Unit 6. Systems of linear equations and inequalities. Outline

Transcription

1 Unit 6 Systems of linear equations and inequalities In this unit, you will learn about systems of equations two or more equations that describe related conditions in a problem situation. You will learn how to solve systems using the same techniques you used in the previous unit tables, graphs, and symbolic methods. You will also learn about systems of inequalities two or more linear inequalities considered at the same time. You will solve systems of inequalities by graphing. Finally, you will review the idea of mindset and how it can affect your success as you practice building your skills in solving multi- step equations. Outline Topic 16: Formulating and solving systems In the last unit, you learned to solve linear equations and inequalities using a variety of methods. These are useful strategies for situations in which a single variable is unknown. But how can you find the values of two different variables in a situation? You solved a problem like this involving bikes and skateboards. To solve such problems, you might need to represent two or more related conditions in the situation. In other words, you may need to work with a system of equations or a system of inequalities. In this topic, you will: Write a system of linear equations in two variables to model a problem situation Verify that a given ordered pair is a solution to a system Solve systems of equations by inspection Solve systems of two linear equations with tables Solve systems of two linear equations with graphs Write a system of linear inequalities in two variables to model a problem situation Solve systems of linear inequalities in two variables by graphing solutions on a coordinate plane Topic 17: Building fluency with equation solving In this topic, you will focus on building fluency and efficiency with solving multi- step equations. You will also continue to develop your problem- solving and algebraic thinking capabilities by working on a non- routine problem called the Speeding Car Problem. This problem will require you to pull together many mathematical ideas. In this topic you will: Practice solving multi- step equations Solve a non- routine problem Topic 18: Other methods for solving systems In this topic, you will continue to explore solution methods for systems of two linear equations. You will be introduced to two algebraic methods for solving systems: the substitution method and the linear combination method. You will begin to see when to use each method and how to interpret the results each method yields. In this topic, you will: Learn how to use substitution to solve a system of linear equations in two variables Learn how to use linear combination to solve a system of linear equations in two variables Learn how to recognize when a system of linear equations has no, one, or many solutions Make connections among solution methods for systems of linear equations Learn how to determine which solution method might be most efficient for a given system of linear equations

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3 Topic 16: Formulating and solving systems 107 FORMULATING AND SOLVING SYSTEMS Lesson 16.1 Introducing systems of linear equations 16.1 OPENER Do you remember the Bike and Skateboard Problem from earlier this year? The problem and its solution are shown here. Review the problem and the solution. Then complete the following steps. 1. State the two mathematical conditions that must be met to solve this problem. Write the two conditions in your own words. 2. Show or explain why the combination of 15 bikes and 6 skateboards is the correct solution to this problem CORE ACTIVITY Ms. Salinas is in charge of sales for Opportunity Company. Ms. Salinas knows that a particular task will take 8 hours to complete. She has budgeted $80 for this task. The supervisor is paid $15 per hour. Her assistant is paid $7 per hour. The supervisor will start the task, so she can plan and organize it. Then, her assistant will take over and complete the task. Ms. Salinas needs to figure out how long each person should work so the company's costs meet her budget and time estimate. With your partner, determine how long each person should work to meet both of Ms. Salinas' conditions. 1. What are you trying to find in this problem? 2. What facts are given in the description of the situation that you need to solve the problem? (These facts are pertinent information.) 3. What facts are given in the description of the situation that you don t need to solve the problem? (These facts are irrelevant information.) 4. Write, in your own words, the two conditions that must be met to solve this problem. 5. Work with your partner to determine how long each person should work to meet both of Ms. Salinas conditions.

4 108 Unit 6 Systems of linear equations and inequalities Show how you found your solution. (Hint: Consider approaches that you or others used to solve the Bike and Skateboard Problem.) Ms. Salinas realizes she can express these two conditions with two equations. To write the equations, she uses two variables: She uses s to represent the number of hours the supervisor works, and a to represent the number of hours the assistant works. 6. Write an equation to show a relationship between the variables s and a and the total hours worked. 7. Write an equation to show a relationship between the variables and the job s total cost, using each person s hourly pay. Together, the two equations you wrote form a system of equations in two variables. In math, the word system is used to describe a set of two or more equations with two or more variables. 8. In question 5, you solved the Supervisor Problem with your partner. Now, express your solution in two ways: a. In words: b. As an ordered pair (s,a), with the number of hours the supervisor works first, and the number of hours that the assistant works second: 9. Check your solution to the Supervisor Problem: Substitute the ordered pair back into the two equations you wrote, and then evaluate each equation. 10. What do you notice after evaluating both equations? 11. In general, what is a solution to a system of equations in two variables? (Hint: In addition to your solution to the Supervisor Problem, think about the solution to the Opener: Why is 15 bikes and 6 skateboards the solution to that problem?)

5 Topic 16: Formulating and solving systems CONSOLIATION ACTIVITY Practice modeling a situation with a system of equations by taking a closer look at the Bike and Skateboard Problem. Uncle Eddie asked McKenna and Lara to order 54 new wheels for the 21 skateboards and bicycles in his repair shop. How many bicycles and how many skateboards are in Uncle Eddie s shop? 1. To solve this problem, you need to find the values of two different quantities and represent two related conditions in the situation. a. Define variables to represent the quantities you need to find. b. There is one condition in the problem related to the total number of bicycles and skateboards. Express that condition as an equation, using the variables you defined in part a. c. There is another condition in the problem related to the number of wheels. Express that condition an equation, using the variables you defined in part a. d. Use your work from parts b and c to write a system of equations that represents both conditions in this problem. e. In the Opener, you showed that 15 bikes and 6 skateboards satisfied both conditions for the Bike and Skateboard Problem. Show algebraically that 15 bikes and 6 skateboards is a solution to the system of equations that represents the Bike and Skateboard problem from part d. 2. Suppose each system of equations below represents a bike and skateboard scenario similar to the Bike and Skateboard Problem. For each system, describe a problem situation that could be represented by that system. Then determine if the combination of 9 bikes and 8 skateboards is a solution. System of Equations What problem situation is represented? Is the combination of 9 bikes and 8 skateboards a solution? Explain. b+ s= 20 2b+ 4s= 50 b+ s= 17 2b+ 4s= 50 b+ s= 17 2b+ 4s= 48

