CHAPTER 6 Systems of Equations In Chapter 4, you studied the connections between the multiple representations of data and learned how to write equations from situations. You also developed a way to solve a system of equations. In this chapter, you will learn how to solve word problems by writing an equation (or a system of equations). Also, unlike previous chapters, where you were limited to certain kinds of systems of equations, in this chapter you will learn how to solve any system of linear equations, regardless of its form. Along the way, you will develop new ways to solve different forms of systems and will learn how to recognize when one method may be most efficient. By the end of this chapter, you will know multiple ways to find the point of intersection of two lines and will be able to solve systems that arise from different contexts. In this chapter, you will learn: Think about these questions throughout this chapter: ? What is a solution? How can I represent it algebraically? How can I solve it? Is there another way? How can I check my answer? What a solution of a system of equations represents. How to solve contextual word problems by writing and solving equations. How to recognize systems of equations that have no solution or infinite solutions. How to solve different forms of systems quickly and efficiently. Section 6.1 In this section, you will write and solve mathematical sentences (such as one- and two-variable equations) to solve contextual word problems. Section 6.2 You will develop methods to solve systems of equations in different forms. You will learn which equations will result in lines when graphed. You will also find ways to know which solving method is most efficient and accurate. Section 6.3 Section 6.3 provides an opportunity for you to review and extend what you learned in Chapters 1 through 6. You will make important connections between solving equations, multiple representations, proportional reasoning, and systems of equations. b + g = 23 y 5 x 5 230 Algebra Connections 6.1.1 How can I write it using algebra? b + g = 23 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Mathematical Sentences Spoken and written languages use sentences to convey information. A sentence has a subject and verb, follows the rules of grammar, and is structured with punctuation. Likewise, algebra uses mathematical sentences, such as b + g = 23 , which also convey information and follow structural rules. During this lesson, you will explore various mathematical sentences and learn how to interpret their meanings. Then you will write mathematical sentences of your own. 6-1. 6-2. How can variables give you new information? Suppose Mr. Titelbaum’s class has b boys and g girls. a. Mr. Titelbaum noticed that b + g = 23 . What does that tell you about his class? b. If b = g ! 3 , what statement can you make about the number of boys and girls? c. How many girls are in Mr. Titelbaum’s class? Explain how you know. The local commuter train has three passenger cars. When it is sold out, each passenger car can hold p people. a. In addition to the passengers, the train has 8 employees. Write an expression that represents the total number of people on this commuter train. b. When it is sold out, the train has a total of 176 people on board. Write an equation that represents this fact. c. Solve your equation to determine how many people a passenger car can hold. Be sure to check your solution when you are finished. Chapter 6: Systems of Equations 231 6-3. MATHEMATICAL SENTENCES A mathematical sentence uses variables and mathematical operations to represent information. For example, if you know that b represents the number of boys in a class and g represents the number of girls in the same class, then the mathematical sentence “ b + g = 23 ” states that if you add the number of boys to the number of girls, you get a result of 23. In other words, there are 23 students in the class. Mathematical Sentence: Same Sentence in English: + b The number of boys = 23 g plus the number of girls is 23 students. While many mathematical sentences contain more than one variable (such as b + g = 23 above), some only contain one variable. For example, if p represents the maximum number of people in a train’s passenger car, then the mathematical sentence 3p + 8 = 176 states that a train with 3 passenger cars and 8 additional people will have 176 people in all. This is shown below. Mathematical Sentence: 3p!!!!+!!!!!!!8!!!!!!!!!!!!!!!!=!!!!!!!!176 Same Sentence in English: Three passenger cars plus 8 employees is 176 people. Mathematical sentences convey information once you understand what each variable represents. Sometimes the structure of the equation or the letter of the variable can reveal its possible meaning. With your team, study the two mathematical sentences below and decide what each could be trying to communicate. Be prepared to share your description with the class. a. 232 0.25q + 0.05n = 5.00 b. l + w + l + w = 30 Algebra Connections 6-4. Mathematical sentences are easier to understand when everyone knows what the variables represent. For example, if you knew that l in part (b) of problem 6-3 represented the length of one side of a rectangle, then it would have been easier to understand that the mathematical sentence l + w + l + w = 30 could have been stating that the perimeter of a rectangle is 30 units. A statement that describes what the variable represents is called a “let” statement. It is called this because it often is stated in the form “Let l = …”. While solving the problems below, examine how “let” statements are used. 6-5. a. Let m = the number of students at Mountain View High School and let m ! 100 = the number of students at neighboring Ferguson High School. Which school has more students? How can you tell? b. Based on the “let” statements in part (a) above, translate this mathematical sentence into English: m + (m ! 100) = 5980 . c. A book called How I Love Algebra has only three chapters. Let p = the number of pages in Chapter 1, p + 12 = the number of pages in p Chapter 2, and 2 = the number of pages in Chapter 3. Which is the longest chapter? Which is the shortest? d. Using the definitions in part (c) above, write and solve a mathematical sentence that states that How I Love Algebra has 182 pages. How many pages are in Chapter 1? With your team, practice translating words into mathematical symbols. For each problem below, write an expression or equation that best represents the given situation. a. Turner rode his bike m miles. If Carolyn rode 10 less than twice the number of miles that Turner rode, how many miles did Carolyn ride? b. Your teacher spent $9.50 on 5 boxes of chalk and 2 boxes of overhead pens. If c represents the price of a box of chalk and p represents the price of a box of overhead pens, write an equation to represent this purchase. c. Each fruit basket comes with a apples, p pears, and b bananas. Wendi orders 4 fruit baskets and gets 84 pieces of fruit. Write an equation that represents this order. Chapter 6: Systems of Equations 233 6-6. In your Learning Log, write your own mathematical sentence. Be sure to state what any variables represent. Title this entry “Writing Mathematical Sentences” and include today’s date. MATH NOTES ETHODS AND MEANINGS Mathematical Sentences A mathematical sentence uses variables and mathematical operations to represent information. An equation is one type of mathematical sentence. When the variables are defined, a mathematical sentence can be translated into a sentence with words. For example, if b represents the number of boys in a class and if g represents the number of girls, then the mathematical sentence b + g = 23 states that the total number of boys and girls is 23. Mathematical Sentence: Same Sentence in English: 6-7. b The number of boys + g = 23 plus the number of girls is 23. Solve the problem below using a Guess and Check table. Note: Be sure to put your work in a safe place, because you will need it for the next lesson. The perimeter of a triangle is 31 cm. Sides #1 and #2 have equal length, while Side #3 is one centimeter shorter than twice the length of Side #1. How long is each side? 234 Algebra Connections 6-8. 