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(2) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 403–404. 405–406. 407. 408. 409–410. 411–412. 413. 414. 447. 415–416. 417–418. 419. 420. 421–422. 423–424. 425. 427–428. 429–430. 431. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. Enr ich me nt. S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice. Ave rag e). Rea di Ma ng to the ma Learn tics. CHAPTER 7 RESOURCE MASTERS. See pages T12–T13.. SC 13. 53. 7-1. 7-1. SM 63–66. 54. 7-2. 7-2. 447, 449. GCS 36. 55. 7-3. 7-3. 426. 448. SC 14. 56. 7-4. 7-4. 432. 448. GCS 35. 57. 7-5. 7-5. 27–28. 433–446, 450–452. Materials. (Follow-Up: graphing calculator). 16. algebra tiles, equation mat. 17. graphing calculator. 58. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual. ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 7 Solving Systems of Linear Equations and Inequalities 366B.
(3) Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 2, students learned that when you add additive inverses, the sum is always 0. They used the Addition, Subtraction, Multiplication, and Division Properties of Equality to solve equations throughout Chapter 3. Students graph linear equations in Chapter 5 and graph linear inequalities in Chapter 6.. Graphing Systems of Equations A solution of a system of equations is the set of points that satisfy each equation in the system. Carefully graph each equation on the same coordinate plane. There is only one solution if the graphs of the lines intersect, since the intersection is at only one point. There is no solution if the lines are parallel. The lines never intersect, so no one point is common to both graphs. If the graphs are the same line, the system has an infinite number of solutions. If there is one solution or infinitely many solutions, the system of equations is described as consistent. Systems with one solution are said to be independent, while those with infinitely many solutions are said to be dependent. If there is no solution, the system is described as inconsistent.. This Chapter Chapter 7 introduces students to systems of linear equations. They first solve the systems by graphing and then classify the systems as consistent or inconsistent, and as independent or dependent. Students also learn to apply the algebraic methods to solving the systems. These methods include substitution, elimination using addition or subtraction, and elimination using multiplication first. Students must determine which method is best for different systems. The chapter ends with students solving systems of inequalities by graphing.. Future Connections Business analysts use systems of linear equations to determine where break-even points are and to analyze trends for predicting future events. There are not only systems of linear equations and inequalities, but also systems of all types of functions including quadratic, absolute value, and sine. These systems can mix any types of functions. The solutions of these systems are not only used in business, but also in science and other fields.. 366C. Chapter 7 Solving Systems of Linear Equations and Inequalities. Substitution It is sometimes difficult to determine the exact solution of a system of equations from a graph. Therefore, an algebraic method may be used to find the exact solution. Substitution is an algebraic method. First solve one equation for one variable in terms of the other. Substitute the expression into the other equation so that one variable is eliminated. Solve for the remaining variable. Substitute this value into either equation and solve for the other variable. The two values make up the solution of the system. They are written in the form (x, y) to represent the point where the two lines intersect if graphed. If a solution results in an identity, for example 2 2, the system has an infinite number of solutions. If the result is a false statement, such as 5 3, there is no solution. If you incur either of these situations during any part of the solution process, you may stop solving and write either infinitely many solutions or no solution..
(4) Elimination Using Addition and Subtraction Elimination is another algebraic method used to solve systems of equations. The objective is to combine the two equations to eliminate one of the variables. If the coefficients of one variable are additive inverses of each other, use addition. If the coefficients of one variable are the same, use subtraction. Because the Addition and Subtraction Properties of Equality state that equal amounts can be added to or subtracted from each side of an equation, you can add or subtract one equation with the other. This step eliminates one variable. Then solve for the other variable. Substitute this value into either original equation to find the value of the variable that was eliminated. The two values are the solution of the system.. Graphing Systems of Inequalities A solution of a system of inequalities is all the points that satisfy both inequalities. Use the methods learned in Lesson 6-6 to graph each inequality. The points that are solutions of both inequalities lie in the region where the graphs overlap, or intersect. A system of inequalities has no solution if the boundary lines are parallel and the shaded regions do not overlap. Otherwise, there are infinitely many solutions since the overlapping shaded region extends on indefinitely.. Elimination Using Multiplication If the coefficients of one variable are neither additive inverses nor equal, one or both equations must be changed so that the elimination method can be applied to solve the system of equations. Change either one, or both, of the equations by applying the Multiplication Property of Equality. Every term of the equation is multiplied by the same number, or both equations are multiplied by different numbers, in order to make one pair of coefficients of a variable either additive inverses or the same. Then follow the steps for solving the system using the elimination method. Five methods for solving systems of equations have been seen in this chapter. They are graphing, substitution, elimination using addition, elimination using subtraction, and elimination using multiplication. Graphing is used only if an estimation is needed, since it is difficult to get an exact solution. Use substitution if one of the variables has a coefficient of 1 or 1. Elimination using addition is used when one of the variables has coefficients that are additive inverses of each other, and elimination using subtraction is best used when one of the variables has the same coefficient. Apply elimination using multiplication if none of the above situations occur.. www.algebra1.com/key_concepts Additional mathematical information and teaching notes are available in Glencoe’s Algebra 1 Key Concepts: Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows. • Graphing Systems of Equations (Lesson 18) • Substitution (Lesson 19) • Elimination Using Addition and Subtraction (Lesson 20) • Elimination Using Multiplication (Lesson 21) Chapter 7 Solving Systems of Linear Equations and Inequalities 366D.
(5) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 367, 374, 381, 386, 392 Practice Quiz 1, p. 381 Practice Quiz 2, p. 392. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 27–28 Quizzes, CRM pp. 447–448 Mid-Chapter Test, CRM p. 449 Study Guide and Intervention, CRM pp. 403–404, 409–410, 415–416, 421–422, 427–428. Mixed Review. pp. 374, 381, 386, 392, 398. Cumulative Review, CRM p. 450. Error Analysis. Find the Error, pp. 384, 396. Find the Error, TWE pp. 384, 396 Unlocking Misconceptions, TWE p. 372 Tips for New Teachers, TWE pp. 367, 384. Standardized Test Practice. pp. 374, 381, 384, 385, 386, 392, 398, 403, 404–405. TWE pp. 404–405 Standardized Test Practice, CRM pp. 451–452. Open-Ended Assessment. Writing in Math, pp. 374, 381, 386, 392, 398 Open Ended, pp. 371, 379, 384, 390, 396 Standardized Test, p. 405. Modeling: TWE pp. 374 Speaking: TWE pp. 381, 392 Writing: TWE pp. 386, 398 Open-Ended Assessment, CRM p. 445. Chapter Assessment. Study Guide, pp. 399–402 Practice Test, p. 403. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 433–438 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 439–444 Vocabulary Test/Review, CRM p. 446. Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters. Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS. TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.. 366E. Chapter 7 Solving Systems of Linear Equations and Inequalities.
(6) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson. AlgePASS Lesson. 7-2. 16 Solving a System of Simultaneous Equations. 7-5. 17 Graphing Linear Inequalities on the Coordinate Plane. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 367 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 371, 379, 384, 390, 396) • Reading Mathematics, p. 393 • Writing in Math questions in every lesson, pp. 374, 381, 386, 392, 398 • WebQuest, pp. 373, 398 Teacher Wraparound Edition. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 53–58 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. • Foldables Study Organizer, pp. 367, 399 • Study Notebook suggestions, pp. 372, 379, 384, 390, 393, 396 • Modeling activities, p. 374 • Speaking activities, pp. 381, 392 • Writing activities, pp. 386, 398 • Differentiated Instruction, (Verbal/Linguistic), p. 389 • ELL Resources, pp. 366, 373, 380, 385, 389, 391, 393, 397, 399 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 7 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 7 Resource Masters, pp. 407, 413, 419, 425, 431) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources • Hot Words/Hot Topics Sections 6.2–6.4, 6.6–6.8, 9.4. For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 7 Solving Systems of Linear Equations and Inequalities 366F.