6 110 Unit 6 Systems of linear equations and inequalities HOMEWORK 16.1 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Justify whether the given ordered pair (x,y) is a solution to the given system of equations. (If you need additional space, record your work on notebook paper.) a. Is (4,6) a solution to the following system of equations? y= x+ 2 y = 10 x b. Is (6,6) a solution to the following system of equations? x+ y= 12 2x = 10 c. Is (0,8) a solution to the following system of equations? 2x = 16 x y = 8 2. Claudia is trying to get the word out about a voter registration rally next week. She wants to use all of her free 90 minutes to make phone calls and send s, and needs to reach 26 people. If it takes her 5 minutes to make a phone call and 3 minutes to personalize an , how many phone calls should she make and how many s should she send? a. What is this question asking you to find? Define the variables in this situation and assign a letter to each one. b. What is the relevant information you need to use to solve this problem? c. Write one equation representing the number of people Claudia reaches with her phone calls and s. d. Write one equation representing the time it will take Claudia to make the phone calls and send the s. e. Why do these two equations make a system of equations? (It may help to write again what a system of equations is.) f. Is (15, 11) a solution to your system of equations? (Does it satisfy the conditions?) Show why or why not. g. Is (6, 20) a solution to your system of equations? (Does it satisfy the conditions?) Show why or why not. h. What does it mean to find a solution to a system of equations?

7 Topic 16: Formulating and solving systems Complete the math journal in the space provided. Concept My understanding of the concept An example that shows the meaning of the concept a. System of equations b. Solution to a system of equations

8 112 Unit 6 Systems of linear equations and inequalities STAYING SHARP 16.1 Practicing algebra skills & concepts 1. Janet sees this problem on a quiz. What is the answer? Answer: (- 6)(- 5)(- 2)(2) =? 2. Joan ran 200 yards in 40 seconds. Bill ran 800 yards in 140 seconds. Who ran at a faster rate? Justify your answer. Answer with supporting work: Preparing for upcoming lessons 3. If 2(4x + 3) 12 = 3x 6, then what value of x makes the equation true? Answer with supporting work: 4. Rewrite the following equation in slope- intercept form: 4x 2y = 6 Answer with supporting work: Reviewing pre- algebra ideas 5. Draw a rectangle that meets both of the following conditions and label the rectangle s length and width: The perimeter of the rectangle is 36 units. The length of the rectangle is 5 times its width. Answer: 6. Draw all three possible rectangles with perimeter of 14 units and length and width that are both whole numbers. Then, circle the rectangle whose length is 2.5 times its width. Answer:

9 Topic 16: Formulating and solving systems 113 Lesson 16.2 Solving systems of linear equations using number sense 16.2 OPENER 1. Use number sense to solve the shape equation puzzle. 2. Rewrite the puzzle as a system of equations using the variables s and t. (Don t forget to define your variables!) 3. Show algebraically that the solution you found in question 1 is the solution for the system of equations you wrote in question CORE ACTIVITY Part I. For each situation, write a system of equations. Then find the solution to the system. 1. A farmer raises chicken and cows. There are 34 animals in all. The farmer counts 110 legs on these animals. How many of each type of animal does the farmer have? a. What are you looking for in this problem? b. Define the variables you will use. i. Let = the number of ii. Let = _the number of c. What conditions are given in the problem? What other pertinent information do you need to solve the problem? d. Write a system of equations to represent the problem. e. Solve the system of equations using whatever method you like (for example, guess- and- check or number sense). Then, check that your solution makes both equations in the system true.

10 114 Unit 6 Systems of linear equations and inequalities 2. You have 14 coins in your pocket that are either quarters or nickels. They total $2.50. How many of each type of coin do you have? a. What is this question asking you to find? b. Define the variables you will use. i. Let = the number of ii. Let = c. What conditions are given in the problem? What other relevant information do you need to use to represent the conditions? d. Write a system of equations to represent the situation in this problem. e. Solve the system of equations using whatever method you like (for example, guess- and- check or number sense). Then, check that your solution makes both equations in the system true. 3. At an ice cream parlor, ice cream cones cost $2.00 and sundaes cost $3.50. One day, the receipts for 114 cones and sundaes total $ How many cones and sundaes were sold? (Assume tax is included in the cost.) a. What is this question asking you to find? b. a. Define the variables you will use and assign letters. i. Let = ii. Let = c. What conditions are given in the problem? Is there d. Write a system of equations to represent the situation in any other relevant information you need to this problem. represent the conditions? e. Solve the system of equations using whatever method you like (for example, guess- and- check or number sense). Then, check that your solution makes both equations in the system true.

11 Topic 16: Formulating and solving systems 115 Part II. Using whatever method you would like, solve each system below. In the last column, check that your solution makes both equations in the system true. System of equations Solution Check 4. x+ y = 10 x y = 2 x = y = 5. y = x+ 4 4x = 12 x = y = 6. x y= 5 xy = 84 x = y = 16.2 REVIEW END-OF-UNIT ASSESSMENT

12 116 Unit 6 Systems of linear equations and inequalities HOMEWORK 16.2 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Think about your performance on the Unit 5 end- of- unit assessment. a. Are you pleased with your performance on the end- of- unit assessment? Circle one: Yes / No b. Does your performance refelct your understanding of the topics in Unit 5? Circle one: Yes / No If you answered No, why do you think this? c. Based on your answers to parts a and b, do you need to revise the goal you wrote at the end of the last unit? If so, write your new goal below along with any enabling goals that will help you reach your goal. 2. Using whatever method you like (for example, guess- and- check or logical thinking), solve each of the following systems. In the last column, check that your solution makes both equations in the system true. a. System of equations Solution Check x+ y = 17 x y = 7 x = y = b. 2x+ 2y= 36 x y = 6 x = y = c. 3x = y x + y = 20 x = y = d. x+ y= 4 xy = 21 x = y =

13 Topic 16: Formulating and solving systems For the following problem situations, choose variables to represent the unknowns and write a system of equations based on the facts in the problem. Then solve the system using whatever method you like. a. A garage contains a combination of 20 bicycles and tricycles. In total, there are 44 wheels. How many bicycles and how many tricycles are in the garage? b. There are 24 questions on a test. Each question is worth either 4 points or 5 points. The total number of points is 100. How many of each type of question are on the test? c. You have 15 coins in your pocket that are either dimes or nickels. The total value of the coins is $1.20. How many dimes and how many nickels do you have? d. The sum of two numbers is 30 and the difference of these numbers is 6. What are the two numbers?