6-9. Write expressions to represent the quantities described below. a. If Thompson Valley High School has x students and if Erwin Middle School has 342 fewer students, how many students does Erwin Middle School have? b. If w represents the width of a rectangle and if its length is twice its width, how long is the rectangle? c. When Mr. Van Exel bought his laptop, he paid $400 more than three times the amount he paid for his camera. If he paid c dollars for his camera, then how much did he pay for his laptop? Solve the system of equations below using the Equal Values Method. a = 12b + 3 a = !2b ! 4 6-10. Ms. Cai’s class is studying a tile pattern. The rule for the tile pattern is y = 10x ! 18 . Kalil thinks that Figure 12 of this pattern will have 108 tiles. Is he correct? Justify your answer. 6-11. Angel is picking blackberries in her backyard for a delicious pie. She can pick 9 blackberries in 2 minutes. If she needs 95 blackberries for the pie, how long will it take her to pick the berries? 6-12. Juan thinks that the graph of 6y + 12x = 4 is a line. a. Solve Juan’s equation for y. b. Is this equation linear? That is, is its graph a line? Explain how you know. c. What are the growth factor and y-intercept of this graph? Chapter 6: Systems of Equations 235 6.1.2 How can I use variables to solve problems? b + g = 23 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Solving Word Problems by Writing Equations In Lesson 6.1.1, you examined mathematical sentences (equations that convey related information). Today you will learn more ways to translate written information into algebraic symbols and will then solve the equations that represent the relationships. 6-13. Match each mathematical sentence on the left with its translation on the right. a. 2z + 12 = 30 b. 12z + 5(z + 2) = 30 c. z + (z ! 2) + 5(z ! 2) = 30 d. 6-14. 1. A zoo has two fewer elephants than zebras and five times more monkeys than elephants. The total number of elephants, monkeys, and zebras is 30. 2. Zola earned $30 by working two hours and receiving a $12 bonus. 3. Thirty ounces of metal is created by mixing zinc with silver. The number of ounces of silver needed is twelve times the number of ounces of zinc. 4. Eddie, who earns $5 per hour, worked two hours longer than Zach, who earns $12 per hour. Together they earned $30. z + 12z = 30 In Lesson 6.1.1, you examined how to translate words into mathematical symbols to form expressions and equations. However, you can also use Guess and Check tables to help you write mathematical sentences. Find your solution for problem 6-7, reprinted below. The perimeter of a triangle is 31 cm. Sides #1 and #2 have equal length, while Side #3 is one centimeter shorter than twice the length of Side #1. How long is each side? 236 a. Add a row to your Guess and Check table. If x represents the length of Side #1, then what is the length of Side #2? Side #3? Fill in the columns for Sides #1, #2, and #3 with these variable expressions. b. Write a mathematical sentence that states that the perimeter is 31 cm. c. If you have not done so already, solve the equation you found in part (b) and determine the length of each side. Does this answer match the one you got for problem 6-7? Algebra Connections 6-15. For the following word problems, write one or two equations and then solve the problem. You may choose to use a Guess and Check table to help you set up equations, although it is not required. Regardless of your method, be sure to define your variable(s) with appropriate “let” statements. a. Herman and Jacquita are each saving money to pay for college. Herman currently has $15,000 and is working hard to save $1000 per month. Jacquita only has $12,000 but is saving $1300 per month. In how many months will they have the same amount of savings? b. There are 21 animals on Farmer Cole’s farm – all sheep and chickens. If the animals have a total of 56 legs, how many of each type of animal lives on his farm? c. When ordering supplies, Mr. Williams accidentally ordered 12 more than twice his usual number of pencils. When the order arrived, he received 60 pencils! How many pencils does Mr. Williams usually order? d. George bought some CDs at his local store. He paid $15.95 for each CD. Nora bought the same number of CDs from a store online. She paid $13.95 for each CD, but had to pay $8 for shipping. In the end, both George and Nora spent the exact same amount of money buying their CDs! How many CDs did George buy? e. After the math contest, Basil noticed that there were four extra-large pizzas that were left untouched. In addition, another three slices of pizza were uneaten. If there were a total of 51 slices of pizza left, how many slices does an extralarge pizza have? Chapter 6: Systems of Equations 237 6-16. Solve for x. Check your solutions, if possible. a. !2(4 ! 3x) ! 6x = 10 b. x!5 !2 = x!1 !3 y = !x + 2 y = 3x + 6 6-17. On the same set of axes, graph the two rules shown at right. Then find the point(s) of intersection, if one (or more) exists. 6-18. Evaluate the expression 6x 2 ! 3x + 1 for x = !2 . 6-19. The basketball coach at Washington High School normally starts each game with the following five players: Melinda, Samantha, Carly, Allison, and Kendra However, due to illness, she needs to substitute Barbara for Allison and Lakeisha for Melinda at this week’s game. What will be the starting roster for this upcoming game? 6-20. When Ms. Shreve solved an equation in class, she checked her solution and it did not make the equation true! Examine her work below and find her mistake. Then find the correct solution. 5(2x ! 1) ! 3x = 5x + 9 10x ! 5 ! 3x = 5x + 9 7x ! 5 = 5x + 9 12x = 4 x = 6-21. 1 3 Determine if the statement below is true or false. Justify your conclusion. 2(3 + 5x) = 6 + 5x 238 Algebra Connections 6.1.3 How can I solve the system? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •b• + • g = 23 Solving Problems by Writing Equations In Lessons 6.1.1 and 6.1.2, you created mathematical sentences that represented word problems. But how can you tell if you can use one variable or two? And is one method more convenient than another? Today you will compare the different ways to represent a word problem with mathematical symbols. You will also explore how to use the Equal Values Method to solve systems containing equations that are not in y = mx + b form. 6-22. ONE EQUATION OR TWO? Review what you learned in Lesson 6.1.2 by answering the questions below. a. Solve the problem below using Guess and Check. Elsie took all of her cans and bottles from home to the recycling plant. The number of cans was one more than four times the number of bottles. She earned 10¢ for each can and 12¢ for each bottle, and ended up earning $2.18 in all. How many cans and bottles did she recycle? b. Use your Guess and Check table to help you write an equation that represents the information in part (a). Be sure to define your variable. c. If you have not done so already, solve your equation from part (b). Does this solution match your answer to part (a)? If not, look for and correct any errors. d. How can this problem be represented using two variables? With your team, write two mathematical sentences that represent this problem. Be sure to state what your variables represent. You do not need to solve the system. e. Show that your solution from part (a) makes both equations in part (d) true. Chapter 6: Systems of Equations 239 6-23. Renard thinks that writing two equations for problem 6-22 was easy, but he’s not sure if he knows how to solve the system of equations. He wants to use two equations with two variables to solve this problem: Ariel bought several bags of caramel candy and taffy. The number of bags of taffy was 5 more than the number of bags of caramels. Taffy bags weigh 8 ounces each, and caramel bags weigh 16 ounces each. The total weight of all of the bags of candy was 400 ounces. How many bags of candy did she buy? a. Renard lets t = the number of taffy bags and c = the number of caramel bags. Help him write two equations to represent the information in the problem. b. Now Renard is stuck. He says, “If both of the equations were in the form ‘t = something,’ I could use the Equal Values Method to find the solution.” Help him change the equations into a form he can solve. c. Solve Renard’s equations to find the number of caramel and taffy bags that Ariel bought. Check to make sure your solution works. 