(7) Solving Systems of Linear Equations and Inequalities. Notes Have students read over the list of objectives and make a list of any words with which they are not familiar.. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson.. • Lesson 7-1 Solve systems of linear equations by graphing. • Lessons 7-2 through 7-4 Solve systems of linear equations algebraically. • Lesson 7-5 Solve systems of linear inequalities by graphing.. Key Vocabulary • • • •. system of equations (p. 369) substitution (p. 376) elimination (p. 382) system of inequalities (p. 394). Business decision makers often use systems of linear equations to model a real-world situation in order to predict future events. Being able to make an accurate prediction helps them plan and manage their businesses. Trends in the travel industry change with time. For example, in recent years, the number of tourists traveling to South America, the Caribbean, and the Middle East is on the rise. You will use a system of linear equations to model the trends in tourism in Lesson 7-2.. Lesson 7-1 Preview 7-1 7-1 Follow-Up 7-2 7-3 7-4 7-5. NCTM Standards. Local Objectives. 1, 2, 6, 9, 10 2, 6, 8, 9, 10 2, 6 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10. Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 366. 366 Chapter 7 Solving Systems of Linear Equations and Inequalities. Vocabulary Builder. ELL. The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 7 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 7 test.. Chapter 7 Solving Systems of Linear Equations and Inequalities.
(8) Prerequisite Skills. To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 7.. For Lesson 7-1. Graph Linear Equations. Graph each equation. (For review, see Lesson 4-5.) 1–6. See pp. 405A–405D. 1. y 1. 2. y 2x. 3. y 4 x. 4. y 2x 3. 5. y 5 2x. 6. y x 2. 1 2. For Lesson 7-2. Solve for a Given Variable. Solve each equation or formula for the variable specified. (For review, see Lesson 3-8.) 16 y a 7. 4x a 6x, for x x 8. 8a y 16, for a a 8 2 120 d 7bc d 7m n 7m n 9. 12, for b b 10. 2m, for q q 10 q 7c 2m For Lessons 7-3 and 7-4. Simplify Expressions. Simplify each expression. If not possible, write simplified. (For review, see Lesson 1-5.) 11. (3x y) (2x y) x. 12. (7x 2y) (7x 4y) 6y 13. (16x 3y) (11x 3y) 27y. 14. (8x 4y) (8x 5y) y 15. 4(2x 3y) (8x y) 13y 16. 3(x 4y) (x 12y) 4x 17. 2(x 2y) (3x 4y) 5x 18. 5(2x y) 2(5x 3y). 19. 3(x 4y) 2(2x 6y) 7x. 11y. Make this Foldable to record information about solving systems of equations and inequalities. Begin with five sheets of grid paper. Fold. Cut. Fold each sheet in half along the width.. Unfold and cut four rows from left side of each sheet, from the top to the crease.. Stack and Staple. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson. For Lesson. Prerequisite Skill. 7-2. Solving Equations for a Specified Value, p. 374 Simplifying Expressions, p. 381 Distributive Property, p. 386 Graphing Inequalities, p. 392. 7-3 7-4 7-5. In addition to reviewing New graphing as in Exercises 1-6, you may want to review slope-intercept form. These are two skills essential for students’ success in this chapter.. Label Solving Systems of Equations and Inequations. Stack the sheets and staple to form a booklet.. This section provides a review of the basic concepts needed before beginning Chapter 7. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 27–28.. 7-1 Graphi Systems of ng Equations. Label each page with a lesson number and title.. As you read and study the chapter, unfold each page and fill the journal with notes, graphs, and examples for systems of equations and inequalities.. Reading and Writing. Chapter 7 Solving Systems of Linear Equations and Inequalities 367. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Organization of Data: Visualization Journal After students make their visualization journal, have them label the top of each front page with a lesson number. Under the tabs of their Foldable, students take notes and define terms presented in each lesson. At the end of each lesson, ask students to design a visual (graph, diagram, picture, chart) that presents the lesson information in a concise, easy-to-study format. Encourage students to clearly label their visuals. Chapter 7 Solving Systems of Linear Equations and Inequalities 367.
(9) Spreadsheet Investigation. A Preview of Lesson 7-1. A Preview of Lesson 7-1. Systems of Equations. Getting Started. You can use a spreadsheet to investigate when two quantities will be equal. Enter each formula into the spreadsheet and look for the time when both formulas have the same result.. Copying Formulas Once the spreadsheet formulas are entered for one sales amount, those formulas can be dragged to copy for all the other sales amounts. Students must take their time creating the first formulas as any errors will be duplicated to all the other cells.. Example Bill Winters is considering two job offers in telemarketing departments. The salary at the first job is $400 per week plus 10% commission on Mr. Winters’ sales. At the second job, the salary is $375 per week plus 15% commission. For what amount of sales would the weekly salary be the same at either job?. Teach • In Excel, you create a graph by using the Chart Wizard button on the standard toolbar. If using another spreadsheet software program, consult the Help feature for instructions on creating graphs (charts). • Remind students that spreadsheet software cannot interpret an expression such as 0.1x. Students must type the multiplication sign implied. • Urge students not to assume that the software will apply the order of operations. Students should use parentheses to insure that operations are performed in the correct order. • You may wish to extend the activity by discussing the pros and cons of different sales commissions. Students who can imagine themselves as salespeople can relate personally to the amounts calculated and to the process of using equations to explore pay rates.. Enter different amounts for Mr. Winters’ weekly sales in column A. Then enter the formula for the salary at the first job in each cell in column B. In each cell of column C, enter the formula for the salary at the second job. The spreadsheet shows that for sales of $500 the total weekly salary for each job is $450.. 4. (500, 450); If Mr. Winters makes $500 in sales, he will make $450 for either job.. Exercises For Exercises 1– 4, use the spreadsheet of weekly salaries above. 1. If x is the amount of Mr. Winters’ weekly sales and y is his total weekly salary, write a linear equation for the salary at the first job. y 400 0.1x 2. Write a linear equation for the salary at the second job. y 375 0.15x 3. Which ordered pair is a solution for both of the equations you wrote for. Exercises 1 and 2? c a. (100, 410). c. (500, 450). d. (900, 510). 4. Use the graphing capability of the spreadsheet program to graph the salary data. using a line graph. At what point do the two lines intersect? What is the significance of that point in the real-world situation? 5. How could you find the sales for which Mr. Winters’ salary will be equal without. using a spreadsheet? Sample answer: Write and graph two linear equations. Find the point where the graphs intersect. 368. Chapter 7 Solving Systems of Linear Equations and Inequalities. Assess Exercises 1–2 Students should write the correct expressions to translate the real-world scenario into an algebraic representation. Exercise 4 Students should be able to translate between the mathematical solution and its real-world meaning. 368. b. (300, 420). Chapter 7 Solving Systems of Linear Equations and Inequalities.