14 118 Unit 6 Systems of linear equations and inequalities STAYING SHARP 16.2 Practicing algebra skills & concepts 1. Mary solved the problem below, but the solution is incorrect. (- 7) (- 4) (2) (- 3) = 42 Explain to Mary why her solution is incorrect, and provide the correct solution. 2. Alex skated 5 meters in 4 seconds. How fast was Alex skating per second? Answer with supporting work: 3. Solve the shape equation puzzle. 4. Graph the line 3x 4y = 12 on the grid provided. Preparing for upcoming lessons 5. The triangles shown below are similar. 6. Fill in the next three terms in the sequence. Reviewing pre- algebra ideas Find the missing side lengths for triangle XYZ. 3, 7, 12, 18,,,... Explain how you determined the missing terms in the sequence.

15 Topic 16: Formulating and solving systems 119 Lesson 16.3 Modeling with systems of linear equations 16.3 OPENER Match each situation in the following table with the variable definitions and equation that describe it. Use the answers choices provided below the table. Situation Variable definitions Equation There are 25 fewer violets than sunflowers in the flower shop window. The number of small pizzas that Mario delivered is 25 less than the number of very large pizzas. Samantha downloaded 25 songs and videos to her MP3 player. Variable definitions s = number of small pizzas v = number of very large pizzas s = number of sunflowers v = number of violets s = number of songs v = number of videos Equations s = v 25 s 25 = v 25 v = s 25v = s s + v = CORE ACTIVITY Part I. A New Bike and Skateboard Problem Uncle Eddie asked McKenna and Lara to order 108 new wheels for the 42 skateboards and bicycles in his repair shop. How many bicycles and how many skateboards are in Uncle Eddie s shop? 1. Record your work and your solution to the problem in the space provided here. 2. Explain how you know that your solution is correct. 3. Think about how you solved this new bike and skateboard problem using a system of equations approach. Then consider how you would describe your problem- solving process to someone else. a. Break your problem- solving process down into small steps. b. Then, list those steps in a logical order.

16 120 Unit 6 Systems of linear equations and inequalities Part II. The Movie and Game Rental Problem The Lupines, a family of werewolves, are heading off for a vacation in the woods of Transylvania. Uncle Harry Lupine has rented a total of 12 horror movie videos and gory games on CDs to amuse the kids, Freddy and Lenore, on the trip. The kids want to know how many games and movies they have. Uncle Harry wants Freddy and Lenore to do some math on the trip, so he tells them, "Games rent for $4.50 each and movies for $3.99 each. The total was $49.92 before tax. Now you can figure out the answer." 4. Step 1: Describe the situation in your own words. What are the important facts? What are you being asked to find? 5. Step 2: Define the variables that will model this situation. 6. Step 3: Write equations to represent the two conditions described by Uncle Harry. 7. Explain why we call this a system of linear equations. Part III. The Swamp Problem On their vacation, the Lupines stay at a hotel that has a rectangular swamp for a swimming pool. Uncle Harry gives Freddy and Lenore a new problem to solve: Suppose the swamp has a perimeter of 124 feet. The length, l, of the swamp is 10 feet less than 5 times its width, w. What are the length and width of the swamp? 8. Write a system of two equations to model this situation. Be sure to define the two variables that you use in your equations.

17 Topic 16: Formulating and solving systems CONSOLIDATION ACTIVITY Identify the two variables in each situation below. Then write a system of two equations to model the situation. Situation 1. A rectangle is 4 times as long as it is wide. The perimeter of a rectangle is 50 cm. What are the dimensions of the rectangle? Identify what you are looking for and assign variables. Write equations to model the situation described in the problem. 2. A farmer grows only corn and lettuce. The farmer plans to plant 455 rows this year. The number of rows of corn will be 2.5 times the number of rows of lettuce. How many rows of each vegetable does the farmer plan to plant? 3. A 100-point test consists of 2-point questions and 5-point questions. There are a total of 44 questions on the test. How many questions of each type are on the test? 4. Maggie and Mia go shopping together. At the Fashion Bee, shirts cost one price and sweaters cost one price. Maggie buys 2 shirts and 2 sweaters for $86. Mia buys 3 shirts and 1 sweater for $81. What is the cost of a shirt and a sweater? 5. The sum of two numbers is 12. The product of those same two numbers is -64. Find the numbers.

18 122 Unit 6 Systems of linear equations and inequalities HOMEWORK 16.3 Notes or additional instructions based on whole- class discussion of homework assignment: Part I: Define variables and create a system of equations to model each situation below. Situation 1. The sum of two numbers is 186. The difference between the same numbers is 32. What are the two numbers? Identify what you are looking for and assign variables Write two equations to model the situation described in the problem 2. The school auditorium seats 310 people. For a particular performance, the number of seats reserved for students is 25 more than twice the amount reserved for adults (faculty, staff, and parents). How many seats are reserved for students? How many seats are reserved for adults? 3. Investment A starts with $1000 and increases its value by $80 each week. Investment B starts with $2000 and loses $50 of its value each week. After how many weeks will the two investments have the same value? 4. Joseph and Patrick purchase school supplies in the school bookstore. Joseph purchases four notebooks and three pens for $ Patrick purchases three notebooks and five pens for $9.50. What is the price of a notebook? What is the price of a pen? 5. An amusement park offers two options. Option 1 involves a $10 admission fee plus $0.50 per ride. Option 2 involves a $6 admission fee plus $0.75 per ride. For how many rides do the two options have the same cost (or, what is the break-even point)? 6. The height of a triangle is 4 inches shorter than its base. The area of the triangle is 198 square inches. Find the dimensions of the triangle.