6-24. 2y + 8x = 10 When you write equations to solve word problems, you y = 5x + 23 sometimes end up with two equations like Renard’s or like the system shown at right. Notice that the second equation is solved for y, but the first is not. Change the first equation into y = mx + b form, and then solve this system of equations. Discuss with your team how you can make sure your solution is correct. 6-25. Solve each system below by first changing each equation so that it is in y = mx + b form. Check that your answer makes both equations true. a. 6-26. 240 x ! 2y = 4 y = ! 12 x + 4 b. x + 2y = 14 !x + 3y = 26 Write expressions to represent the quantities described below. a. Geraldine is 4 years younger than Tom. If Tom is t years old, how old is Geraldine? Also, if Steven is twice as old as Geraldine, how old is he? b. 150 people went to see “Ode to Algebra” performed in the school auditorium. If the number of children that attended the performance was c, how many adults attended? c. The cost of a new CD is $14.95, and the cost of a video game is $39.99. How much would c CDs and v video games cost? Algebra Connections 6-27. Nina has some nickels and 9 pennies in her pocket. Her friend, Maurice, has twice as many nickels as Nina. Together, these coins are worth 84¢. How many nickels does Nina have? Solve using any method, but show all of your work. 6-28. To count the number of endangered falcons in the local county, Fernando first tagged each of the 8 falcons he saw one day. Then, days later, he counted 11 falcons and noticed that only 3 were tagged. What is a good estimate of how many falcons exist in his county? Show how you know. 6-29. As Sachiko solved the equation (x + 2) + 3 = 9 , she showed her work in the table below. Copy the table and provide justification for each step. Statement 1. (x + 2) + 3 = 9 2. x + (2 + 3) = 9 3. x+5=9 4. x+5!5=9!5 5. x=4 Reason Given 6-30. A prime number is defined as a number with exactly two integer factors: itself and 1. Jeannie thinks that all prime numbers are odd. Is she correct? If so, state how you know. If not, provide a counterexample. 6-31. In an “If…then…” statement, the “if” portion is called the hypothesis, while the “then” portion is called the conclusion. For example, in the statement “If x = 3 , then x 2 = 9 ,” the hypothesis is “ x = 3 ” while the conclusion is “ x 2 = 9 .” Identify the hypothesis and conclusion of each of the following statements. Then decide if you think the statement is true or not. Justify your decision. a. If !x = 8 then x = !8 . b. If 3x + y = !11 , then 6x + 2y = !22 . c. If Tomas runs at a constant rate of 4 meters every five seconds, then he will run 50 meters in 1 minute. Chapter 6: Systems of Equations 241 6.2.1 How can I solve the system? y ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 5 Solving Systems of Equations Using Substitution x 5 In Chapter 4, you learned that a set of two or more equations that go together is called a system of equations. In Lesson 6.1.3, you helped Renard develop a method for solving a system of equations when one of the equations was not solved for a variable. Today you will develop a more efficient method of solving systems that are too messy to solve with the Equal Values Method. 6-32. Review what you learned in Lesson 6.1.3 as you solve the system of equations below. Check your solution. y = !x ! 7 5y + 3x = !13 6-33. AVOIDING THE MESS A new method, called the Substitution Method, can help you solve the system in problem 6-32 without getting involved in messy fractions. This method is outlined below. a. b. If y = !x ! 7 , then does !x ! 7 = y ? That is, can you switch the y and the !x ! 7 ? Why or why not? y = ! x! 7 5y + 3x = !13 Since you know that y = !x ! 7 , can you switch the y in the second equation with !x ! 7 from the top equation? Why or why not? y = ! x!7 5 y + 3x = !13 c. Once you replace the y in the second equation with !x ! 7 , you have an equation with only one variable, as shown below. This is called substitution because you are substituting for (replacing) y with an expression that it equals. Solve this new equation for x and then use that result to find y in either of the original equations. 5(!x ! 7) + 3x = !13 242 Algebra Connections 6-34. 6-35. Use the Substitution Method to solve the systems of equations below. a. y = 3x 2y ! 5x = 4 b. x!4= y !5y + 8x = 29 c. 2x + 2y = 18 x = 3! y d. c = !b ! 11 3c + 6 = 6b When Mei solved the system of equations below, she got the solution x = 4 , y = 2 . Without solving the system yourself, can you tell her whether this solution is correct? How do you know? 4x + 3y = 22 x ! 2y = 0 6-36. HAPPY BIRTHDAY! You’ve decided to give your best friend a bag of marbles for her birthday. Since you know that your friend likes green marbles better than red ones, the bag has twice as many green marbles as red. The label on the bag says it contains a total of 84 marbles. How many green marbles are in the bag? Write an equation (or system of equations) for this problem. Then solve the problem using any method you choose. Be sure to check your answer when you are finished. 6-37. Solve each equation for the variable. Check your solutions, if possible. a. 8a + a ! 3 = 6a ! 2a ! 3 c. x 2 +1 = 6 Chapter 6: Systems of Equations b. 8(3m ! 2) ! 7m = 0 d. 4t ! 2 + t 2 = 6 + t 2 243 6-38. The Fabulous Footballers scored an incredible 55 points at last night’s game. Interestingly, the number of field goals was 1 more than twice the number of touchdowns. The Fabulous Footballers earned 7 points for each touchdown and 3 points for each field goal. a. b. 6-39. 6-40. Multiple Choice: Which system of equations below best represents this situation? Explain your reasoning. Assume that t represents the number of touchdowns and f represents the number of field goals. i. t = 2 f +1 7t + 3 f = 55 ii. f = 2t + 1 7t + 3 f = 55 iii. t = 2 f +1 3t + 7 f = 55 iv. f = 2t + 1 3t + 7f = 55 Solve the system you selected in part (a) and determine how many touchdowns and field goals the Fabulous Footballers earned last night. Yesterday Mica was given some information and was asked to find a linear equation. But last night her cat destroyed most of the information! At right is all she has left: (-3, 3) x –3 –2 –1 0 1 2 3 y 1 a. Complete the table and graph the line that represents Mica’s rule. b. Mica thinks the equation for this graph could be 2x + y = !3 . Is she correct? Explain why or why not. If not, find your own algebraic rule to match the graph and x → y table. Kevin and his little sister, Katy, are trying to solve the system of equations shown below. Kevin thinks the new equation should be 3(6x ! 1) + 2y = 43 , while Katy thinks it should be 3x + 2(6x ! 1) = 43 . Who is correct and why? y = 6x ! 1 3x + 2y = 43 6-41. Create a table and graph the rule y = 10 ! x 2 + 3x . Label its x- and y-intercepts. 6-42. Maurice thinks that x = !2 is a solution to x 2 ! 3x ! 8 = 0 . Is he correct? Explain. 244 Algebra Connections 6.2.2 How does a graph show a solution? y 5 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Making Connections: Systems, Solutions, and Graphs x 5 In this chapter you have practiced writing mathematical sentences to represent situations. Often, these sentences give you a system of equations, which you can solve using substitution. Today you will start to represent these situations in an additional way: on a graph. You will also examine more closely what makes a solution to a two-variable equation. 