(10) Graphing Systems of Equations. Lesson Notes. • Determine whether a system of linear equations has 0, 1, or infinitely many solutions. • Solve systems of equations by graphing.. Vocabulary system of equations consistent inconsistent independent dependent. During the 1990s, sales of cassette singles decreased, and sales of CD singles increased. Assume that the sales of these singles were linear functions. If x represents the years since 1991 and y represents the sales in millions of dollars, the following equations represent the sales of these singles.. These equations are graphed at the right. The point at which the two graphs intersect represents the time when the sales of cassette singles equaled the sales of CD singles. The ordered pair of this point is a solution of both equations.. 1 Focus 5-Minute Check Transparency 7-1 Use as a quiz or review of Chapter 6. Mathematical Background notes are available for this lesson on p. 366C.. Cassette singles: y 69 6.9x CD singles: y 5.7 6.3x Cassette and CD Singles Sales 70 Sales (millions of dollars). • • • • •. can you use graphs to compare the sales of two products?. 60. y 69 6.9x. y. 50 40. Sales of cassette singles equals sales of CD singles.. 30 20. y 5.7 6.3x. 10 0 1. 2 3 4 5 6 7 Years Since 1991. x 8. NUMBER OF SOLUTIONS Two equations, such as y 69 6.9x and. y 5.7 6.3x, together are called a system of equations . A solution of a system of equations is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have 0, 1, or an infinite number of solutions. • If the graphs intersect or coincide, the system of equations is said to be consistent. That is, it has at least one ordered pair that satisfies both equations. • If the graphs are parallel, the system of equations is said to be inconsistent. There are no ordered pairs that satisfy both equations. • Consistent equations can be independent or dependent . If a system has exactly one solution, it is independent. If the system has an infinite number of solutions, it is dependent.. Systems of Equations Intersecting Lines. Parallel Lines. Same Line. y. can you use graphs to compare the sales of two products? Ask students: • The graph that is sloping downward represents sales of which product? cassette singles • The graph that is sloping upward represents sales of which product? CD singles • In what year were the sales of cassette and CD singles equal? 1996; Note: The x-value of 0 represents the year 1991. • You are told to assume that the sales are linear functions. What does this mean? It means that sales either increase or decrease by the same amount every year. Also, the graphs of linear functions are straight lines.. y. y. Graph of a System O. Number of Solutions Terminology. x. O. x. O. x. exactly one solution. infinitely many. no solutions. consistent and independent. consistent and dependent. inconsistent. Lesson 7-1. Graphing Systems of Equations 369. Resource Manager Workbook and Reproducible Masters Chapter 7 Resource Masters • Study Guide and Intervention, pp. 403–404 • Skills Practice, p. 405 • Practice, p. 406 • Reading to Learn Mathematics, p. 407 • Enrichment, p. 408. Parent and Student Study Guide Workbook, p. 53 School-to-Career Masters, p. 13. Transparencies 5-Minute Check Transparency 7-1 Real-World Transparency 7 Answer Key Transparencies. Technology Interactive Chalkboard. Lesson x-x Lesson Title 369.
(11) 2 Teach. Example 1 Number of Solutions Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.. Building on Prior Knowledge. SOLVE BY GRAPHING One method of solving systems of equations is to carefully graph the equations on the same coordinate plane.. Example 2 Solve a System of Equations Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. a. y x 8 y 4x 7. Study Tip Look Back. To review graphing linear equations, see Lesson 4-5.. y x 1. CHECK y x 8. y y x 8 (3, 5). y 4x 7. 5 3 8. 5 4(3) 7. 55 ⻫. 5 12 7. y 4x 7. b. x 2y 5 2x 4y 2 The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions to this system of equations. Notice that the lines have the same slope but different y-intercepts. Recall that a system of equations that. The graphs are parallel, so there are no solutions.. y x 2y 5 x O. has no solution is said to be inconsistent.. b. 3x 3y 9 y x 1. 2x 4y 2. 370. Chapter 7 Solving Systems of Linear Equations and Inequalities. The graphs coincide, so there are infinitely many solutions.. Differentiated Instruction Logical Before students graph a system of equations, have them write the two equations in slope-intercept form and compare slopes and intercepts. Different slopes mean the lines intersect and there is one solution. Same slope, same intercept means they are the same line and there are infinite solutions. Same slope, different intercepts indicates parallel lines and no solution.. 370. x. O. The solution is (3, 5).. a. y x 1 y x 4. c. x y 3 3x 3y 9. The graphs appear to intersect at the point with coordinates (3, 5). Check this estimate by replacing x with 3 and y with 5 in each equation.. 55 ⻫. x. The graphs intersect, so there is one solution.. yx3. Since the graphs of 2x 2y 8 and y x 4 coincide, there are infinitely many solutions.. 1 Use the graph to determine. O xy3. y x 4. c. 2x 2y 8 y x 4. Tell students to pay close attention to the labels on the graph. If any lines are labeled with more than one equation, then the two equations will have infinitely many solutions.. 3x 3y 9. x O. Since the graphs of y x 5 and 2x 2y 8 are parallel, there are no solutions.. Power Point®. y x 4. 2x 2y 8. b. y x 5 2x 2y 8. Reading Tip. y. y x 5. Since the graphs of y x 5 and y x 3 intersect, there is one solution.. NUMBER OF SOLUTIONS. whether each system has no solution, one solution, or infinitely many solutions.. y 2x 14. a. y x 5 yx3. In Chapter 4, students learned to graph linear equations. In this lesson, they should recognize that graphing systems of equations simply involves graphing more than one linear equation on the same coordinate grid.. In-Class Example. y. Chapter 7 Solving Systems of Linear Equations and Inequalities.
(12) SOLVE BY GRAPHING. Example 3 Write and Solve a System of Equations WORLD RECORDS Use the information on Guy Delage’s swim at the left. If Guy can swim 3 miles per hour for an extended period and the raft drifts about 1 mile per hour, how many hours did he spend swimming each day?. In-Class Examples. Words. You have information about the amount of time spent swimming and floating. You also know the rates and the total distance traveled.. 2 Graph each system of equa-. Variables. Let s the number of hours Guy swam, and let f the number of hours he floated each day. Write a system of equations to represent the situation.. tions. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.. Equations. Source: Banner Aerospace, Inc.. s f 24. 3s f 44. 10 14 24. 3(10) 14 44. 24 24 ⻫. 24. equals. . 1f 28 24 20 16 12 8 4 4. f. the total number of hours in a day.. . . the daily miles traveled floating. Graph the equations s f 24 and 3s f 44. The graphs appear to intersect at the point with coordinates (10, 14). Check this estimate by replacing s with 10 and f with 14 in each equation. CHECK. . f. plus. 3s. equals. . The daily miles traveled swimming. the number of hours floating. . . . . s. World Records In 1994, Guy Delage swam 2400 miles across the Atlantic Ocean from Cape Verde to Barbados. Everyday he would swim awhile and then rest while floating with the current on a huge raft. He averaged 44 miles per day.. plus. . The number of hours swimming. Power Point®. a. 2x y 3 infinitely many 8x 4y 12 solutions y. the total miles traveled in a day.. 2x y 3. 44. x. O. 3s f 44. 8x 4y 12. (10, 14). s f 24 O. b. x 2y 4 x 2y 2 no solution. s 4 8 12 16 20 24 28. 30 14 44 44 44 ⻫. y x 2y 2. Guy Delage spent about 10 hours swimming each day.. x. O. x 2y 4. Concept Check 1–3. See pp. 405A–405D.. 1. OPEN ENDED Draw the graph of a system of equations that has one solution at (2, 3).. 3 BICYCLING Tyler and Pearl. 2. Determine whether a system of equations with (0, 0) and (2, 2) as solutions sometimes, always, or never has other solutions. Explain. 3. Find a counterexample for the following statement. If the graphs of two linear equations have the same slope, then the system of equations has no solution.. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 4–7 8–13 14. 1 2 3. Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions. 4. y x 4 y 1x 2 one 3. 6. x y 4 yx4. infinitely many. www.algebra1.com/extra_examples. 1 3 1 y x 2 3. 5. y x 2 no. solution. y y 13 x 4 y 13 x 2. 7. x y 4 y 1x 4 3. xy4. O. one y x 4 Lesson 7-1. x y 13 x 2. Graphing Systems of Equations. 371. went on a 20-kilometer bike ride that lasted 3 hours. Because there were many steep hills on the bike ride, they had to walk for most of the trip. Their walking speed was 4 kilometers per hour. Their riding speed was 12 kilometers per hour. How much time did they spend walking? Let r the number of hours they rode and w the number of hours they walked. rw3 12r 4w 20 w. Interactive. Chalkboard PowerPoint® Presentations. This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools. 12r 4w 20. rw3 O. r. They walked for 2 hours. Lesson 7-1 Graphing Systems of Equations 371.
