 NAME DATE PERID 6- Stud Guide and Intervention (continued) Substitution Solve Real-World Problems Substitution can also be used to solve real-world problems involving sstems of equations. It ma be helpful to use tables, charts, diagrams, or graphs to help ou organize data. Eample CHEMISTRY How much of a 10% saline solution should be mied with a 0% saline solution to obtain 1000 milliliters of a 1% saline solution? Let s = the number of milliliters of 10% saline solution. Let t = the number of milliliters of 0% saline solution. Use a table to organize the information. 10% saline 0% saline 1% saline Total milliliters s t 1000 Milliliters of saline 0.10 s 0.0 t 0.1(1000) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Write a sstem of equations. s + t = s + 0.0t = 0.1(1000) Use substitution to solve this sstem. s + t = 1000 First equation s = t Solve for s. 0.10s + 0.0t = 0.1(1000) Second equation 0.10( t) + 0.0t = 0.1(1000) s = t t + 0.0t = 0.1(1000) Distributive Propert t = 0.1(1000) Combine like terms. 0.10t = 0 Simplif. 0.10t 0.10 = Divide each side b t = 00 Simplif. s + t = 1000 First equation s + 00 = 1000 t = 00 s = 800 Solve for s. 800 milliliters of 10% solution and 00 milliliters of 0% solution should be used. Eercises 1. SPRTS At the end of the football season, 38 Super Bowl games had been plaed with the current two football leagues, the American Football Conference (AFC) and the National Football Conference (NFC). The NFC won two more games than the AFC. How man games did each conference win?. CHEMISTRY A lab needs to make 100 gallons of an 18% acid solution b miing a 1% acid solution with a 0% solution. How man gallons of each solution are needed? Lesson 6-3. GEMETRY The perimeter of a triangle is 4 inches. The longest side is 4 inches longer than the shortest side, and the shortest side is three-fourths the length of the middle side. Find the length of each side of the triangle. Chapter 6 13 Glencoe Algebra 1

20 NAME DATE PERID 6- Skills Practice Substitution Use substitution to solve each sstem of equations. 1. = 4. = + = = = 3 4. = -4 + = = 0 5. = = = = 7. = = = = = = 5-8 = = = = = = = = 8-3 = -9-5 = = = = 0 - = = = 7-3 = = -3 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc = = - - = = 0 Chapter 6 14 Glencoe Algebra 1

21 NAME DATE PERID 6- Practice Substitution Use substitution to solve each sstem of equations. 1. = 6. = 3 3. = = = 1 = = - 5. = = 1 = + - = = = = = = = 1 + = = = = = 18-7 = -7 - = = = = = = 4 - = = = = -4 + = 5 = = 5 Lesson 6- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 19. EMPLYMENT Kenisha sells athletic shoes part-time at a department store. She can earn either $500 per month plus a 4% commission on her total sales, or $400 per month plus a 5% commission on total sales. a. Write a sstem of equations to represent the situation. b. What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pa scale? c. Which is the better offer? 0. MVIE TICKETS Tickets to a movie cost $7.5 for adults and $5.50 for students. A group of friends purchased 8 tickets for $5.75. a. Write a sstem of equations to represent the situation. b. How man adult tickets and student tickets were purchased? Chapter 6 15 Glencoe Algebra 1