19 Topic 16: Formulating and solving systems 123 Part II: Using whatever method you would like (for example, guess- and- check or logical thinking), solve each system below. In the last column, check that your solution makes both equations in the system true. Problem Solution Check x = 51 2x+ y= 40 m+ n= 100 m n= 30 a+ b= 6 ab = 40 y = x+ 13 xy = 30 2l+ 2w= 38 l w= 78 x = y = m = n = a = b = x = y = l = w = Part III: Complete these questions. 12. List at least five combinations of nickels and dimes such that the number of nickels is double the number of dimes. 13. List at least five combinations of nickels and dimes such that the total value of the coins is 80 cents. 14. Find a combination of nickels and dimes that meets both of the conditions stated above (in Questions 12 and 13). Explain how you found your answer. 15. Write a system of equations that models the two conditions described in Questions 12 and 13.

20 124 Unit 6 Systems of linear equations and inequalities STAYING SHARP 16.3 Practicing algebra skills & concepts 1. One of the equations below is not correct. Circle the incorrect equation. Then explain the error that was made and write the correct answer. Explanation: ( 14)(20)= = 7 6( 2)(3)( 4)(1) = 144! 5 ( 2)(3)( 1) ( 7) = John is driving across the country. He used 75 gallons of gas in 5 days. If he keeps using gas at the same rate, how many gallons will he use in 8 days? Answer with supporting work: Correction: 3. The graph below shows miles driven vs. hours driven. What does the slope of the line represent? 4. What is the value of - 3x 2 y 3 when x = 2 and y = - 1? Preparing for upcoming lessons Answer: y Distance (miles) Time (hours) x Answer with supporting work: Reviewing pre- algebra ideas 5. A spinner has 5 equal sections labeled A, B, C, B, A. What is the theoretical probability of landing on a B if the spinner is spun once? Answer with supporting work: 6. Four friends go to dinner. Each chooses a different meal. When the check comes, they decide they will each pay the same amount. If the costs of the meals (including tax and tip) were $12.20, $11.00, $8.50, and $7.50, what should each friend pay so they cover the whole bill? Answer with supporting work:

21 Topic 16: Formulating and solving systems 125 Lesson 16.4 Solving systems of equations with tables 16.4 OPENER Sarah is trying to use tables to find the solution to some linear equations. She creates a table for each of the linear equations. 1. Use or adjust her tables to help her find the solution to each equation. a. 3x + 4 = 16 b. 2x + 6 = 11 c. 4x + 2 = 30 d. 5x + 9 = 4 x 3x + 4 x 2x + 6 x 4x + 2 x 5x x = x = x = x = 2. Think about these questions and discuss them with your class: How can a table help you find the solution to an equation? What makes an equation easy to solve with a table? When is an equation not so easy to solve with a table? Could you apply this approach to finding the solution to a system of equations? 16.4 CORE ACTIVITY Part I. The Hiker and Cyclist Problem Ama and Joan plan to hike on the Riverside Trail, from their neighborhood to the lake. But Joan has to work at the library in the morning, so she texts Ama that she will start later and catch up on her bicycle. Joan leaves 4 hours after Ama. Ama hikes at 3 miles per hour, and Joan cycles at 7 miles per hour. When and where will Joan catch up with Ama? 1. Complete the statements and the tables to help you answer this question. Ama s Information Starts after hours and hikes at miles per hour Time from Ama s start (hrs) Distance down trail (miles) Joan s Information Starts after hours and cycles at miles per hour Time from Ama s start (hrs) Distance down trail (miles)

22 126 Unit 6 Systems of linear equations and inequalities 2. At what time are Ama and Joan at the same distance down the Riverside Trail? 3. What is their distance down the trail when they meet? 4. How can you identify the solution by looking at the two tables? 5. Why are the first four rows in the Distance down trail column of Joan s table grayed out? Part II. The Painting Problem Your uncle needs the walls of his storage room painted. He is a smart shopper, so he asks his friends for recommendations of painters. He finds two he thinks will do a good job: Evelyn and Rico. Evelyn charges an initial fee of $80 for any job and $1.20 per square foot. Rico charges no initial fee, but charges $1.90 per square foot. Which painter s deal is better for your uncle? 6. Complete the statements to help you answer this question. a. Evelyn charges a fee of and per square foot. b. Rico charges a fee of and per square foot. 7. Using the variables c for the total cost in dollars and a for the area in square feet, write rules for Evelyn and Rico s deals. a. Evelyn: b. Rico: 8. Use your rules for Evelyn and Rico s deals and your number sense to complete these tables. Area in square feet (a) Evelyn s total cost in dollars (c) Area in square feet (a) Rico s total cost in dollars (c)

23 Topic 16: Formulating and solving systems Use your equations and tables to give your uncle specific advice about which painter to hire. For what wall area would Evelyn s deal be better? For what wall area would Rico s deal be better? For what area would the costs be equal? Explain your reasoning. 10. If you graphed each painter s total cost in dollars as a function of the area in square feet, what would the graphs look like? Part III. The Workout Problem Hans and Franz go to the gym. Hans likes to get right to the exercise cycle, while Franz thinks that stretching first will make his cycling more effective. Hans burns 10 calories per minute on the exercise cycle, starting right away. Franz s stretching takes 10 minutes and doesn t burn any calories, but then his cycling burns 15 calories per minute. Hans and Franz want to find out who will burn more calories in a 40- minute workout; when the two of them will have burned exactly the same number of calories. Answer these questions to help them solve the problem. 11. Using the variables c for the total calories burned and t for the total minutes exercised, write rules for Hans and Franz. a. Hans: b. Franz: 12. Together with your partner, make tables for Hans and Franz on your whiteboards for times from 0 to 40 minutes, counting up by 5 minutes in each row. One of you should create the table for Hans, and the other should create the table for Franz. (Make the rows of your tables the same height on each whiteboard so you can compare them easily.) 13. Do you notice that the input (time) column of both your tables have exactly the same values? Overlap your whiteboards to turn your two tables into one. Can you see the point at which Hans and Franz have burned the same number of calories?