(13) Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 8–13. See pp. 405A–405D for graphs.. 3 Practice/Apply. 8. y x y 2x one; (0, 0) 11. x y 4 x y 1 no solution. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 7. • include examples of graphs that have exactly one solution, infinitely many solutions, and no solution. • include any other item(s) that they find helpful in mastering the skills in this lesson.. Application. 9. x y 8 x y 2 one; (5, 3). 10. 2x 4y 2 infinitely 3x 6y 3 many. 12. x y 2 13. x y 2 3y 2x 9 one; (3, 1) y 4x 7 one; (1, 3). 14. RESTAURANTS The Rodriguez family and the Wong family went to a brunch buffet. The restaurant charges one price for adults and another price for children. The Rodriguez family has two adults and three children, and their bill was $40.50. The Wong family has three adults and one child, and their bill was $38.00. Determine the price of the buffet for an adult and the price for a child.. $10.50; $6.50 ★ indicates increased difficulty. Practice and Apply Homework Help For Exercises. See Examples. 15–22 23–40 41–54. 1 2 3. Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.. y 2x 1 y x 3. 15. x 3 16. y x 2 y 2x 4 one y 2x 1 one. Extra Practice See page 835.. 19. y 3x 6 20. 2y 4x 2 y 2x 4 no solution y 2x 4 one. About the Exercises…. y 2x 4. y x 2. 17. y x 2 inf. 18. y 2x 1 y 2x 4 no solution y x 2 many. O 2y 4x 2. 21. 2y 4x 2 22. 2y 4x 2 y 2x 1 infinitely many y 3x 6 one. Organization by Objective • Number of Solutions: 15–22 • Solve by Graphing: 23–54. y x 2. Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 23–40. See pp. 405A–405D for graphs.. Odd/Even Assignments Exercises 15–40 are structured so that students practice the same concepts whether they are assigned odd or even problems.. 1. 23. y 6 24. x 2 25. y x 2 3x y 8 one; (2, 2) 4x y 2 one; (2, 6) 2x y 10 one; (4, 2) 26. y x 27. y 3x 4 28. y 2x 6 y 3x 4 one; (0, 4) y x 3 one; (3, 0) y 2x 6 one; (2, 2) 29. x 2y 2 30. x y 2 31. 3x 2y 12 2y x 10 one; (2, 4) 3x 2y 6 no solution 3x y 6 one; (2, 0). Assignment Guide. 32. 2x 3y 4 infinitely 33. 2x y 4 34. 4x 3y 24 one; (3, 4) 4x 6y 8 many 5x 3y 6 one; (6, 8) 5x 8y 17. Basic: 15–41 odd, 42–47, 55–68 Average: 15–41 odd, 42–47, 55–68 Advanced: 16–40 even, 48–64 (optional: 65–68). 35. 3x y 3 infinitely 2y 6x 6 many. ★ 38. y 23x 5 3y 2x no solution. 36. y x 3 37. 2x 3y 17 3y x 5 one; (1, 2) y x 4 one; (1, 5) 3 8. 39. 6 y x infinitely 2 1 x y 4 3 4. many. 41. GEOMETRY The length of the rectangle at the right is 1 meter less than twice its width. What are the dimensions of the rectangle?. 13 m by 7 m. 1 1 40. x y 6 2. 1. 3. y 2x 2 one; (8, 6) . Perimeter 40 m. 372 Chapter 7 Solving Systems of Linear Equations and Inequalities. Unlocking Misconceptions Infinitely Many Solutions Though two equations may look different because variables are on different sides of the equals sign, or because all terms are multiplied by a common factor, any two equations that have the same graph are equivalent. The easiest way to tell whether two equations have the same graph, and thus infinitely many solutions, is to rewrite both equations in slope-intercept form.. 372. x y 3x 6. Chapter 7 Solving Systems of Linear Equations and Inequalities. w.
(14) GEOMETRY For Exercises 42 and 43, use the graphs of y 2x 6, 3x 2y 19, and y 2, which contain the sides of a triangle.. NAME ______________________________________________ DATE. p. 403 (shown) and p. 404 Graphing Systems of Equations. 42. Find the coordinates of the vertices of the triangle. (2, 2), (5, 2), (1, 8). Number of Solutions Two or more linear equations involving the same variables form a system of equations. A solution of the system of equations is an ordered pair of numbers that satisfies both equations. The table below summarizes information about systems of linear equations. Graph of a System. intersecting lines. same line. y. BALLOONING For Exercises 44 and 45, use the information in the graphic at the right.. O. 44. In how many minutes will the balloons be at the same height? 4 min. SAVINGS For Exercises 46 and 47, use the following information. Monica and Michael Gordon both want to buy a scooter. Monica has already saved $25 and plans to save $5 per week until she can buy the scooter. Michael has $16 and plans to save $8 per week.. exactly one solution. infinitely many solutions. no solution. Terminology. consistent and independent. consistent and dependent. inconsistent. y yx1. 50. If the profit patterns continue, will the profits of the two companies ever be equal? Explain. No; the graphs are. y x 2 x. O. 3x 3y 3. Exercises Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions. 1. y x 3 yx1. y 3x y 3 2 x 2y 4. 2. 2x 2y 6 y x 3. one. yx1. 4. 2x 2y 6 3x y 3. none. x. O 2x 2y 6. infinitely many. 3. y x 3 2x 2y 4. y x 3. one. NAME ______________________________________________ DATE. Skills Practice, 7-1 Practice (Average). ____________ PERIOD _____. p. 405 and Practice, p. 406 (shown) Graphing Systems of Equations. Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions. 1. x y 3 x y 3. 8 Profit (millions of dollars). 49. Which company had a greater rate of growth? neither. y x 1. c. 3x 3y 3 y x 1 Since the graphs of 3x 3y 3 and y x 1 coincide, there are infinitely many solutions.. Balloon 2 is 150 meters above the ground, descending 20 meters per minute.. Yearly Profits. 48. Which company had the greater profit during the ten years? Widget Company. x. Number of Solutions. b. y x 2 3x 3y 3 Since the graphs of y x 2 and 3x 3y 3 are parallel, there are no solutions.. Balloon 1 is 10 meters above the ground, rising 15 meters per minute.. BUSINESS For Exercises 48–50, use the graph at the right.. 4. xy3 x. O. infinitely many. 3. x 3y 3 x y 3 one. 4x 2y 6. x y 3. 4. x 3y 3 2x y 3 one. Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.. Gadget Company. Widget Company. 2x y 3. 2. 2x y 3 4x 2y 6. no solution. 6. y x 3y 3. 5. 3x y 2 no 3x y 0 solution. 6. y 2x 3 infinitely 4x 2y 6 many. 7. x 2y 3 one; 3x y 5 (1, 2). y. y. y. 3x y 2. 3x y 5 3x y 0. 2. x. O. x. O y 2x 3. x 2y 3 (–1, 2). 4 x 2y 6. x O. 0 6. BUSINESS For Exercises 8 and 9, use the following. 8. information. Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 in operating costs per week plus $0.50 to produce each treat. He plans to sell each treat for $1.50.. 10. Write a system of equations to represent the situation.. 0. y 1.5x. c v 40 4c 6v 180. 40. 5 10 15 20 25 30 35 40 45 Sales ($). CD and Video Sales. 35. c v 40. 30 25 20 15 10. 4c 6v 180. 0. NAME ______________________________________________ DATE. Mathematics, p. 407 Graphing Systems of Equations. Pre-Activity. (30, 10). 5 5 10 15 20 25 30 35 40 45 CD Sales ($). ____________ PERIOD _____. Reading 7-1 Readingto to Learn Learn Mathematics. ★ 53. Graph the population equations. See pp. 405A–405D. ★ 54. Assume that the rate of growth of each of these areas remains the same.. ELL. How can you use graphs to compare the sales of two products? Read the introduction to Lesson 7-1 at the top of page 369 in your textbook. • What is meant by the term linear function? Sample answer:. a function whose graph is a line. Estimate when the population of the West would be equal to the population of the Midwest. in about 11.7 years or sometime in 2001. • What does it mean to say that two lines intersect? Sample answer:. The lines cross.. 