22 NAME DATE PERID 6- Word Problem Practice Substitution 1. BUSINESS Mr. Randolph finds that the suppl and demand for gasoline at his station are generall given b the following equations. - =- + = 10 Use substitution to find the equilibrium point where the suppl and demand lines intersect. 4. PPULATIN Sanja is researching population trends in South America. He found that the population of Ecuador to increased b 1,000,000 and the population of Chile to increased b 600,000 from 004 to 009. The table displas the information he found. Countr 004 Population 5-Year Population Change Ecuador 13,000,000 +1,000,000 Chile 16,000, ,000 Source: World Almanac. GEMETRY The measures of complementar angles have a sum of 90 degrees. Angle A and angle B are complementar, and their measures have a difference of 0. What are the measures of the angles? A B 3. MNEY Harve has some $1 bills and some $5 bills. In all, he has 6 bills worth $. Let be the number of $1 bills and let be the number of $5 bills. Write a sstem of equations to represent the information and use substitution to determine how man bills of each denomination Harve has. If the population growth for each countr continues at the same rate, in what ear are the populations of Ecuador and Chile predicted to be equal? 5. CHEMISTRY Shelb and Calvin are doing a chemistr eperiment. The need 5 ounces of a solution that is 65% acid and 35% distilled water. There is no undiluted acid in the chemistr lab, but the do have two flasks of diluted acid: Flask A contains 70% acid and 30% distilled water. Flask B contains 0% acid and 80% distilled water. a. Write a sstem of equations that Shelb and Calvin could use to determine how man ounces the need to pour from each flask to make their solution. b. Solve our sstem of equations. How man ounces from each f lask do Shelb and Calvin need? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 16 Glencoe Algebra 1

23 NAME DATE PERID 6- Enrichment Intersection of Two Parabolas Substitution can be used to find the intersection of two parabolas. Replace the -value in one of the equations with the -value in terms of from the other equation. Eample Find the intersection of the two parabolas. = = Graph the equations Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc From the graph, notice that the two graphs intersect in one point. Use substitution to solve for the point of intersection = = = 3 = -3 The graphs intersect at = -3. Subtract from each side. Subtract 4 from each side. Subtract 6 from each side. Replace with -3 in either equation to find the -value. = riginal equation = (-3) + 5(-3) + 6 Replace with -3. = or 0 The point of intersection is (-3, 0). Eercises Simplif. Use substitution to find the point of intersection of the graphs of each pair of equations. 1. = = = = = = Lesson 6- Chapter 6 17 Glencoe Algebra 1

24 NAME DATE PERID 6-3 Stud Guide and Intervention Elimination Using Addition and Subtraction Elimination Using Addition In sstems of equations in which the coefficients of the or terms are additive inverses, solve the sstem b adding the equations. Because one of the variables is eliminated, this method is called elimination. Eample 1 Use elimination to solve Eample the sstem of equations. - 3 = = 9 Write the equations in column form and add to eliminate. + = 70-3 = 7 (+) - = 4 (+) = 9 = 94 4 = 16 Solve for. 4 4 = 16 4 = 4 Substitute 4 for in either equation and solve for. 4-3 = = = = 3-3 = -1 The solution is (4, -1). Eercises Use elimination to solve each sstem of equations. The sum of two numbers is 70 and their difference is 4. Find the numbers. Let represent one number and represent the other number. = 94 = 47 Substitute 47 for in either equation = = = 3 The numbers are 47 and = = = -9 - = + 3 = = = = = = -4 - = 6 - = = = = 3 - = = - + = = = = = = = 0. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 13. Rema is older than Ken. The difference of their ages is 1 and the sum of their ages is 50. Find the age of each. 14. The sum of the digits of a two-digit number is 1. The difference of the digits is. Find the number if the units digit is larger than the tens digit. Chapter 6 18 Glencoe Algebra 1

25 NAME DATE PERID 6-3 Stud Guide and Intervention (continued) Elimination Using Addition and Subtraction Elimination Using Subtraction In sstems of equations where the coefficients of the or terms are the same, solve the sstem b subtracting the equations. Eample - 3 = = 14 Use elimination to solve the sstem of equations. - 3 = 11 (-) 5-3 = 14-3 = -3 Subtract the two equations. is eliminated = -3-3 Divide each side b -3. Write the equations in column form and subtract. = 1 Simplif. (1) - 3 = 11 Substitute 1 for in either equation. - 3 = 11 Simplif = 11 - Subtract from each side. -3 = 9 Simplif = 9-3 = -3 The solution is (1, -3). Divide each side b -3. Simplif. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eercises Use elimination to solve each sstem of equations = 4. 3m - 4n = a + b = = -0 3m + n = - a + b = = = = = - 3 = 16 + = 3 7. a - 3b = = = 6 a + b = = = = = = = 4-3 = = The sum of two numbers is 70. ne number is ten more than twice the other number. Find the numbers. Lesson GEMETRY Two angles are supplementar. The measure of one angle is 10 more than three times the other. Find the measure of each angle. Chapter 6 19 Glencoe Algebra 1