24 128 Unit 6 Systems of linear equations and inequalities 16.4 CONSOLIDATION ACTIVITY Part I. Use tables to investigate two linear functions: y 1 = 3x + 5 and y 2 = 3x Use the first two tables to compute the function values. Then, merge these two tables into a three- column table. x Process y 1 x Process y 2 x y 1 y For what value(s) of x will these two functions have the same value? Explain your reasoning. 3. What does your observation in question 2 mean about the relationship between the graphs of these two functions? 4. How does your conclusion in question 3 relate to their slopes? Part II. Now investigate two more linear functions: y 1 = 2x + 1 and y 2 = 5x + 7. On your whiteboard or on separate paper, make a three- column table using input values (in the first column) of x = 0, x = 1, x = 2, x = 3, and x = What do you notice about the difference between the values of the two functions as x increases? 6. In which direction from x = 2 will the intersection of the graphs of these functions occur? Why? 7. Estimate the x value at which the graphs of these functions will intersect. Part III. Nicholas has short hair, but it grows quickly. The length of his hair in inches is modeled by the function y 1 = x, where x is the time measured in months. Salima s hair is longer, but grows more slowly. The length of her hair in inches is modeled by the function y 2 = x. 8. How can you use tables to figure out when their hair will be the same length? Discuss with your partner and show any related work here.

25 Topic 16: Formulating and solving systems 129 HOMEWORK 16.4 Notes or additional instructions based on whole- class discussion of homework assignment: Part I. Two swimming pools are draining at constant rates. The draining of the pools begins at the same time. The above- ground pool is 72 inches deep, and is draining so that the water depth decreases at 4 inches per hour. The in- ground pool is 92 inches deep, but is being pumped out so the water depth is decreasing at 8 inches per hour. Does the water in the two pools ever have the same depth at the same point in time? If so, when does this happen, and what is the depth? 1. To answer these questions, start by completing these statements and tables The above- ground pool has a depth of inches and is draining at a rate of inches per hour. Time (hours) Depth (inches) The in- ground pool has a depth of inches and is draining at a rate of inches per hour. Time (hours) Depth (inches) Using the variables d for the depth in inches and t for the time in minutes, write a rule for each pool. a. Above- ground pool: b. In- ground pool: 3. When do the two pools have the same depth? What is the depth when this happens? 4. When does each pool become empty?

26 130 Unit 6 Systems of linear equations and inequalities Part II. In this part, you will read two statements about solutions to linear systems of equations with tables. Each statement contains a mistake. First, identify and explain the mistake. Then make a correct statement based on the tables provided. 5. Fred makes these two tables to solve a system of equations. He says, I see y values of 11 in both tables, so the solution to the system must be y = 11. x y x y a. What is the mistake in Fred s statement? Explain. b. Make a correct statement about the solution of the system represented by these two tables. 6. Rachel makes these two tables to solve a system of equations problem that asks her where lines A and B intersect. She says, Line A is always above line B, so the lines never intersect. They must be parallel. Line A Line B x y x y a. What is the mistake in Rachel s statement? Explain. b. Make a correct statement about what these two tables can tell you about Line A and line B.

27 Topic 16: Formulating and solving systems 131 STAYING SHARP 16.4 Practicing algebra skills & concepts 1. On a recent test, Edward incorrectly wrote: 4(x + 3) = 4x + 3 a. Pick a value of x. Show that it does not make the equation true. b. Identify or explain Edward s mistake. (Hint: What property did Edward apply incorrectly in this statement?) 2. Victoria wants to figure out if she can run faster than an elephant. She reads that an elephant can run 100 yards in 20 seconds. She knows she can run 60 feet in 10 seconds. Victoria says, Since = 6 and = 5, my speed is faster than an elephant s! Does her statement make sense? Explain why or why not. Answer and explanation: Preparing for upcoming lessons 3. Solve for x in the equation 2 x + 1 = Answer with supporting work: 4. Find the value of the following expression when a = - 2, b = 3, and c = - 5: a 2 + bc 7a Answer with supporting work: 5. Your last bag of Halloween candy contains 3 fruit candies, 2 chocolates, 5 caramels, and 6 sour chews. If you pick a candy at random, what is the probability it will be caramel or chocolate? 6. A kennel at a dog rescue has dogs with the following weights in pounds: {11, 11, 11, 11, 11, 11, 16, 100} Reviewing pre- algebra ideas Answer: Find the mean and mode of the dogs weights. If a kennel volunteer is trying to decide how to describe the dogs weights to someone who might adopt a dog, which one makes more sense: the mean or the mode? Answer and explanation:

28 132 Unit 6 Systems of linear equations and inequalities

29 Topic 16: Formulating and solving systems 133 Lesson 16.5 Solving other systems of equations with tables 16.5 OPENER In a previous lesson, you created a process of modeling situations with systems of equations. You wrote equations using the variables p (number of pencils) and b (number of ballpoint pens) to represent the two conditions in this problem: Pencils cost 25 cents and ballpoint pens cost 75 cents. A total of 9 writing instruments were purchased. If the total cost for the writing instruments was $3.75, find the number of pencils and ballpoint pens purchased. Now, solve the system of equations by using tables. 1. Fill in this table of combinations meeting the first condition. Equation 1 (from information about number of writing instruments purchased): p + b = 9 p b To save you time, the table for the other condition is provided. Answer parts a and b below the table. Equation 2 (from information about price and total cost of instruments purchased):.25p +.75b = 3.75 p b a. Verify that (3,4) is a solution to this equation. b. Pick another combination from the table and verify that it also has a cost of $ Explain how you can use the tables to find a solution to the system of equations CORE ACTIVITY Part I. Unpacking the Opener Rewriting equations in y = form can help you solve a system of equations. 1. Rewrite the equation p + b = 9 to show b by itself. 2. In comparison to rewriting the equation shown in question 1, would it be easier or harder to rewrite the equation 0.25p b = 3.75 to show b by itself? Explain. Part II. Solving the Movie and Game Rental Problem In Lesson 16.3 you completed steps 1 3 to model the situation in the Movie and Game Rental Problem with a system of equations. Now you will complete steps 4 6 to solve the problem.