55. CRITICAL THINKING The solution of the system of equations Ax y 5 and Ax By 20 is (2, 3). What are the values of A and B? A 4, B 4. Reading the Lesson 1. Each figure shows the graph of a system of two equations. Write the letter of the figures that illustrate each statement. A.. Graphing Systems of Equations. B.. y. 373. ____________ PERIOD _____. p. 408. y. x. O O. NAME ______________________________________________ DATE. 15. 12. How many CDs and videos did the store sell in the first week? 30 CDs and 10 videos. area increased an average of about 1 million per year during the 1990s. Write an equation to represent the population of the West for the years since 1990.. 7-1 Enrichment Enrichment,. 20. 5. 11. Graph the system of equations.. ★ 52. The population of the West was about 53 million in 1990. The population of this. y 0.5x 20. 25. 10. SALES For Exercises 10–12, use the following information. 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 $180.00. population of this area increased an average of about 0.4 million per year. Write an equation to represent the population of the Midwest for the years since 1990.. (20, 30). 30. 9. How many treats does Nick need to sell per week to break even? 20. ★ 51. In 1990, the population of the Midwest was about 60 million. During the 1990s, the. Lesson 7-1. 35. 8. Graph the system of equations y 0.5x 20 and y 1.5x to represent the situation.. POPULATION For Exercises 51–54, use the following information. The U.S. Census Bureau divides the country into four sections. They are the Northeast, the Midwest, the South, and the West.. Dog Treats. 40. C.. x. D.. y. O. y. x O. x. Graphing a Trip The distance formula, d rt, is used to solve many types of problems. If you graph an equation such as d 50t, the graph is a model for a car going at 50 mi/h. 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 you drive to a nearby town and return. You average 50 mi/h on the trip out but only 25 mi/h on the trip home. The round trip takes 5 hours. How far away is the town? The graph at the right represents your trip. Notice that the return trip is shown with a negative slope because you are driving in the opposite direction. Solve each problem. 1. Estimate the answer to the problem in the above example. About how far away is the town?. a. A system of two linear equations can have an infinite number of solutions. D. d. b. A system of equations is consistent if there is at least one ordered pair that satisfies both equations. B, C, D. slope is 50 slope is –25. 50 O. 2. t. c. If two graphs are parallel, there are no ordered pairs that satisfy both equations. A d. If a system of equations has exactly one solution, it is independent. B, C e. If a system of equations has an infinite number of solutions, it is dependent. D. Helping You Remember 2. Describe how you can solve a system of equations by graphing. Sample answer:. Graph the equations on the same coordinate plane. Locate any points of intersection.. about 80 miles. Lesson 7-1 Graphing Systems of Equations 373. Lesson 7-1. 4 Year. Cost ($). 2. Video Sales ($). parallel so the lines will never meet and there is no year when the profits will be equal.. www.algebra1.com/self_check_quiz. O. a. y x 2 yx1 Since the graphs of y x 2 and y x 1 intersect, there is one solution.. 47. How much will each person have saved at that time? $40. 52. p 53 1t or p 53 t. y x. Example Use the graph at the right to determine whether the system has no solution, one solution, or infinitely many solutions.. 45. How high will the balloons be at that time? 70 m. 46. In how many weeks will Monica and Michael have saved the same amount of money? 3 weeks. 51. p 60 0.4t. parallel lines. y O. x. Lesson 7-1. 43. Find the area of the triangle. 21 units2. You can graph a system of equations to predict when men’s and women’s Olympic times will be the same. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.. ____________ PERIOD _____. Study Guide andIntervention Intervention, 7-1 Study Guide and.
(15) 56. WRITING IN MATH. 4 Assess. How can you use graphs to compare the sales of two products? Include the following in your answer: • an estimate of the year in which the sales of cassette singles equaled the sales of CD singles, and • an explanation of why graphing works.. Open-Ended Assessment Modeling Model a line on a coordinate plane with string, spaghetti, or a similar item. Then ask volunteers to come up and model another line that would represent a system of equations with one solution. Do the same for systems of equations with no solutions and with infinitely many solutions.. Standardized Test Practice. 374. A. B. y. y. y 13 x 2. x. y 13 x 1. y x. C. x. O. O. D. y. y y 3x 5. xy1. PREREQUISITE SKILL Students will learn to solve systems of equations by substitution in Lesson 7-2. The process of substitution involves solving equations for a specific variable. Use Exercises 65–68 to determine your students’ familiarity with solving equations for a specific variable.. 56. Graphs can show when the sales of one item is greater than the sales of the other item and when the sales of the items are equal. Answers should include the following. • The sales of cassette singles equaled the sales of CD singles in about 5 years or by the end of 1995. • The graph of each equation contains all of the points whose coordinates satisfy the equation. If a point is contained in both lines, then its coordinates satisfy both equations.. 57. Which graph represents a system of equations with no solution? B. yx2. Getting Ready for Lesson 7-2. Answer. Answer the question that was posed at the beginning of the lesson. See margin.. x. x. O. O. y x 1 y 2x 5. 58. How many solutions exist for the system of equations below? B 4x y 7 3x y 0 A C. no solution infinitely many solutions. B D. one solution cannot be determined. Maintain Your Skills Mixed Review. Determine which ordered pairs are part of the solution set for each inequality. (Lesson 6-6). 59. y 2x, {(1, 4), (1, 5), (5, 6), (7, 0)} {(5, 6)} 60. y 8 3x, {(4, 2), (3, 0), (1, 4), (1, 8)} {(4, 2), (3, 0), (1, 4)} 61. MANUFACTURING The inspector at a perfume manufacturer accepts a bottle if it is less than 0.05 ounce above or below 2 ounces. What are the acceptable numbers of ounces for a perfume bottle? (Lesson 6-5) {n1.95 n 2.05} Write each equation in standard form. (Lesson 5-5) 62. y 1 4(x 5). 4x y 19. Getting Ready for the Next Lesson. 374. x 3y 3. 64. y 4 6(x 2). 6x y 8. PREREQUISITE SKILL Solve each equation for the variable specified. (To review solving equations for a specified variable, see Lesson 3-8.). 65. 12x y 10x, for y y 2x 7m n 7m n 67. 10, for q q q 10. Chapter 7 Solving Systems of Linear Equations and Inequalities. Chapter 7 Solving Systems of Linear Equations and Inequalities. 1 3. 63. y 2 (x 3). 1 66. 6a b 2a, for a a 4b 5tz s 12 s 68. 6, for z z 2 5t.