26 NAME DATE PERID 6-3 Skills Practice Elimination Using Addition and Subtraction Use elimination to solve each sstem of equations = = 1 + = 3 + = = = 6-6 = = = = = 33-4 = = = = = = = = 30 + = = = = = = = = = = = = = The sum of two numbers is 8 and their difference is 4. What are the numbers? 18. Find the two numbers whose sum is 9 and whose difference is The sum of two numbers is 4 and their difference is. What are the numbers? 0. Find the two numbers whose sum is 54 and whose difference is 4. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. Two times a number added to another number is 5. Three times the first number minus the other number is 0. Find the numbers. Chapter 6 0 Glencoe Algebra 1

27 NAME DATE PERID 6-3 Practice Elimination Using Addition and Subtraction Use elimination to solve each sstem of equations = 1. p + q = = 3 + = -9 p - q = = = = = + = = = = = c - d = = = -6 c - d = = = = = = = = = m - 8n = 3-4 = = 1.9 m - 8n = -3 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc a + b = = = 8 1 4a + 3b = = = The sum of two numbers is 41 and their difference is 5. What are the numbers? 0. Four times one number added to another number is 36. Three times the first number minus the other number is 0. Find the numbers. 1. ne number added to three times another number is 4. Five times the first number added to three times the other number is 36. Find the numbers.. LANGUAGES English is spoken as the first or primar language in 78 more countries than Farsi is spoken as the first language. Together, English and Farsi are spoken as a first language in 130 countries. In how man countries is English spoken as the first language? In how man countries is Farsi spoken as the first language? Lesson DISCUNTS At a sale on winter clothing, Cod bought two pairs of gloves and four hats for $ Tori bought two pairs of gloves and two hats for $ What were the prices for the gloves and hats? Chapter 6 1 Glencoe Algebra 1

28 NAME DATE PERID 6-3 Word Problem Practice Elimination Using Addition and Subtraction 1. NUMBER FUN Ms. Simms, the sith grade math teacher, gave her students this challenge problem. Twice a number added to another number is 15. The sum of the two numbers is 11. Lorenzo, an algebra student who was Ms. Simms aide, realized he could solve the problem b writing the following sstem of equations. + = 15 + = 11 Use the elimination method to solve the sstem and find the two numbers.. GVERNMENT The Teas State Legislature is comprised of state senators and state representatives. The sum of the number of senators and representatives is 181. There are 119 more representatives than senators. How man senators and how man representatives make up the Teas State Legislature? 3. RESEARCH Melissa wondered how much it would cost to send a letter b mail in 1990, so she asked her father. Rather than answer directl, Melissa s father gave her the following information. It would have cost $3.70 to send 13 postcards and 7 letters, and it would have cost $.65 to send 6 postcards and 7 letters. Use a sstem of equations and elimination to find how much it cost to send a letter in SPRTS As of 010, the New York Yankees had won more World Series Championships than an other team. In fact, the Yankees had won 3 fewer than 3 times the number of World Series championships won b the second mostwinning team, the St. Louis Cardinals. The sum of the two teams World Series championships is 37. How man times has each team won the World Series? 5. BASKETBALL In 005, the average ticket prices for Dallas Mavericks games and Boston Celtics games are shown in the table below. The change in price is from the 004 season to the 005 season. Team Average Ticket Price Source: Team Marketing Report Change in Price Dallas $53.60 $0.53 Boston $ $1.08 a. Assume that tickets continue to change at the same rate each ear after 005. Let be the number of ears after 005, and be the price of an average ticket. Write a sstem of equations to represent the information in the table. b. In how man ears will the average ticket price for Dallas approimatel equal that of Boston? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 Glencoe Algebra 1