30 134 Unit 6 Systems of linear equations and inequalities You developed the following system of equations to model the Lupines situation: Let g = number of games rented, and let m = number of movies rented. g+ m= g+ 3.99m= Now you are ready for the next steps in solving the system of equations. Step 4: Solve the system of equations 3. Generate a table of combinations for the simpler equation, g + m = 12. Then test the combinations in the other equation (4.50g m = 49.92) to find the combination that satisfies both conditions. This combination is a solution to the system. (Hints: Test in whatever order makes sense based on your results. You can stop once you have found the solution!) g m Test: 4.50g m The solution to the system of equations is: g = m = or (, ) Step 5: Check the solution in both equations 5. Check the solution by substituting the values into each equation in the system. Step 6: Write your answer in a sentence 6. Write the solution to the movie and game rental situation in a sentence.

31 Topic 16: Formulating and solving systems 135 Part III. Solving the Swamp Problem Now return to the Swamp Problem and complete the entire six- step process to solve the problem. On their vacation, the Lupines stay at a hotel that has a rectangular swamp for a swimming pool. The swamp has a perimeter of 124 feet. The length, l, of the swamp is 10 feet less than 5 times its width, w. What are the length and width of the swamp? Step 1. Read the problem carefully and understand the situation. 7. Step 2. Identify what you are looking for and assign variables Let l represent and w represent. 8. Step 3. Write two equations to model the situation described in the problem. 9. Step 4. Solve the system with a table: Generate combinations with the simpler equation, then test them in the other equation to find the combination that satisfies both conditions. W l Test 10. The solution to the system of equations is: l = w = or (, ) 11. Step 5. Check the solution in both equations. (How can you use your previous work on this problem to verify the solution?) 12. Step 6. Write the solution in a sentence. Part IV. The Pool Problem When the Lupine kids come home from vacation, they miss swimming in the swamp. Luckily, they find that there is a public pool near their home! This pool has a perimeter of 58 feet. The pool s length, l, is 4 feet more than twice its width, w. Help them find the length and width of the pool. 13. Write two equations to model the situation described in this pool problem. 14. Examine the table in the animation. Where is the solution in the table? 15. Consider your answer to question 14. Does this mean the system doesn t have a solution? 16. What conclusion can you draw from the table about the solution to this new system of equations? 17. What do you think about using tables to solve systems of equations in general?

32 136 Unit 6 Systems of linear equations and inequalities 16.5 CONSOLIDATION ACTIVITY Part I. Rewrite each equation in y= form using your algebraic skills. 1. 6x + y = x + 2y = x y = x 2y = Graph the equations you wrote for questions 1 4 with the graphing calculator. a. What do you notice about the graphs? b. When a linear equation in standard form has x and y coefficients with the same sign, which way does the line slope? c. When a linear equation in standard form has x and y coefficients with opposite signs, which way does the line slope? Again, rewrite the following equations in y= form. 6. 3x + 5y = x + 7y = 9 Part II. Graph the following equations on the coordinate grid provided. Pick an efficient method for graphing each equation. Since you will graph all of the equations on the same coordinate grid, label each equation when you graph it. 8. y = 2x 3 graphing method: 10 y x + 3y = 12 graphing method: y = x + 6 graphing method: x x + 3y = 10 graphing method:

33 Topic 16: Formulating and solving systems 137 HOMEWORK 16.5 Notes or additional instructions based on whole- class discussion of homework assignment: Complete all the steps of the process to solve these systems of equations problems. Show all of your work for each problem. If you need additional space, use notebook paper. In Lesson 16.3 you modeled the situations in these problems with systems of equations. Use that work to help you complete Steps 2 and 3. In Step 4, use a table to solve the system. 1. A rectangle is 4 times as long as it is wide. The perimeter of a rectangle is 50 cm. What are the dimensions of the rectangle? 2. A farmer grows only corn and lettuce. The farmer plans to plant 455 rows this year. The number of rows of corn will be 2.5 times the number of rows of lettuce. How many rows of each vegetable does the farmer plan to plant? 3. A 100- point test consists of 2- point questions and 5- point questions. There are a total of 44 questions on the test. How many questions of each type are on the test? 4. Maggie and Mia go shopping together. At the Fashion Bee, all shirts cost one price and all sweaters cost one price. Maggie buys 2 shirts and 2 sweaters for $86. Mia buys 3 shirts and 1 sweater for $81. What is the cost of a shirt and a sweater? 5. The sum of two numbers is 12. The product of those same two numbers is Find the numbers.

34 138 Unit 6 Systems of linear equations and inequalities STAYING SHARP 16.5 Practicing algebra skills & concepts 1. Maria and Katia disagree over the value of the expression Maria thinks the value is 36, but Katia thinks that answer is Who is correct? Explain why. Answer: Explanation: 2. The graph shows the relationship between the depth of snow in inches measured at a weather station and the time it has been snowing. During what one- hour time period did the greatest amount of snow fall? Answer: How do you know?: Preparing for upcoming lessons 3. If 3x 4 = - 25, then x =? Answer with supporting work: 4. If x = 6 and y = - 2, what is the value of the following expression? 3x + xy + xy 2 + xy Answer with supporting work: 5. A coin is flipped and a spinner with three equal sections labeled A, B, and C is spun. What is the probability that the coin will come up heads AND the spinner lands on C? 6. In the dot plot shown, each dot represents a data point. What is the median for this set of data? Reviewing pre- algebra ideas Answer with supporting work: Answer: Explanation