(16) A Follow-Up of Lesson 7-1. A Follow-Up of Lesson 7-1. Systems of Equations. Getting Started. You can use a TI-83 Plus graphing calculator to solve a system of equations.. Entering Equations The equations must be solved for y before they are entered in the calculator.. Example Solve the system of equations. State the decimal solution to the nearest hundredth. 2.93x y 6.08 8.32x y 4.11. Teach. Solve each equation for y to enter them into the calculator. 2.93x y 6.08. First equation. 2.93x y 2.93x 6.08 2.93x. • Step 2 Remind students to clear all previous equations from the Y= list first. Have students graph each system using the standard viewing window. If the intersection is not visible, have students adjust the window to an area suggested by the directions of the lines. • Step 3 The GUESS feature that appears after the second ENTER gives students an opportunity to use the arrow keys to estimate the solution and then check their estimate by pressing ENTER the third time.. Subtract 2.93x from each side.. y 6.08 2.93x. Simplify.. 8.32x y 4.11. Second equation. 8.32x y 8.32x 4.11 8.32x. Subtract 8.32x from each side.. y 4.11 8.32x. Simplify.. (1) y (1)(4.11 8.32x) Multiply each side by 1.. y 4.11 8.32x. Simplify.. Enter these equations in the Y= list and graph. KEYSTROKES:. Review on pages 224–225.. Use the CALC menu to find the point of intersection. KEYSTROKES: 2nd. [CALC] 5 ENTER ENTER ENTER. The solution is approximately (0.91, 3.43).. [10, 10] scl: 1 by [10, 10] scl: 1. Exercises Use a graphing calculator to solve each system of equations. Write decimal solutions to the nearest hundredth. 1. y 3x 4 2. y 2x 5 y 0.2x 4 (4.09, 3.18) y 0.5x 6 (2.86, 4.57) 3. x y 5.35. 4. 0.35x y 1.12. 5. 1.5x y 6.7. 6. 5.4x y 1.8. 7. 5x 4y 26. 8. 2x 3y 11. 3x y 3.75 (2.28, 3.08) 5.2x y 4.1 (1.61, 4.28) 4x 2y 53.3 (10.2, 6.25). 9. 0.22x 0.15y 0.30. 0.33x y 6.22 (2.35, 5.44). Graphing Calculator Investigation. Assess Ask students how they can verify that their solution is correct. They should respond that substitution of values into the original equations will confirm solutions.. 2.25x y 4.05 (1.13, 1.51) 6.2x y 3.8 (0.17, 2.73) 4x y 6 (2.9, 5.6). 10. 125x 200y 800. 65x 20y 140 (1.14, 3.29). www.algebra1.com/other_calculator_keystrokes Investigating Slope-Intercept Form 375 Graphing Calculator Investigation Systems of Equations 375. Graphing Calculator Investigation Systems of Equations 375.
(17) Lesson Notes. 5-Minute Check Transparency 7-2 Use as a quiz or review of Lesson 7-1.. • Solve systems of equations by using substitution. • Solve real-world problems involving systems of equations.. Vocabulary • substitution. Americans spend more time online than they spend reading daily newspapers. If x represents the number of years since 1993 and y represents the average number of hours per person per year, the following system represents the situation.. Mathematical Background notes are available for this lesson on p. 366C. can a system of equations be used to predict media use? Ask students: • Which is changing at a greater rate: the number of newspaper readers, or the number of people online? the number of people online • According to the graph, in about what year will the number of hours online per person equal time spent reading newspapers? in about 2003 • If the two equations weren’t labeled, how would you know which was which? The one with negative slope represents newspaper readers because the time spent reading newspapers is declining.. can a system of equations be used to predict media use?. y 2.8x 170 y 14.4x 2. reading daily newspapers: online:. The solution of the system represents the year that the number of hours spent on each activity will be the same. To solve this system, you could graph the equations and find the point of intersection. However, the exact coordinates of the point would be very difficult to determine from the graph. You could find a more accurate solution by using algebraic methods.. Media Usage Hours per Person per Year. 1 Focus. Substitution. 250 200 150 100 50 0. Daily Newspapers. Online 1 2 3 4 5 6 7 8 9 10 11 12 1314 Years Since 1993. SUBSTITUTION The exact solution of a system of equations can be found by using algebraic methods. One such method is called substitution.. Using Substitution Use algebra tiles and an equation mat to solve the system of equations. 3x y 8 and y x 4 Model and Analyze. 4. On an equation mat, use algebra tiles to model 4x 3y 10 using 1 positive x tile and 1 positive 1 tile to represent each y. Use what you know about equation mats to solve for x. Use the value of x and y x 2 to solve for y. The solution is (1, 2). 376. Since y x 4, use 1 positive 3x x x x tile and 4 negative 1 tiles to represent y. Use algebra tiles to represent 3x y 8. 1 1. Use what you know about x y 1 equation mats to solve for x. What is the value of x? 3 2. Use the y x 4 to solve for y. 1 3. What is the solution of the system of equations?. x. 1. . 1. 1. 1. 1. 1. 1. 1. 1. (3, 1). Make a Conjecture. 4. Explain how to solve the following system of equations using algebra tiles. 4x 3y 10 and y x 1 5. You substitute a representation of y for y. 5. Why do you think this method is called substitution?. Chapter 7 Solving Systems of Linear Equations and Inequalities. Resource Manager Workbook and Reproducible Masters Chapter 7 Resource Masters • Study Guide and Intervention, pp. 409–410 • Skills Practice, p. 411 • Practice, p. 412 • Reading to Learn Mathematics, p. 413 • Enrichment, p. 414 • Assessment, p. 447. 1. Parent and Student Study Guide Workbook, p. 54 Prerequisite Skills Workbook, pp. 27–28 Science and Mathematics Lab Manual, pp. 63–66 Teaching Algebra With Manipulatives Masters, pp. 10–11, 16, 125. Transparencies 5-Minute Check Transparency 7-2 Answer Key Transparencies. Technology AlgePASS: Tutorial Plus, Lesson 16 Interactive Chalkboard.
(18) Example 1 Solve Using Substitution. 2 Teach. Use substitution to solve the system of equations. y 3x x 2y 21. Study Tip Look Back. To review solving linear equations, see Lesson 3-5.. SUBSTITUTION. Since y 3x, substitute 3x for y in the second equation. x 2y 21 x 2(3x) 21 x 6x 21 7x 21. Combine like terms. Divide each side by 7.. Explain that the purpose of the substitution method is that once you find one of the values (either x or y), you can then substitute it into either of the original equations to find the other value. In Example 1, the value of y is given in terms of x because y 3x. So, substitute 3x for y in the second equation and solve.. Simplify.. Use y 3x to find the value of y. y 3x. First equation. y 3(3) x 3 y 9. Power Point®. Teaching Tip. y 3x Simplify.. 7x 21 7 7. x 3. In-Class Examples. Second equation. The solution is (3, 9).. 1 Use substitution to solve the system of equations. x 4y 4x y 75 (20, 5). Example 2 Solve for One Variable, Then Substitute Use substitution to solve the system of equations. x 5y 3 3x 2y 8. Teaching Tip. You must solve for one variable first because neither equation gives one variable in terms of the other as in Example 1. The easiest choice in Example 2 is to solve the first equation for x by subtracting 5y from both sides.. Solve the first equation for x since the coefficient of x is 1. x 5y 3. First equation. x 5y 5y 3 5y x 3 5y. Subtract 5y from each side. Simplify.. Find the value of y by substituting 3 5y for x in the second equation. 3x 2y 8. Second equation. 3(3 5y) 2y 8. x 3 5y. 9 15y 2y 8. Distributive Property. 9 17y 8. Combine like terms.. 9 17y 9 8 9. Add 9 to each side.. 17y 17. Simplify.. 17y 17 17 17. Divide each side by 17.. y 1. 2 Use substitution to solve the system of equations. 4x y 12 2x 3y 14 (5, 8). Simplify.. Substitute 1 for y in either equation to find the value of x. Choose the equation that is easier to solve.. x 5y 3 x 5(1) 3 x 5 3 x2. First equation y 1 Simplify.. y 3x 2y 8 O. x x 5y 3. (2, 1). Add 5 to each side.. The solution is (2, 1). The graph verifies the solution.. www.algebra1.com/extra_examples. Lesson 7-2 Substitution 377. Algebra Activity Materials: algebra tiles, equation mat • Remind students that anything they add to one side of the equation mat must also be added to the other side of the mat. • Remind students that the purpose is to eliminate zero pairs. • Once a value is found for x, have students use that value to find y. Lesson 7-2 Substitution 377.