29 NAME DATE PERID 6-3 Enrichment Solving Sstems of Equations in Three Variables Sstems of equations can involve more than equations and variables. It is possible to solve a sstem of 3 equations and 3 variables using elimination. Eample Solve the following sstem. + + z = z = 8 - z = Step 1: Use elimination to get rid of the in the first two equations. + + z = z = z = 14 Step : Use the equation ou found in step 1 and the third equation to eliminate the z. 4 + z = 14 z = Multipl the second equation b so that the z s will eliminate. 4 + z = 14 z = 4 6 = 18 So, = 3. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 3: Replace with 3 in the third original equation to determine z. 3 z =, so z = 1. Step 4: Replace with 3 and z with 1 in either of the first two original equation to determine the value of = 6 or 4 + = 6. So, =. So, the solution to the sstem of equations is (3,, 1). Eercises Solve each sstem of equations z = z = z = 7 + z + 1 = z = z = 10 3 = 0 z = 4 + z = 5 Lesson 6-3 Chapter 6 3 Glencoe Algebra 1

30 NAME DATE PERID 6-4 Stud Guide and Intervention Elimination Using Multiplication Elimination Using Multiplication Some sstems of equations cannot be solved simpl b adding or subtracting the equations. In such cases, one or both equations must first be multiplied b a number before the sstem can be solved b elimination. Eample 1 Use elimination to solve Eample the sstem of equations = = 5 If ou multipl the second equation b -, ou can eliminate the terms = 3 (+) = = = -7-7 = 1 Substitute 1 for in either equation = = = = 10 = 1 5 The solution is (1, 1. 5) Eercises Use elimination to solve each sstem of equations. Use elimination to solve the sstem of equations. 3 - = -7-5 = 10 If ou multipl the first equation b and the second equation b -3, ou can eliminate the terms. 6-4 = -14 (+) = = = = -4 Substitute -4 for in either equation. 3 - (-4) = = = = = = -5 The solution is (-5, -4) = 6. m + 3n = a - b = + = 5 -m + n = 5 a + b = = = = = -0 - = = = a - 3b = = = 8 a + b = = = = = = = =.5 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 13. GARDENING The length of Sall s garden is 4 meters greater than 3 times the width. The perimeter of her garden is 7 meters. What are the dimensions of Sall s garden? Chapter 6 4 Glencoe Algebra 1

31 NAME DATE PERID 6-4 Stud Guide and Intervention (continued) Elimination Using Multiplication Solve Real-World Problems Sometimes it is necessar to use multiplication before elimination in real-world problems. Eample CANEING During a canoeing trip, it takes Ramond 4 hours to paddle 1 miles upstream. It takes him 3 hours to make the return trip paddling downstream. Find the speed of the canoe in still water. Read You are asked to find the speed of the canoe in still water. Solve Let c = the rate of the canoe in still water. Let w = the rate of the water current. r t d r t = d Against the Current c w 4 1 (c w)4 = 1 With the Current c + w 3 1 (c + w)3 = 1 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. So, our two equations are 4c - 4w = 1 and 3c + 3w = 1. Use elimination with multiplication to solve the sstem. Since the problem asks for c, eliminate w. 4c - 4w = 1 Multipl b 3 1c - 1w = 36 3c + 3w = 1 Multipl b 4 (+) 1c + 1w = 48 4c = 84 w is eliminated. 4c = Divide each side b 4. c = 3.5 Simplif. The rate of the canoe in still water is 3.5 miles per hour. Eercises 1. FLIGHT An airplane traveling with the wind flies 450 miles in hours. n the return trip, the plane takes 3 hours to travel the same distance. Find the speed of the airplane if the wind is still.. FUNDRAISING Benji and Joel are raising mone for their class trip b selling gift wrapping paper. Benji raises $39 b selling 5 rolls of red wrapping paper and rolls of foil wrapping paper. Joel raises $57 b selling 3 rolls of red wrapping paper and 6 rolls of foil wrapping paper. For how much are Benji and Joel selling each roll of red and foil wrapping paper? Lesson 6-4 Chapter 6 5 Glencoe Algebra 1