6-43. THE HILLS ARE ALIVE The Alpine Music Club is going on its annual music trip. The members of the club are yodelers, and they like to play the xylophone. This year they are taking their xylophones on a gondola to give a performance at the top of Mount Monch. The gondola conductor charges $2 for each yodeler and $1 for each xylophone. It costs $40 for the entire club, including the xylophones, to ride the gondola. Two yodelers can share a xylophone, so the number of yodelers on the gondola is twice the number of xylophones. How many yodelers and how many xylophones are on the gondola? Your Task: • Represent this problem with a system of equations. Solve the system and explain how its solution relates to the yodelers on the music trip. • Represent this problem with a graph. Identify how the solution to this problem appears on the graph. How can the given information be represented with equations? What is a solution to a two-variable equation? How can this problem be represented on a graph? How does the solution appear on the graph? Chapter 6: Systems of Equations 245 6-44. 6-45. 246 Start by focusing on one aspect of the problem: the cost to ride the gondola. The conductor charges $2 for each yodeler and $1 for each xylophone. It costs $40 for the entire club, with instruments, to ride the gondola. a. Write an equation with two variables that represents this information. Be sure to define your variables. b. Find a combination of xylophones and yodelers that will make your equation from part (a) true. Is this is the only possible combination? c. List five additional combinations of xylophones and yodelers that could ride the gondola if it costs $40 for the trip. With your team, decide on a good way to organize and share your list. d. Jon says, “I think there could be 28 xylophones and 8 yodelers on the gondola.” Is he correct? Use the equation you have written to explain why or why not. e. Helga says, “Each correct combination we found is a solution to our equation.” Is this true? Explain what it means for something to be a solution to a twovariable equation. Now consider the other piece of information: The number of yodelers is twice the number of xylophones. a. Write an equation (mathematical sentence) that expresses this piece of information. b. List four different combinations of xylophones and yodelers that will make this equation true. c. Put the equation you found in part (a) together with your equation from problem 6-44 and use substitution to solve this system of equations. d. Is the answer you found in part (c) a solution to the first equation you wrote (the equation in part (a) of problem 6-44)? How can you check? Is it a solution to the second equation you wrote (the equation in part (a) of this problem)? Why is this a solution to the system of equations? Algebra Connections 6-46. The solution to “The Hills are Alive” problem can also be represented graphically. a. On graph paper, graph the equation you wrote in part (a) of problem 6-44. The points you listed for that equation may help. What is the shape of this graph? Label your graph with its equation. b. Explain how each point on the graph represents a solution to the equation. c. Now graph the equation you wrote in part (a) of problem 6-45 on the same set of axes. The points you listed for that equation may help. Label this graph with its equation. d. Find the intersection point of the two graphs. What is special about this point? e. With your team, find as many ways as you can to express the solution to “The Hills are Alive” problem. Be prepared to share all the different forms you found for the solution with the class. Further Guidance section ends here. 6-47. Consider this system of equations: 2x + 2y = 18 y= x!3 a. Use substitution to solve this system. b. With your team, decide how to fill in the rest of the table at right for the equation 2x + 2y = 18 . c. Use your table to make an accurate graph of the equation 2x + 2y = 18 . d. Now graph y = x ! 3 on the same set of axes. Find the point of intersection. e. Does the point of intersection you found in part (a) agree with what you see on your graph? x –2 –1 0 1 2 3 y 11 6-48. If you had an equation with three variables, how would you write its solutions? 6-49. What is a solution to a two-variable equation? Answer this question in complete sentences in your Learning Log. Then give an example of a two-variable equation followed by two different solutions to it. Finally, make a list of all of the ways to represent solutions to twovariable equations. Title your entry “Solutions to Two-Variable Equations” and label it with today’s date. Chapter 6: Systems of Equations 247 ETHODS AND MEANINGS MATH NOTES The Substitution Method The Substitution Method is a way to change two equations with two variables into one equation with one variable. It is convenient to use when only one equation is solved for a variable. x = !3y + 1 4x ! 3y = !11 For example, to solve the system: Use substitution to rewrite the two equations as one. In other words, replace x with (!3y + 1) to get 4(!3y + 1) ! 3y = !11 . This equation can then be solved to find y. In this case, y = 1. To find the point of intersection, substitute to find the other value. Substitute y = 1 into x = !3y + 1 and write the answer for x and y as an ordered pair. To test the solution, substitute x = !2 and y = 1 into 4x ! 3y = !11 to verify that it makes the equation true. Since 4(!2) ! 3(1) = !11 , the solution must be correct. x = !3y + 1 4 (! ) ! 3y = !11 4(!3y + 1) ! 3y = !11 !12y + 4 ! 3y = !11 !15y + 4 = !11 !15y = !15 y =1 x = !3(1) + 1 = !2 ( !2,!1) 6-50. Camila is trying to find the equation of a line that passes through the points (–1, 16) and (5, 88). Does the equation y = 12x + 28 work? Justify your answer. 6-51. Solve the systems of equations below using the method of your choice. Check your solutions, if possible. a. 248 y = 7 ! 2x 2x + y = 10 b. 3y ! 1 = x 4x ! 2y = 16 Algebra Connections 6-52. 6-53. Hotdogs and corndogs were sold at last night’s football game. Use the information below to write mathematical sentences to help you determine how many corndogs were sold. a. The number of hotdogs sold was three fewer than twice the number of corndogs. Write a mathematical sentence that relates the number of hotdogs and corndogs. Let h represent the number of hotdogs and c represent the number of corndogs. b. A hotdog costs $3 and a corndog costs $1.50. If $201 was collected, write a mathematical sentence to represent this information. c. How many corndogs were sold? Show how you found your answer. Examine the balanced scales in Figures 1 and 2 shown below. Figure 1 shows that two candies balance three dice. Figure 2 shows that one rubber ball balances two jacks. Figure 1 Figure 2 Figure 3 Determine what could be placed on the right side of the scale in Figure 3 to balance with the left side. Justify your solution in complete sentences. 6-54. Rianna thinks that if a = b and if c = d , then a + c = b + d . Is she correct? 6-55. For each of the following generic rectangles, find the dimensions (length and width) and write the area as the product of the dimensions and as a sum. a. b. 3y 2 Chapter 6: Systems of Equations !12y 3y 2 !12y 5y !20 249 6.2.3 Can I solve without substituting? y ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 5 Solving Systems Using Elimination x 5 In this chapter, you have learned the Substitution Method for solving systems of equations. In Chapter 4, you learned the Equal Values Method. But are these methods the best to use for all types of systems? Today you will develop a new solution method that can save time for systems of equations in standard form. 6-56. Jeanette is trying to find the intersection point of these two equations: 2x + 3y = !2 5x ! 3y = 16 She has decided to use substitution to find the point of intersection. Her plan is to solve the first equation for y, and then to substitute the result into the second equation. Use Jeanette’s idea to solve the system. 