35 Topic 16: Formulating and solving systems 139 Lesson 16.6 Using graphs to solve systems of equations 16.6 OPENER Sasha is trying to solve the following problem: The Botany Club wants to fence off a new garden. The perimeter of the garden is 25 feet. The length of the garden is twice its width. What are the dimensions of the garden? Sasha defines variables and sets up a system of equations. Let l represent the length of the garden, and w represent the width of the garden. l = 2w 2 l + 2w = 25 Then she makes a table of combinations satisfying the l = 2w condition, and tests each combination with the other. w l Test for 2 l + 2w = 25 condition 1 2 2(1) + 2(2) = (2) + 2(4) = (3) + 2(6) = (4) + 2(8) = (5) + 2(10) = 30 Now, I m stuck! says Sasha. I know the solution is the pair of numbers that gives a perimeter of 25, but I don t see any pair that gives me a perimeter of 25! Does this mean there is no solution for this system? What should I do now? 1. Show how Sasha can finish solving this problem. Find a solution. 2. What challenges did you encounter as you used the table to solve the problem? 16.6 CORE ACTIVITY Think about the problems that you have solved by building tables. You may have found that using tables was more challenging for some systems than for others. 1. What drawbacks do you see to using a table to solve a system of equations? The Roses Problem Remember the Roses Problem? You compared offers from Roses- R- Red and Flower Power to decide which flower shop would give the best deal for a Valentine s Day fundraiser. You used tables, graphs, and algebraic rules to explore each shop s offer. 2. What quantities did you compare as you explored each offer? It turns out that you can explore the two offers together by thinking of them as a single system of equations. Here are the facts of the problem. Roses- R- Red charges $20 plus 75 per rose. Flower Power charges $60 plus 50 per rose. 3. Define variables and write a system of equations to represent the facts of the Roses Problem.

36 140 Unit 6 Systems of linear equations and inequalities Here is a graph that shows both offers on the same set of axes. 4. At what point do the two lines intersect? Cost of Roses from Roses-R-Red and Flower Power c What does the intersection point mean in the context of the problem situation? Cost (dollars) 60 Roses-R-Red Flower Power Number of roses n Here is a single table that includes costs for both offers. Compare the table to the graph of the offers. 6. How does each representation give you similar information about the offers? 7. What information do you see in the graph that you do not see in the table? The Swamp Problem Recall the Swamp Problem from earlier in this topic: Number of roses Roses- R- Red Flower Power 1 $20.75 $ $21.50 $ $23.75 $ $27.50 $ $95.00 $ $ $ $ $ The swamp that the Lupines use as a swimming pool on vacation has a perimeter of 124 feet. The length, l, of the swamp is 10 feet less than 5 times its width, w. What are the dimensions of the swamp? 8. Explain how you would solve this system using a graph. 9. Define variables and write a system of equations to represent the situation. 10. Solve this system by graphing and write your solution. How does it compare to the solution you found when you solved the system using a table?

37 Topic 16: Formulating and solving systems CONSOLIDATION ACTIVITY Solve the following systems of equations by graphing. Graph the equations on the coordinate grid provided and then report the solution. Check that your solution is correct. 1. y = 3x 9 x + 2y = 10 Solution (, ) Check: 2. y = - x + 30 y = - 2x + 15 Solution (, ) Check: 3. n = 3m + 12 m + 3n = - 9 Solution (, ) Check: 4. How did you graph the systems of equations in questions 1-3? Did you make a table of values, or use another method? 5. What challenges did you encounter as you used graphs to solve the systems problems?

38 142 Unit 6 Systems of linear equations and inequalities HOMEWORK 16.6 Notes or additional instructions based on whole- class discussion of homework assignment: Solve the following systems of equations by graphing. Graph the system and report the solution. Then, check to verify that your solution is correct. 2. y = x 1. y = 5x 2 y = x + 6 Solution (, ) Check: y = 3x + 2 Solution (, ) Check: 3. y = 4x + 7 y = - 3x 4. y = - x + 3 y = - 2x + 8 Solution (, ) Check: Solution (, ) Check: 5. y + 3 = 4x y = - 3x 3 6. x y = 1 y = 3 4 x + 1 Solution (, ) Check: Solution (, ) Check:

39 Topic 16: Formulating and solving systems You worked on the following problem earlier in the unit. Now, use the six- step process to solve the problem. You will need to set up a system of equations. Use a graphing approach to solve the system. You have 14 coins in your pocket that are either quarters or nickels. The total value of the coins is $2.50. How many of each type of coin do you have?

40 144 Unit 6 Systems of linear equations and inequalities STAYING SHARP 16.6 Practicing algebra skills & concepts 1. Find the value of each expression. a Answer with supporting work: b Answer with supporting work: 2. Greg swims 100 yards in 52 seconds. Kelly plans to swim 150 yards at the same rate. How many seconds will it take her to complete her swim? Answer with supporting work: Preparing for upcoming lessons 3. If 2x 4 = - 16, then x =? Answer with supporting work: 4. Simplify this expression: - 2x 3 x 2 (3x 3 2x 2 ) Answer with supporting work: Reviewing pre- algebra ideas 5. Mary wants to make the softball team. The coach chooses players by giving each player three scores, for catching, throwing, and batting. Players must earn a mean score of 80 or greater to make the team. Mary scores 78 for catching and 75 for throwing. What is the lowest batting score she could get and still make the team? Answer with supporting work: 6. During each of the past three months, Tyler grew by the following amounts: 1 8 in., 3 16 in., and 1 4 in. What is Tyler s total amount of growth over these three months? Answer with supporting work:

41 Topic 16: Formulating and solving systems 145 Lesson 16.7 Using graphs to solve systems of equations, continued 16.7 OPENER Use this graph below to answer the following questions 1. Provide an interpretation of the graph. In other words, what story does the graph tell? 2. At what point do the two lines intersect on the graph? 3. What is the meaning of the intersection point in terms of the situation depicted by the graph? 4. Write a system of two linear equations to represent the situation depicted by the graph CORE ACTIVITY 1. On your calculator, graph the system of equations you wrote to model the Swamp Problem. How should you rewrite these equations so that they can be entered into the calculator? l = 5w 10 2l + 2w = Explore how you can use your calculator's built- in capabilities to find the intersection point. Then report the intersection point. 3. Solve the following system of equations by graphing, using your graphing calculator s built- in capabilities. Sketch the graph and state the solution. 2y 2 = x y = - 2x + 2 Solution (, )