32 NAME DATE PERID 6-4 Skills Practice Elimination Using Multiplication Use elimination to solve each sstem of equations = = = 3 - = = = = = = = = = = = = = = = = = = = = = Two times a number plus three times another number equals 13. The sum of the two numbers is 7. What are the numbers? 14. Four times a number minus twice another number is -16. The sum of the two numbers is -1. Find the numbers. 15. FUNDRAISING Trisha and Bron are washing and vacuuming cars to raise mone for a class trip. Trisha raised $38 washing 5 cars and vacuuming 4 cars. Bron raised $8 b washing 4 cars and vacuuming cars. Find the amount the charged to wash a car and vacuum a car. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 6 Glencoe Algebra 1

33 NAME DATE PERID 6-4 Practice Elimination Using Multiplication Use elimination to solve each sstem of equations = = = = = = = = = = = = = = = = = = 0 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc = = = = + 6 = = Eight times a number plus five times another number is -13. The sum of the two numbers is 1. What are the numbers? 14. Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7. Find the numbers. 15. FINANCE Gunther invested $10,000 in two mutual funds. ne of the funds rose 6% in one ear, and the other rose 9% in one ear. If Gunther s investment rose a total of $684 in one ear, how much did he invest in each mutual fund? 16. CANEING Laura and Brent paddled a canoe 6 miles upstream in four hours. The return trip took three hours. Find the rate at which Laura and Brent paddled the canoe in still water. Lesson NUMBER THERY The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 more than the original number. Find the number. Chapter 6 7 Glencoe Algebra 1

34 NAME DATE PERID 6-4 Word Problem Practice Elimination Using Multiplication 1. SCCER Suppose a outh soccer field has a perimeter of 30 ards and its length measures 40 ards more than its width. Ms. Hughe asks her plaers to determine the length and width of their field. She gives them the following sstem of equations to represent the situation. Use elimination to solve the sstem to find the length and width of the field. 4. TRAVEL Antonio flies from Houston to Philadelphia, a distance of about 1340 miles. His plane travels with the wind and takes hours and 0 minutes. At the same time, Paul is on a plane from Philadelphia to Houston. Since his plane is heading against the wind, Paul s flight takes hours and 50 minutes. What was the speed of the wind in miles per hour? L + W = 30 L W = 40. SPRTS The Fan Cost Inde (FCI) tracks the average costs for attending sporting events, including tickets, drinks, food, parking, programs, and souvenirs. According to the FCI, a famil of four would spend a total of $59.30 to attend two Major League Baseball (MLB) games and one National Basketball Association (NBA) game. The famil would spend $ to attend one MLB and two NBA games. Write and solve a sstem of equations to find the famil s costs for each kind of game according to the FCI. 3. ART Mr. Santos, the curator of the children s museum, recentl made two purchases of cla and wood for a visiting artist to sculpt. Use the table to find the cost of each product per kilogram. Cla (kg) Wood (kg) Total Cost 5 4 $ $ BUSINESS Suppose ou start a business assembling and selling motorized scooters. It costs ou $1500 for tools and equipment to get started, and the materials cost $00 for each scooter. Your scooters sell for $300 each. a. Write and solve a sstem of equations representing the total costs and revenue of our business. b. Describe what the solution means in terms of the situation. c. Give an eample of a reasonable number of scooters ou could assemble and sell in order to make a profit, and find the profit ou would make for that number of scooters. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 8 Glencoe Algebra 1