6-57. AVOIDING THE MESS: THE ELIMINATION METHOD Your class will now discuss a new method, called the Elimination Method, to find the solution to Jeanette’s problem without the complications and fractions of the previous problem. Your class discussion is outlined below. + a. + x _ Verify that each equation mat at right represents one of Jeanette’s equations. _ x x x 2x + 3y = !2 5x ! 3y = 16 b. + y y x y x + y _ x y _ y Can these two equations be merged onto one equation mat as shown below? That is, can the left sides and right sides of two equations be added together to create a new equation? Why or why not? + x y x x x x x y x y + y _ y y _ This is the result when the equations are combined. Problem continues on next page → 250 Algebra Connections 6-57. 6-58. 6-59. Problem continued from previous page. c. Write a new equation for the result of merging Jeanette’s equations. Simplify and then solve this new equation for the remaining variable. Notice that you now have only one equation with one variable. What happened to the y-terms? d. Use your solution for x to find y. Check to be sure your solution makes both original equations true. e. How can you record this process on paper? That is, when solving this type of system, how can you show that you are combining the equations? f. Now use the Elimination Method to solve the system of equations at right for x and y. Check your solution. 2x ! y = !2 !2x + 3y = 10 Pat was in a fishing competition at Lake Pisces. She caught some bass and some trout. Each bass weighed 3 pounds, and each trout weighed 1 pound. Pat caught a total of 30 pounds of fish. She got 5 points in the competition for each bass, but since trout are endangered in Lake Pisces, she lost 1 point for each trout. Pat scored a total of 42 points. a. Write a system of equations representing the information in this problem. b. Is this system a good candidate for the Elimination Method? Why or why not? c. Solve this system to find out how many bass and trout Pat caught. Be sure to record your work and check your answer by substituting your solution into the original equations. ANNIE NEEDS YOUR HELP Annie was all ready to “push together” the two equations below to eliminate the x-terms when she noticed a problem: Both x-terms are positive! 2x + 7y = 13 2x + 3y = 5 With your team, figure out something you could do that would allow you to put these equations together and eliminate the x-terms. As you try out different ideas, ask your teacher for some algebra tiles and an equation mat if you think they will help. Once you have figured out a method, solve the system and check your solution. Be ready to share your method with the class. Chapter 6: Systems of Equations 251 6-60. Find the point of intersection of each pair of lines below. If you use an equation mat, be sure to record your process on paper. Otherwise, show your steps algebraically. Check each solution when you are finished. a. 2y ! x = 5 !3y + x = !9 b. 2x ! 4y = 14 4y ! x = !3 3x + 4y = 1 2x + 4y = 2 c. MATH NOTES ETHODS AND MEANINGS Systems of Linear Equations y = 2x y = !3x + 5 A system of linear equations is a set of two or more linear equations that are given together, such as the example at right: If the equations come from a real-world context, then each variable will represent some type of quantity in both equations. For example, in the system of equations above, y could stand for a number of dollars in both equations. To represent a system of equations graphically, you can simply graph each equation on the same set of axes. The graph may or may not have a point of intersection, as shown circled at right. y 6 4 2 Sometimes two lines have no points of intersection. This happens when the two lines are parallel. It is also possible for two lines to have an infinite number of intersections. This happens if they are simply the same equation in different forms. Such lines are said to coincide. Also notice that the point of intersection lies on both graphs in the system of equations. This means that the point of intersection is a solution to both equations in the system. For example, the point of intersection of the two lines graphed above is (1, 2). This point of intersection makes both equations true, as shown at right. –1 –2 y = 2x (2) = 2(1) 2=2 1 2 3 4 x y = !3x + 5 (2) = !3(1) + 5 2 = !3 + 5 2=2 The point of intersection makes both equations true; therefore the point of intersection is a solution to both equations. For this reason, the point of intersection is sometimes called a solution to the system of equations. 252 Algebra Connections 6-61. Find the point of intersection of each pair of lines, if one exists. If you use an equation mat, be sure to record your process on paper. Check each solution, if possible. a. 6-62. x = !2y ! 3 4y ! x = 9 b. x + 5y = 8 !x + 2y = !1 c. 4x ! 2y = 5 y = 2x + 10 Jai was solving the system of equations below when something strange happened. y = !2x + 5 2y + 4x = 10 6-63. a. Solve the system. Explain to Jai what the solution should be. b. Graph the two lines on the same set of axes. What happened? c. Explain how the graph helps to explain your answer in part (a). On Tuesday the cafeteria sold pizza slices and burritos. The number of pizza slices sold was 20 less than twice the number of burritos sold. Pizza sold for $2.50 a slice and burritos for $3.00 each. The cafeteria collected a total of $358 for selling these two items. a. Write two equations with two variables to represent the information in this problem. Be sure to define your variables. b. Solve the system from part (a). Then determine how many pizza slices were sold. 6-64. A local deli sells 6-inch sub sandwiches for $2.95. It has decided to sell a “family sub” that is 50 inches long. How much should it charge? Show all work. 6-65. Represent the tile pattern below with a table, a rule, and a graph. Figure 1 6-66. Figure 2 Figure 3 Use generic rectangles to multiply each of the following expressions. a. ( x + 2)( x ! 5) b. (y + 2x)(y + 3x) c. (3y ! 8)(! x + y) d. ( x ! 3y )( x + 3y ) Chapter 6: Systems of Equations 253 6.2.4 How can I eliminate a variable? y ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• More Elimination 5 x 5 In Lesson 6.2.3, you learned how to use the Elimination Method to solve systems of equations. In this method, you combined two equations in a way that made one variable disappear. This method is particularly useful for solving systems of equations where neither equation is in y = mx + b form. Today you will practice using the Elimination Method while learning to deal with various complications that systems of equations sometimes present. As you solve these systems, ask your teammates these questions: How can you create one equation with only one variable? How can you eliminate one variable? How do you know your solution is correct? 6-67. Which system of equations below would be easiest to solve using the Elimination Method? Once you have explained your decision, use the Elimination Method to solve this system of equations. (You do not need to solve the other system!) Record your steps and check your solution. a. 6-68. 5x ! 4y = 37 !8x + 4y = !52 b. 4 ! 2x = y 3y + x = 11 Rachel is trying to solve this system: 2x + y = 10 3x ! 2y = 1 254 a. Combine these equations. What happened? b. Is 2x + y = 10 the same line as 4x + 2y = 20 ? That is, do they have the same solutions? Are their graphs the same? Justify your conclusion! Be ready to share your reasoning with the class. c. Since you can rewrite 2x + y = 10 as 4x + 2y = 20 , perhaps this equivalent form of the original equation can help solve this system. Combine 4x + 2y = 20 and 3x ! 2y = 1. Is a variable eliminated? If so, solve the system for x and y. If not, brainstorm another way to eliminate a variable. Be sure to check your solution. d. Why was the top equation changed? Would a variable have been eliminated if the bottom equation were multiplied by 2 on both sides? Test this idea. Algebra Connections 6-69. 