(19) In-Class Example. Example 3 Dependent System. Power Point®. Use substitution to solve the system of equations.. Teaching Tip. If there are infinitely many solutions, then the two equations represent the same line. If you solve both equations for y, then you will see this is true. If there are no solutions, then the equations represent two parallel lines. If you solve both equations for y, the equations will have the same slope but a different y-intercept.. 6x 2y 4 y 3x 2 Since y 3x 2, substitute 3x 2 for y in the first equation. 6x 2y 4 First equation 6x 2(3x 2) 4 y 3x 2 6x 6x 4 4 Distributive Property 4 4 Simplify. The statement 4 4 is true. This means that there are infinitely many solutions of the system of equations. This is true because the slope-intercept form of both equations is y 3x 2. That is, the equations are equivalent, and they have the same graph.. 3 Use substitution to solve the system of equations. 2x 2y 8 x y 2 no solution. Study Tip Alternative Method. REAL-WORLD PROBLEMS. In-Class Example. In general, if you solve a system of linear equations and the result is a true statement (an identity such as 4 4), the system has an infinite number of solutions. However, if the result is a false statement (for example, 4 5), the system has no solution.. Power Point®. Using a system of equations is an alternative method for solving the weighted average problems that you studied in Lesson 3-9.. REAL-WORLD PROBLEMS Sometimes it is helpful to organize data before solving a problem. Some ways to organize data are to use tables, charts, different types of graphs, or diagrams.. Example 4 Write and Solve a System of Equations METAL ALLOYS A metal alloy is 25% copper. Another metal alloy is 50% copper. How much of each alloy should be used to make 1000 grams of a metal alloy that is 45% copper? Let a the number of grams of the 25% copper alloy and b the number of grams of the 50% copper alloy. Use a table to organize the information.. 4 GOLD Gold is alloyed with different metals to make it hard enough to be used in jewelry. The amount of gold present in a gold alloy is measured in 24ths called karats.. 25% Copper. 50% Copper. a. b. 1000. 0.25a. 0.50b. 0.45(1000). Total Grams Grams of Copper. 45% Copper. The system of equations is a b 1000 and 0.25a 0.50b 0.45(1000). Use substitution to solve this system.. 24 24. 24-karat gold is or 100%. a b 1000. gold. Similarly, 18-karat gold 18 is or 75% gold. How many 24. First equation. a b b 1000 b Subtract b from each side. a 1000 b Simplify.. ounces of 18-karat gold should be added to an amount of 12-karat gold to make 4 ounces of 14-karat gold?. 0.25a 0.50b 0.45(1000) 0.25(1000 b) 0.50b 0.45(1000). 1 3 2 2 ounces of 12-karat gold 3. 1 ounces of 18-karat gold and. a 1000 b. 250 0.25b 0.50b 450. Distributive Property. 250 0.25b 450. Combine like terms.. 250 0.25b 250 450 250. Subtract 250 from each side.. 0.25b 200. Simplify.. 200 0.25b 0.25 0.25. Divide each side by 0.25.. b 800 378. Second equation. Simplify.. Chapter 7 Solving Systems of Linear Equations and Inequalities. Differentiated Instruction Intrapersonal If students have difficulty solving systems of equations by using substitution, ask them to indicate where they get confused in the process before doing another example. Encourage students to think through each step.. 378. Chapter 7 Solving Systems of Linear Equations and Inequalities.
(20) a b 1000. First equation. a 800 1000. 3 Practice/Apply. b 800. a 800 800 1000 800 Subtract 800 from each side. a 200. Simplify.. 200 grams of the 25% copper alloy and 800 grams of the 50% copper alloy should be used.. Concept Check. 1. Explain why you might choose to use substitution rather than graphing to solve a system of equations. Substitution may result in a more accurate solution. 2. Describe the graphs of two equations if the solution of the system of equations yields the equation 4 2. They are parallel lines. 3. OPEN-ENDED Write a system of equations that has infinitely many solutions.. Sample answer: y x 3, 2y 2x 6. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 4–9 10. 1 –3 4. Application. Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 4. x 2y 4x 2y 15 (3, 1.5). 5. y 3x 8 y 4 x (3, 1). 6. 2x 7y 3 x 1 4y (5, 1). 7. 6x 2y 4 8. x 3y 12 y 3x 2 infinitely many x y 8 (9, 1). 9. y x no solution 3x 5y 15. 10. TRANSPORTATION The Thrust SSC is the world’s fastest land vehicle. Suppose the driver of a car whose top speed is 200 miles per hour requests a race against the SSC. The car gets a head start of one-half hour. If there is unlimited space to race, at what distance will the SSC pass the car?. About the Exercises… Organization by Objective • Substitution: 11–28 • Real–World Problems: 29–37. Thrust SSC top speed is 763 mph.. ★ indicates increased difficulty. Practice and Apply For Exercises. See Examples. 11–28 29–37. 1–3 4. Extra Practice See page 835.. Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 11. y 5x 2x 3y 34 (2, 10). 12. x 4y 2x 3y 44 (16, 4). 13. x 4y 5 x 3y 2 (23, 7). 14. y 2x 3 y 4x 1 (2, 7). 15. 4c 3d 3 c d 1 (6, 7). 16. 4x 5y 11 y 3x 13 (4, 1). 17. 8x 2y 13 18. 2x y 4 19. 3x 5y 11 3x y 9 (13, 30) x 3y 1 (7, 2) 4x y 11 no solution 20. 2x 3y 1 21. c 5d 2 2c d 4 (2, 0) 3x y 15 (4, 3) 23. 3x 2y 12 1 3 x 2y 6 4, 2 4 26. 0.5x 2y 17 2x y 104 (50, 4). . www.algebra1.com/self_check_quiz. . 22. 5r s 5 4r 5s 17 (2, 5). 24. x 3y 0 1 7 3x y 7 2, 10 10 1 27. y x 3 2 2 1 y 2x 1 2, 4 3 3. . Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 7. • include examples of how to solve a system of equations by using substitution. • include any other item(s) that they find helpful in mastering the skills in this lesson.. 3 5. about 135.5 mi. Homework Help. Study Notebook. . . 25. 0.3x y 0.5 0.5x 0.3y 1.9 (5, 2) 1 2. 28. x y 3 2x y 6 infinitely. many. Odd/Even Assignments Exercises 11–32 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercise 38 requires the Internet or other research materials.. Assignment Guide Basic: 11–33 odd, 34, 35, 39–53 Average: 11–33 odd, 34, 35, 37, 39–53 Advanced: 12–32 even, 36–49 (optional: 50–53) All: Practice Quiz 1 (1–5). Lesson 7-2 Substitution 379. Lesson 7-2 Substitution 379.