35 NAME DATE PERID 6-4 Enrichment George Washington Carver and Perc Julian In 1990, George Washington Carver and Perc Julian became the first African Americans elected to the National Inventors Hall of Fame. Carver ( ) was an agricultural scientist known worldwide for developing hundreds of uses for the peanut and the sweet potato. His work revitalized the econom of the southern United States because it was no longer dependent solel upon cotton. Julian ( ) was a research chemist who became famous for inventing a method of making a snthetic cortisone from sobeans. His discover has had man medical applications, particularl in the treatment of arthritis. There are dozens of other African American inventors whose accomplishments are not as well known. Their inventions range from common household items like the ironing board to comple devices that have revolutionized manufacturing. The eercises that follow will help ou identif just a few of these inventors and their inventions. Match the inventors with their inventions b matching each sstem with its solution. (Not all the solutions will be used.) 1. Sara Boone + = A. (1, 4) automatic traffic signal - = 10. Sarah Goode = - B. (4, -) eggbeater + = 9 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3. Frederick M. = + 6 C. (-, 3) fire etinguisher Jones = J. L. Love + 3 = 8 D. (-5, 7) folding cabinet bed - = T. J. Marshall - 3 = 9 E. (6, -4) ironing board + = 4 6. Jan Matzeliger + 4 = F. (-, 4) pencil sharpener 6-3 = 1 7. Garrett A. 3 - = -5 G. (-3, 0) portable -ra machine Morgan 3-4 = 8 8. Norbert Rillieu 3 - = 1 H. (, -3) plaer piano - 3 = 15 I. no solution evaporating pan for refining sugar J. infinitel lasting (shaping) man machine for solutions manufacturing shoes Lesson 6-4 Chapter 6 9 Glencoe Algebra 1

36 NAME DATE PERID 6-5 Stud Guide and Intervention Appling Sstems of Linear Equations Determine The Best Method You have learned five methods for solving sstems of linear equations: graphing, substitution, elimination using addition, elimination using subtraction, and elimination using multiplication. For an eact solution, an algebraic method is best. Eample At a baseball game, Henr bought 3 hotdogs and a bag of chips for $14. Scott bought hotdogs and a bag of chips for $10. Each of the bos paid the same price for their hotdogs, and the same price for their chips. The following sstem of equations can be used to represent the situation. Determine the best method to solve the sstem of equations. Then solve the sstem. 3 + = 14 + = 10 Since neither the coefficients of nor the coefficients of are additive inverses, ou cannot use elimination using addition. Since the coefficient of in both equations is 1, ou can use elimination using subtraction. You could also use the substitution method or elimination using multiplication The following solution uses elimination b subtraction to solve this sstem. 3 + = 14 Write the equations in column form and subtract. (-) + (-) = (-)10 = 4 The variable is eliminated. 3(4) + = 14 Substitute the value for back into the first equation. = Solve for. This means that hot dogs cost $4 each and a bag of chips costs $. Eercises Determine the best method to solve each sstem of equations. Then solve the sstem = = = = = = = = 50 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 6 30 Glencoe Algebra 1

37 NAME DATE PERID 6-5 Stud Guide and Intervention (continued) Appling Sstems of Linear Equations Appl Sstems f Linear Equations When appling sstems of linear equations to problem situations, it is important to analze each solution in the contet of the situation. Eample BUSINESS A T-shirt printing compan sells T-shirts for $15 each. The compan has a fied cost for the machine used to print the T-shirts and an additional cost per T-shirt. Use the table to estimate the number of T-shirts the compan must sell in order for the income to equal epenses. T-shirt Printing Cost printing machine $ blank T-shirt $5.00 Understand You know the initial income and the initial epense and the rates of change of each quantit with each T-shirt sold. Plan Solve Write an equation to represent the income and the epenses. Then solve to find how man T-shirts need to be sold for both values to be equal. Let = the number of T-shirts sold and let = the total amount. total amount initial amount rate of change times number of T-shirts sold income = epenses = Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Check Eercises You can use substitution to solve this sstem. = = = 3000 = 300 The fi rst equation. Substitute the value for into the second equation. Subtract 10 from each side and simplif. Divide each side b 10 and simplif. This means that if 300 T-shirts are sold, the income and epenses of the T-shirt compan are equal. Does this solution make sense in the contet of the problem? After selling 300 T-shirts, the income would be about 300 $15 or $4500. The costs would be about $ $5 or $4500. Refer to the eample above. If the costs of the T-shirt compan change to the given values and the selling price remains the same, determine the number of T-shirts the compan must sell in order for income to equal epenses. 1. printing machine: $ ;. printing machine: $100.00; T-shirt: $10.00 each T-shirt: $8.00 each 3. printing machine: $ ; 4. printing machine: $100.00; T-shirt: $4.00 each T-shirt: $1.00 each Lesson 6-5 Chapter 6 31 Glencoe Algebra 1