6-70. For each system below, determine: • Is this system a good candidate for the Elimination Method? Why or why not? • What is the best way to get one equation with one variable? Carry out your plan and solve the system for both variables. • Is your solution correct? Verify by substituting your solution into both original equations. a. 5m + 2n = !10 3m + 2n = !2 b. 6a ! b = 3 b + 4a = 17 c. 7x + 4y = 17 3x ! 2y = !15 d. !18x + 3y = !12 !!! 6x !!!y = 4 A NEW CHALLENGE Carefully examine this system: 4x + 3y = 10 9x ! 4y = 1 With your team, propose a way to combine these equations so that you eventually have one equation with one variable. Be prepared to share your proposal with the class. MATH NOTES ETHODS AND MEANINGS Coefficients and Constants A coefficient is the numerical part of a term that includes a variable. For example, in the expression below, the coefficient of 7x 2 is the number 7, the coefficient of 4x is 4, and the coefficient of –y is –1. Note that the 9 in the expression below is called a constant. A constant is a term that does not include a variable. Chapter 6: Systems of Equations 7x 2 + 4x ! y + 9 255 6-71. 6-72. Solve these systems of equations using any method. Check each solution, if possible. a. 2x + 3y = 9 !3x + 3y = !6 b. x = 8 ! 2y y! x = 4 c. y = ! 12 x + 7 y= x!8 d. 9x + 10y = 14 7x + 5y = !3 For each line below, make a table and graph. What do you notice? a. y = 23 x ! 1 b. 2x ! 3y = 3 6-73. Consecutive numbers are integers that are in order without skipping, such as 3, 4, and 5. Find three consecutive numbers with a sum of 54. 6-74. Identify the hypothesis and conclusion for each of the following statements. Then decide if the statement is true or false. Justify your decision. You may want to review the meanings of hypothesis and conclusion from problem 6-31. 6-75. a. If y = 23 x ! 5 , then the point (6, –1) is a solution. b. If Figure 2 of a tile pattern has 13 tiles and Figure 4 of the same pattern has 15 tiles, then the pattern grows by 2 tiles each figure. c. If (3x + 1)(x ! 2) = 4 , then 3x 2 ! 5x ! 2 = 4 . Aimee thinks the solution to the system below is (– 4, –6). Eric thinks the solution is (8, 2). Who is correct? Explain your reasoning. 2x ! 3y = 10 6y = 4x ! 20 6-76. 256 Figure 3 of a tile pattern has 11 tiles, while Figure 4 has 13 tiles. If the tile pattern grows at a constant rate, how many tiles will Figure 50 have? Algebra Connections 6.2.5 What is the best method? y 5 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Choosing a Strategy for Solving Systems x 5 When you have a system of equations to solve, how do you know which method to use? Focus today on how to choose a strategy that is the most convenient, efficient, and accurate for a system of equations. 6-77. Erica works in a soda-bottling factory. As bottles roll past her on a conveyer belt, she puts caps on them. Unfortunately, Erica sometimes breaks a bottle before she can cap it. She gets paid 4 cents for each bottle she successfully caps, but her boss deducts 2 cents from her pay for each bottle she breaks. Erica is having a bad morning. Fifteen bottles have come her way, but she has been breaking some and has only earned 6 cents so far today. How many bottles has Erica capped and how many has she broken? 6-78. a. Write a system of equations representing this situation. b. Solve the system of equations using two different methods: substitution and elimination. Demonstrate that each method gives the same answer. For each system below, decide which strategy to use. That is, which method would be the most efficient, convenient, and accurate: the Substitution Method, the Elimination Method, or the Equal Values Method? Do not solve the systems yet! Be prepared to justify your reasons for choosing one strategy over the others. a. x = 4 ! 2y 3x ! 2y = 4 b. 3x + y = 1 4x + y = 2 c. x = !5y + 2 x = 3y ! 2 d. 2x ! 4y = 10 x = 2y + 5 e. y = 12 x + 4 y = !2x + 9 f. !6x + 2y = 76 !!3x !!y = !38 g. 5x + 3y = !6 2x ! 9y = 18 h. x!3= y 2(x ! 3) ! y = 7 Chapter 6: Systems of Equations 257 6-79. Your teacher will assign you a variety of systems from problem 6-78 to solve. With your team, use the best strategy to solve each system assigned by your teacher. Be sure to check your solution. 6-80. In your Learning Log, write down everything you know about solving systems of equations. Include examples and explain your reasoning. Title this entry “Solving Systems of Equations” and label it with today’s date. MATH NOTES ETHODS AND MEANINGS Intersection, Parallel, and Coincide When two lines lie on the same flat surface (called a plane), they may intersect (cross each other) once, an infinite number of times, or never. For example, if the two lines are parallel, then they never intersect. Examine the graph of two parallel lines at right. Notice that the distance between the two lines is constant. However, what if the two lines lie exactly on top of each other? When this happens, we say that the two lines coincide. When you look at two lines that coincide, they appear to be one line. Since these two lines intersect each other at all points along the line, coinciding lines have an infinite number of intersections. While some systems contain lines that are parallel and others coincide, the most common case for a system of equations is when the two lines intersect once, as shown at right. 258 parallel lines y x intersecting lines y x Algebra Connections 6-81. Solve the following systems of equations using any method. Check each solution, if possible. a. !2x + 3y = 1 2x + 6y = 2 b. y = 13 x + 4 x = !3y c. 3x ! y = 7 y = 3x ! 2 d. x + 2y = 1 3x + 5y = 8 6-82. The Math Club is baking pies for a bake sale. The fruit-pie recipe calls for twice as many peaches as nectarines. If it takes a total of 168 pieces of fruit for all of the pies, how many nectarines are needed? 6-83. Candice is solving this system: 2x ! 1 = 3y 5(2x ! 1) + y = 32 6-84. a. She notices that each equation contains the expression 2x ! 1 . Can she substitute 3y for 2x ! 1 ? Why or why not? b. Substitute 3y for 2x ! 1 in the second equation to create one equation with one variable. Then solve for x and y. Examine the diagram at right. The smaller triangle is similar to the larger triangle. Write and solve a proportion to find x. 8 x 10 6-85. 6-86. Figure 2 of a tile pattern is shown at right. If the pattern grows linearly and if Figure 5 has 15 tiles, then find a rule for the pattern. 8 Figure 2 Given the hypothesis that line l is parallel to line m and that line m is parallel to line n, what can you conclude? Justify your conclusion. Chapter 6: Systems of Equations 259 6.3.1 What can I do now? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Pulling It All Together This lesson contains many problems that will require you to use the algebra content you have learned so far in new ways. It will require you to use all five Ways of Thinking (justifying, making connections, applying and extending, reversing thinking, and generalizing) and will help you solidify your understanding. Your teacher will describe today’s activity. As you solve the problems below, remember to make connections between all of the different subjects you have studied in Chapters 1 through 6. If you get stuck, think of what the problem reminds you of. Decide if there is a different way to approach the problem. Most importantly, discuss your ideas with your teammates. 6-87. Brianna has been collecting insects and measuring the lengths of their legs and antennae. Below is the data she has collected so far. Length of Antenna (x) Length of Leg (y) 260 Ant 2 mm 4 mm Beetle 6 mm 10 mm Grasshopper 20 mm 31 mm a. Graph the data Brianna has collected. Put the antenna length on the x-axis and leg length on the y-axis. b. Brianna thinks that she has found an algebraic rule relating antenna length and leg length: 4y ! 