(21) NAME ______________________________________________ DATE. 29. GEOMETRY The base of the triangle is 4 inches longer than the length of one of the other sides. Use a system of equations to find the length of each side of the triangle. 14 in., 14 in., 18 in.. ____________ PERIOD _____. Study Guide andIntervention Intervention, 7-2 Study Guide and p. 409 (shown) and p. 410 Substitution. One method of solving systems of equations is substitution.. Example 1. Example 2. Solve for one variable, then. Use substitution to solve the system of equations. y 2x 4x y 4. substitute. x 3y 7 2x 4y 6. Substitute 2x for y in the second equation. 4x y 4 Second equation 4x 2x 4 y 2x 2x 4 Combine like terms.. Solve the first equation for x since the coefficient of x is 1. x 3y 7 First equation x 3y 3y 7 3y Subtract 3y from each side. x 7 3y Simplify.. 2x 4 2 2. x 2. Find the value of y by substituting 7 3y for x in the second equation. 2x 4y 6 Second equation 2(7 3y) 4y 6 x 7 3y 14 6y 4y 6 Distributive Property 14 10y 6 Combine like terms. 14 10y 14 6 14 Subtract 14 from each side. 10y 20 Simplify.. Divide each side by 2. Simplify.. Use y 2x to find the value of y. y 2x First equation y 2(2) x 2 y 4 Simplify. The solution is (2, 4).. x in.. x in.. Perimeter 46 in.. 10y 20 10 10. Divide each side by 10.. y2. Simplify.. y in.. 30. FUND-RAISING The Future Teachers of America Club at Paint Branch High School is making a healthy trail mix to sell to students during lunch. The mix will have three times the number of pounds of raisins as sunflower seeds. Sunflower seeds cost $4.00 per pound, and raisins cost $1.50 per pound. If the group has $34.00 to spend on the raisins and sunflower seeds, how many pounds of each should they buy? 4 lb of sunflower seeds, 12 lb of raisins. Lesson 7-2. Substitution. Use y 2 to find the value of x. x 7 3y x 7 3(2) x1 The solution is (1, 2).. 31. CHEMISTRY MX Labs needs to make 500 gallons of a 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. How many gallons of each solution should be mixed to make the 34% solution?. Exercises Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 1. y 4x 3x y 1 (1, 4). 2. x 2y y x 2 (4, 2). 3. x 2y 3 x 2y 4 no solution. 4. x 2y 1 3y x 4 (5, 3). 5. c 4d 1 infinitely 2c 8d 2 many. 6. x 2y 0 3x 4y 4 (4, 2). 7. 2b 6a 14 infinitely 3a b 7 many. 8. x y 16 9. y x 3 no 2y 2x 2 no solution 2y 2x 4 solution. 10. x 2y (20, 10) 0.25x 0.5y 10. 11. x 2y 5 x 2y 1 (3, 1). 320 gal of 25% acid, 180 gal of 50% acid 32. GEOMETRY Supplementary angles are two angles whose measures have the sum of 180 degrees. Angles X and Y are supplementary, and the measure of angle X is 24 degrees greater than the measure of angle Y. Find the measures of angles X and Y. m ⬔X 102, m ⬔Y 78. 12. 0.2x y 0.5 0.4x y 1.1 (1, 0.7). NAME ______________________________________________ DATE. Skills Practice, 7-2 Practice (Average). ____________ PERIOD _____. p. 411 and Practice, p. 412 (shown) Substitution. 33. SPORTS At the end of the 2000 baseball season, the New York Yankees and the Cincinnati Reds had won a total of 31 World Series. The Yankees had won 5.2 times as many World Series as the Reds. How many World Series did each team win? Yankees: 26, Reds: 5. Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 1. y 6x 2x 3y 20 (1, 6). 2. x 3y 3x 5y 12 (9, 3). 3. x 2y 7 x y 4 (1, 3). 4. y 2x 2 y x 2 (4, 6). 5. y 2x 6 2x y 2 no solution. 6. 3x y 12 y x 2 (7, 9). 7. x 2y 13 (3, 8) 2x 3y 18. 8. x 2y 3 infinitely 4x 8y 12 many. 9. x 5y 36 (4, 8) 2x y 16. 10. 2x 3y 24 x 6y 18 (6, 4). 11. x 14y 84 2x 7y 7 (14, 5). 12. 0.3x 0.2y 0.5 x 2y 5 (5, 5). 13. 0.5x 4y 1 x 2.5y 3.5 (6, 1). 14. 3x 2y 11. 15. x 2y 12. 1 3. 16. x y 3. 13 , 132 . (12, 1). 1 2. 1. x y 4 (5, 2) 2 17. 4x 5y 7 y 5x. 2x y 25. JOBS For Exercises 34 and 35, use the following information. Shantel Jones has two job offers as a car salesperson. At one dealership, she will receive $600 per month plus a commission of 2% of the total price of the automobiles she sells. At the other dealership, she will receive $1000 per month plus a commission of 1.5% of her total sales.. x 2y 6 (12, 3). 34. What is the total price of the automobiles that Ms. Jones must sell each month to make the same income from either dealership? $80,000. 18. x 3y 4 2x 6y 5. 1 43 , 1 12 . EMPLOYMENT For Exercises 19–21, use the following information. 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. 19. Write a system of equations to represent the situation.. y 0.04x 500 and y 0.05x 400. 20. What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pay scale? $10,000 21. Which is the better offer? the first offer if she expects to sell less than. $10,000 in shoes, and the second offer if she expects to sell more than $10,000 in shoes MOVIE TICKETS For Exercises 22 and 23, use the following information. Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased 8 tickets for $52.75. 22. Write a system of equations to represent the situation.. 35. Explain which job offer is better. The second offer is better if she sells less. than $80,000. The first offer is better if she sells more than $80,000.. Tourism Every year, multitudes of visitors make their way to South America to stand in awe of Machu Picchu, the spectacular ruins of the Lost City of the Incas.. ★ 36. LANDSCAPING A blue spruce grows an average of 6 inches per year. A hemlock grows an average of 4 inches per year. If a blue spruce is 4 feet tall and a hemlock is 6 feet tall, when would you expect the trees to be the same height? in 12 years 37. TOURISM In 2000, approximately 40.3 million tourists visited South America ★ and the Caribbean. The number of tourists to that area had been increasing at an average rate of 0.8 million tourists per year. In the same year, 17.0 million tourists visited the Middle East. The number of tourists to the Middle East had been increasing at an average rate of 1.8 million tourists per year. If the trend continues, when would you expect the number of tourists to South America and the Caribbean to equal the number of tourists to the Middle East?. Source: www.about.com. x y 8 and 7.25x 5.5y 52.75. 23. How many adult tickets and student tickets were purchased? 5 adult and 3 student NAME ______________________________________________ DATE. ____________ PERIOD _____. Reading 7-2 Readingto to Learn Learn Mathematics Mathematics, p. 413 Substitution. Pre-Activity. ELL. How can a system of equations be used to predict media use? Read the introduction to Lesson 7-2 at the top of page 376 in your textbook.. during the year 2023. • What is the system of equations? y 2.8x 170, y 14.4x 2 • Based on the graph, are there 0, 1, or infinitely many solutions of the system? 1 solution. 38. RESEARCH Use the Internet or other resources to find the pricing plans for various cell phones. Determine the number of minutes you would need to use the phone for two plans to cost the same amount of money. Support your answer with a table, a graph, and/or an equation. See students’ work.. Reading the Lesson 1. Describe how you would use substitution to solve each system of equations. a. y 2x x 3y 15. Substitute 2x for y in the second equation. Simplify and solve for x. Then use the value of x to find the value of y.. b. 3x 2y 12 x 2y. Substitute 2y for x in the first equation. Simplify and solve for y. Then use the value of y to find the value of x.. c. x 2y 7 2x 8y 8. Solve the first equation for x. Substitute the expression you find for x in the second equation. Simplify and solve for y. Then use the value of y to find the value of x.. d. 3x 5y 81 2x y 24. Solve the second equation for y. Substitute the expression you find for y in the first equation. Simplify and solve for x. Then use the value of x to find the value of y.. 2. Jess solved a system of equations and her result was 8 8. All of her work was correct. Describe the graph of the system. Explain. It is a line. There are infinitely. many solutions, since 8 8 is always true.. 3. Miguel solved a system of equations and his result was 5 2. All of his work was correct. Describe the graph of the system. Explain. The graph has two parallel. 380. Chapter 7 Solving Systems of Linear Equations and Inequalities. NAME ______________________________________________ DATE. 7-2 Enrichment Enrichment,. Equations of Lines and Planes in Intercept Form One form that a linear equation may take is intercept form. The constants a and b are the x- and y-intercepts of the graph.. z. x y 1 a b. In three-dimensional space, the equation of a plane takes a similar form.. lines, since 5 2 is always false.. Helping You Remember. ____________ PERIOD _____. p. 414. O. x y z 1 a b c. y x – 8. Here, the constants a, b, and c are the points where the plane meets the x, y, and z-axes.. y. –7 –6z 1. x. 4. What is usually the first step in solving a system of equations by substitution? Sample. answer: Solve one of the equations for one variable in terms of the other.. Solve each problem. x y z 1. Graph the equation 1. 3 2 1 z. 2. For the plane in Exercise 1, write an equation for the line where the plane intersects the xy-plane. Use intercept forms.. x. 380. Chapter 7 Solving Systems of Linear Equations and Inequalities. y.
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