6x = 4 . If x represents the length of the antenna and y represents the leg length, could Brianna’s rule be correct? If not, find your own algebraic rule relating antenna length and leg length. c. If a ladybug has an antenna 1 mm long, how long does Brianna’s rule say its legs will be? Use both the rule and the graph to justify your answer. Algebra Connections 6-88. Barry is helping his friend understand how to solve systems of equations. He wants to give her a problem to practice. He wants to give her a problem that has two lines that intersect at the point (–3, 7). Help him by writing a system of equations that will have (–3, 7) as a solution and demonstrate how to solve it. 6-89. Examine the generic rectangle at right. Determine the missing attributes and then write the area as a product and as a sum. 2xy !9 x !3 !5 y 6-91. One evening, Gemma saw three different phonecompany ads. TeleTalk boasted a flat rate of 8¢ per minute. AmeriCall charges 30¢ per call plus 5¢ per minute. CellTime charges 60¢ per call plus only 3¢ per minute. a. Gemma is planning a phone call that will take about 5 minutes. Which phone plan should she use and how much will it cost? b. Represent each phone plan with a table and a rule. Then graph each plan on the same set of axes, where x represents time in minutes and y represents the cost of the call in cents. If possible, use different colors to represent the different phone plans. c. How long would a call need to be to cost the same with TeleTalk and AmeriCall? What about AmeriCall and CellTime? d. Analyze the different phone plans. How long should a call be so that AmeriCall is cheapest? Lashayia is very famous for her delicious brownies, which she sells at football games. The graph at right shows the relationship between the number of brownies she sells and the amount of money she earns. a. b. How much should she charge for 10 brownies? Be sure to demonstrate your reasoning. During the last football game, Lashayia made $34.20. How many brownies did she sell? Show your work. Chapter 6: Systems of Equations Sales (in dollars) 6-90. 40 30 (12,21.6) 20 10 10 Number of Brownies Sold 261 6-92. How many solutions does each equation below have? How can you tell? a. 4x ! 1+ 5 = 4x + 3 b. 6t ! 3 = 3t + 6 c. 6(2m ! 3) ! 3m = 2m ! 18 + m d. 10 + 3y ! 2 = 4y ! y + 8 6-93. Anthony has the rules for three lines: A, B, and C. When he solves a system with lines A and B, he gets no solution. When he solves a system with lines B and C, he gets infinite solutions. What solution will he get when he solves a system with lines A and C? Justify your conclusion. 6-94. Complete the Guess and Check table below and find a solution. Then write a possible word problem that would fit the table. Stevie 3 10 7.50 6-95. Joan 5 19 14 Julio 8.50 22.50 17.50 31.50? Check Too low Too high Too high Normally, the longer you work for a company, the higher your salary per hour. Hector surveyed the people at his company and placed his data in the table below. Number of Years at Company Salary per Hour 262 Total 16.50 51.50 39.00 a. Use Hector’s data to estimate how much he makes, assuming he has worked at the company for 12 years. b. Hector is hiring a new employee who will work 20 hours a week. How much should the new employee earn for the first week? 1 $7.00 3 $8.50 6 7 $10.75 $11.50 Algebra Connections 6-96. Dexter loves to find shortcuts. He has proposed a few new moves to help simplify and solve equations. Examine his work below. For each, decide if his move is “legal.” That is, decide if the move creates an equivalent equation. Justify your conclusions using the “legal” moves you already know. a. b. + x x + _ 6-97. + _ _ x + x x _ c. + y d. + _ _ + x x _ + _ Solve the problem below using two different methods. The Math Club sold roses and tulips this year for Valentine’s Day. The number of roses sold was 8 more than 4 times the number of tulips sold. Tulips were sold for $2 each and roses for $5 each. The club made $414.00. How many roses were sold? 6-98. Use substitution to find where the two parabolas below intersect. Then confirm your solution by graphing both on the same set of axes. y = x2 + 5 y = x 2 + 2x + 1 Chapter 6: Systems of Equations 263 ETHODS AND MEANINGS MATH NOTES The Elimination Method for Solving Systems of Equations One method of solving systems of equations is the Elimination Method. This method involves adding or subtracting both sides of two equations to eliminate a variable. Equations can be combined this way because balance is maintained when equal amounts are added to both sides of an equation. For example, if a = b and c = d , then if you add a and c you will get the same result as adding b and d. Thus, a + c = b + d . Consider the system of linear equations shown at right. Notice that when both sides of the equations are added together, the sum of the x-terms is zero and so the x-terms are eliminated. (Be sure to write both equations so that x is above x, y is above y, and the constants are similarly matched.) Now that you have one equation with one variable ( 7y = 28 ), you can solve for y by dividing both sides by 7. To find x, you can substitute the answer for y into one of the original equations, as shown at right. You can then test the solution for x and y by substituting both values into the other equation to verify that !3x + 5y = 14 . 3x + 2y = 14 !3x + 5y = 14 7y 7 = 278 y =4 3x + 2(4) = 14 3x + 8 = 14 3x = 6 x=2 !3(2) + 5(4) = 14 Since x = 2 and y = 4 is a solution to both equations, it can be stated that the two lines cross at the point (2, 4). 6-99. 264 Find the point of intersection for each set of equations below using any method. Check your solutions, if possible. a. 6x ! 2y = 10 3x ! 5 = y b. 6x ! 2y = 5 3x + 2y = !2 c. 5 ! y = 3x y = 2x d. y = 14 x + 5 y = 2x ! 9 Algebra Connections 6-100. Consider the equation !6 x = 4 ! 2y . a. If you graphed this equation, what shape would the graph have? How can you tell? b. Without changing the form of the equation, find the coordinates of three points that must be on the graph of this equation. Then graph the equation on graph paper. c. Solve the equation for y. Does your answer agree with your graph? If so, how do they agree? If not, check your work to find the error. 6-101. A tile pattern has 10 tiles in Figure 2 and increases by 2 tiles for each figure. Find a rule for this pattern and then determine how many tiles are in Figure 100. 6-102. Make a table and graph the rule y = ! x 2 + x + 2 on graph paper. Label the x-intercepts. 6-103. Mr. Greer solved an equation below. However, when he checked his solution, it did not make the original equation true. Find his error and then find the correct solution. 4x = 8(2x ! 3) 4x = 16x ! 3 !12x = !3 x= x= 6-104. !3 !12 1 4 Thirty coins, all dimes and nickels, are worth $2.60. How many nickels are there? Chapter 6: Systems of Equations 265 6-105. 6-106. 6-107. Multiple Choice: Martha’s equation has the graph shown at right. Which of these are solutions to Martha’s equation? (Remember that more than one answer may be correct.) a. (– 4, –2) b. (–1, 0) c. x = 0 and y = 1 d. x = 2 and y = 2 Copy and complete the table below. Then write the corresponding rule. IN (x) 2 OUT (y) –7 10 6 7 –3 0 18 3 –10 100 x Solve the following equations for x, if possible. Check your solutions. a. !(2 ! 3x) + x = 9 ! x b. 6 x+2 c. 5 ! 2(x + 6) = 14 d. 1 2 = 3 4 x ! 4 + 1 = !3 ! 12 x 6-108. Using the variable x, write an equation that has no solution. Explain how you know it has no solution. 6-109. Given the hypothesis that 2x ! 3y = 6 and x = 0 , what can you conclude? Justify your conclusion. 6-110. Multiple Choice: Which equation below could represent a tile pattern that grows by 3 and has 9 tiles in Figure 2? 266 a. 3x + y = 3 b. !3x + y = 9 c. !3x + y = 3 d. 2x + 3y = 9 Algebra Connections Chapter 6 Closure What have I learned? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Reflection and Synthesis The activities below offer you a chance to reflect on what you have learned during this chapter. As you work, look for concepts that you feel very comfortable with, ideas that you would like to learn more about, and topics you need more help with. Look for connections between ideas as well as connections with material you learned previously. TEAM BRAINSTORM With your team, brainstorm a list for each of the following topics. Be as detailed as you can. How long can you make your list? Challenge yourselves. Be prepared to share your team’s ideas with the class. Topics: What have you studied in this chapter? What ideas and words were important in what you learned? Remember to be as detailed as you can. Connections: What topics, ideas, and words that you learned before this chapter are connected to the new ideas in this chapter? Again, make your list as long as you can. MAKING CONNECTIONS The following is a list of the vocabulary used in this chapter. The words that appear in bold are new to this chapter. Make sure that you are familiar with all of these words and know what they mean. Refer to the glossary or index for any words that you do not yet understand. coefficients coincide Elimination Method Equal Values Method equation graph “let” statement linear equation mathematical sentence ordered pair parallel point of intersection situation solution standard form Substitution Method system of equations variable y = mx + b Make a concept map showing all of the connections you can find among the key words and ideas listed above. To show a connection between two words, draw a line between them and explain the connection, as shown in the example below. A word can be connected to any other word as long as there is a justified connection. For each key word or idea, provide a sketch that illustrates the idea (see the example on the following page). Continues on next page → Chapter 6: Systems of Equations 267 Continued from previous page. Word A Example: These are connected because… Word B Example: Your teacher may provide you with vocabulary cards to help you get started. If you use the cards to plan your concept map, be sure either to re-draw your concept map on your paper or to glue the vocabulary cards to a poster with all of the connections explained for others to see and understand. While you are making your map, your team may think of related words or ideas that are not listed here. Be sure to include these ideas on your concept map. SUMMARIZING MY UNDERSTANDING This section gives you an opportunity to show what you know about certain math topics or ideas. Your teacher will give you directions for exactly how to do this. Your teacher may give you a “GO” page to work on. The “GO” stands for “Graphic Organizer,” a tool you can use to organize your thoughts and communicate your ideas clearly. WHAT HAVE I LEARNED? This section will help you evaluate which types of problems you have seen with which you feel comfortable and those with which you need more help. Even if your teacher does not assign this section, it is a good idea to try these problems and find out for yourself what you know and what you need to work on. Solve each problem as completely as you can. The table at the end of the closure section has answers to these problems. It also tells you where you can find additional help and practice on problems like these. CL 6-111. Solve these systems of equations using any method. a. c. 268 y = 3x + 7 y = ! 4x + 21 b. x = 3y ! 5 2x + 12y = ! 4 d. 3x ! y = 17 !x + y = !7 2x ! 3y = !16 ! 4x + 2y = ! 4 Algebra Connections CL 6-112. Bob climbed down a ladder from his roof, while Rob climbed up another ladder next to him. Each ladder had 30 rungs. Their friend Jill recorded the following information about Bob and Rob: Bob went down 2 rungs every second. Rob went up 1 rung every second. At some point, Bob and Rob were at the same height. Which rung were they on? CL 6-113. Solve for x. a. 6x ! 11 = 4x + 12 b. 2(3x ! 5) = 6x ! 4 c. (x ! 3)(x + 4) = x 2 + 4 d. x 25 7 = 10 CL 6-114. Solve the equations in parts (a) and (b) for y. Then name the growth factor and the y-intercept of each equation in part (c). a. !6x ! 2y = 8 c. For each of the two solved equations, find the y-intercept and growth factor. Justify your answers. b. 2x 2 + 2y = 4x + 2x 2 ! 7 CL 6-115. Florida ecologists sampled Lake George to estimate the number of rainbow trout in the lake. Out of 156 fish, 18 were rainbow trout. About how many rainbow trout should they expect to find in a sample of 500 fish? CL 6-116. As treasurer of his school’s 4H club, Kenny wants to buy gifts for all 18 members. He can buy t-shirts for $9 and sweatshirts for $15. The club has only $180 to spend. If Kenny wants to spend all of the club’s money, how many of each type of gift can he buy? a. Write a system of equations representing this problem. b. Solve your system of equations and figure out how many of each type of gift Kenny should buy. CL 6-117. Simplify each expression. a. 3(x 2 ! 7x) + 5xy ! (x ! 4xy) ! 2x 2 + 21x b. 3y ! (4x + 7) ! y + 11 + (2x ! y + 12) Chapter 6: Systems of Equations 269 CL 6-118. Rewrite each expression below as a product and as a sum. a. (x + 7)(2x ! 5) b. 5x(y ! 7) c. (3x ! 7)(x 2 ! 2x + 11) CL 6-119. Each part (a) through (d) below represents a different tile pattern. For each, find the growth factor and the number of tiles in Figure 0. a. y b. x Figure 2 c. Figure 3 Figure 4 y = 3x ! 14 d. x y –3 18 –2 13 –1 8 0 3 1 –2 2 –7 3 –12 CL 6-120. Check your answers using the table at the end of the closure section. Which problems do you feel confident about? Which problems were hard? Use the table to make a list of topics you need help on and a list of topics you need to practice more. HOW AM I THINKING? This course focuses on five different Ways of Thinking: reversing thinking, justifying, generalizing, making connections, and applying and extending understanding. These are some of the ways in which you think while trying to make sense of a concept or to solve a problem (even outside of math class). During this chapter, you have probably used each Way of Thinking multiple times without even realizing it! Review each of the Ways of Thinking that are described in the closure sections of Chapters 1 through 5. Then choose three of these Ways of Thinking that you remember using while working in this chapter. For each Way of Thinking that you choose, show and explain where you used it and how you used it. Describe why thinking in this way helped you solve a particular problem or understand something new. (For instance, explain why you wanted to generalize in this particular case, or why it was useful to see these particular connections.) Be sure to include examples to demonstrate your thinking. 270 Algebra Connections Answers and Support for Closure Activity #4 What Have I Learned? Problem CL 6-111. Solution a. x = 2 , y = 13 b. x = 5 , y = !2 c. x = ! 4 , y = 1 3 Need Help? More Practice Lessons 6.2.2, 6.2.3, and 6.3.1 Math Notes boxes Problems 6-24, 6-25, 6-32, 6-34, 6-51, 6-56, 6-61, 6-62, 6-71, and 6-81 d. x = 11 , y=9 2 CL 6-112. They were on the 10th rung. Lessons 6.2.2, 6.2.3, and 6.3.1 Math Notes boxes Problems 6-38, 6-43, 6-52, 6-58, 6-77, 6-90, and 6-97 CL 6-113. a. x = 11.5 b. no solution c. x = 16 d. x = 17.5 Lesson 5.1.3 Math Notes box, Lesson 5.1.4 Problems 6-16, 6-37, and 6-107 CL 6-114. a. y = !3x ! 4 Lesson 5.1.5, Lesson 5.1.5 Math Notes box Problems 6-12 and 6-100 b. y = 2x ! 27 c. (a) y-intercept: (0, – 4), growth: –3 (b) y-intercept: (0, –3.5), growth: 2 CL 6-115. approximately 58 rainbow trout Lesson 5.2.1, Lesson 5.2.1 Math Notes box Problems 6-11, 6-28, and 6-64 CL 6-116. a. 9x + 15y = 180 , x + y = 18 b. 15 t-shirts, 3 sweatshirts Lessons 6.2.2, 6.2.3, and 6.3.1 Math Notes boxes Problems 6-38, 6-43, 6-52, 6-58, 6-77, 6-90, and 6-97 Chapter 6: Systems of Equations 271 Problem Solution Need Help? More Practice CL 6-117. a. x 2 + 9xy ! x b. y ! 2x + 16 Lessons 2.1.5 and 5.1.3 Math Notes boxes Problems 4-6, 3-15, and 3-75 CL 6-118. a. 2x 2 + 9x ! 35 b. 5xy ! 35x c. 3x 3 ! 13x 2 + 47x ! 77 Lessons 5.1.3 and 5.2.3 Math Notes boxes Problems 6-66 and 6-103 CL 6-119. a. growth: 5, Figure 0: 3 tiles b. growth: –2, Figure 0: 3 tiles c. growth: 3, Figure 0: –14 tiles d. growth: –5, Figure 0: 3 tiles Sections 3.1 and 4.1, Lesson 4.1.7 Math Notes box Problems 6-10, 6-76, 6-85, 6-101, and 6-110 272 Algebra Connections

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