42 146 Unit 6 Systems of linear equations and inequalities 4. What system of equations is graphed here? Write the system below. Equation for line l: Equation for line m: 5. By looking at the graph in question 4, what do you think the solution is? (, ) 6. Use your graphing calculator to verify that your solution is correct. Have your partner verify that you have obtained the intersection point, then have your partner sign his or her name below. I verify that my partner obtained the intersection point (1,2) on the graphing calculator. Signature 7. What are some advantages of graphing to solve a system of linear equations instead of using tables? 16.7 CONSOLIDATION ACTIVITY 1. Desmond lost his record sheet for his second week of work. But he remembers that he received $240 for mowing 12 lawns that week. In the neighborhood where Desmond and his customers live, the houses are built on lots that come in two sizes: standard- sized interior lots and larger corner lots. Desmond charges $15 per standard- sized lot and $30 per large corner lot for his mowing services. Let x represent the number of standard- sized lots mowed that week and y represents the number of large corner lots mowed that week. Create a system of two linear equations to model this situation x + 30y 12 x + y Equation 1: Equation 2:

43 Topic 16: Formulating and solving systems You found a system of equations to represent Desmond s missing records situation. a. How is this system different from the systems of equations you solved in the last lesson? b. Will this difference prevent you from using a table and/or a graph to solve the system? c. How do you know whether to use slope- intercept form or standard form for the equations in a system of linear equations? 3. Sketch the graph of each equation. x + y = 12 15x + 30y = As you saw in both the table and graph, the solution to the system is the point (8,4). What does this solution tell you about Desmond s missing records problem? How many lots of each size were mowed during the week? 5. Suppose Desmond charges $20 for the standard- sized interior lots and $25 for the larger corner lots. He received a total of $300 for mowing 14 lots during the past week. Let x represent the number of interior lots mowed and y represent the number of corner lots mowed. Set up a system of two linear equations to model this situation. Then use your graphing calculator to help you determine how many lots of each type Desmond mowed during the week.

44 148 Unit 6 Systems of linear equations and inequalities HOMEWORK 16.7 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Complete the triple- entry log below. In the first column are listed the three main methods you have learned so far for solving systems of equations. In the middle column, describe (in your own words) how to use each method to solve a system of equations. Describe important things to keep in mind when using that method. In the last column, provide an example, with all of the steps, that shows how to use that method. A method for solving systems of equations Guess and check/logic A description of how to use that method Example showing how that method is used Tables Graphs 2. Two worked examples of solutions to linear systems of equations are provided. The work for each problem contains a mistake. First, identify the mistake. Then solve the problem correctly, using the same method that was used in the original problem. a. Solve the following system using tables. 1 2 y = x y = x + 1 x y -2-1 x y -2-1 Point common to both tables/solution: (- 2,- 1)

45 Topic 16: Formulating and solving systems 149 b. Solve the following system using graphs. y = 2x + 6 y + x = 3 à y = 2x + 6 y = x y y = x + 3 Intersection point/solution: (- 3,0) y = 2x x You worked on the following problem earlier in the unit. Now, use the six- step process to solve the problem. You will need to set up a system of equations. Use a graphing approach to solve the system. There are 24 questions on a test. Each question is worth either 4 points or 5 points. The total number of points is 100. How many of each type of question are on the test?

46 150 Unit 6 Systems of linear equations and inequalities STAYING SHARP 16.7 Practicing algebra skills & concepts 1. Find the value of each expression. a = b = c. (- 2) = d. 2-3 (- 3) 2 = 2. In which quadrant of the standard (x,y) coordinate plane is the point (2,- 4) located? Answer: Preparing for upcoming lessons 3. In the standard coordinate plane, what is the slope of the line passing through the points (- 1,1) and (3,4)? Answer with supporting work: 4. If x = 3 and y = - 2, find the value of the following expression: xy 2 + x 2 2y Answer with supporting work: 5. Mr. Gonzales recorded the height in centimeters of nine students in his class: 6. What is the perimeter of the rectangle shown? Reviewing pre- algebra ideas 147.3, 147.3, 147.3, 152.4, 156.4, 157.4, 160.0, 165.1, Then he found the median height. Now a new student has joined the class. The student is centimeters tall. Describe how this will change the median. (Will it increase, decrease, or stay the same?) Description and explanation: in in. Answer with supporting work:

47 Topic 16: Formulating and solving systems 151 Lesson 16.8 Systems of inequalities 16.8 OPENER 1. Graph each inequality on the coordinate plane provided. a. y 5 b. x - 2 c. y 2x 3 2. What do you think a graph of the compound inequality y 5 and x - 2 and y 2x 3 would look like? Explain CORE ACTIVITY 1. Graph the solution region for the system of inequalities: y 5 x 2 y 2x 3

48 152 Unit 6 Systems of linear equations and inequalities 2. Use the graph provided to answer the questions. a. Write an equation of line r. b. Write an equation of line s. c. Write the two inequalities that form the shaded region A. d. Write the two inequalities that form the shaded region B. e. Write the two inequalities that form the shaded region C. 3. Recall the Snack Bar Problem: Suppose you and some friends go to the movies and buy some snacks. The snack bar charges $2 for a box of candy and $6 for the combo. The combo is a medium drink and popcorn. The only spending restriction you have is you must bring home some change from the money you have been given. After buying the tickets, you have $12 left to spend for snacks. Look at the graph used to model the Snack Bar Problem. a. What three inequalities produce this graph? b. What does each of these inequalities mean in the context of the problem? c. What does the shaded region mean in the context of the problem? 16.8 ONLINE ASSESSMENT Today you will take an online assessment.

49 Topic 16: Formulating and solving systems 153 HOMEWORK 16.8 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Graph the systems of inequalities on the coordinate planes provided. Identify the solution region (the part of the graph where the two inequalities overlap) by shading in the region. a. y < 8 y 3x 1 b. x 4 y > 2x + 2 c. y < 4x 8 y 2x 2. Graph the following systems of inequalities in the plane. Identify the solution region (the part of the graph where all three inequalities overlap) by shading in the region. a. x 1 y 0 2x+ y 6 b. y < x y < 2x y > 6

50 154 Unit 6 Systems of linear equations and inequalities STAYING SHARP 16.8 The graph shows the inequality y > 3 2 x Practicing algebra skills & concepts 1. Does point A satisfy the inequality? Explain your answer. 2. Does point B satisfy the inequality? Explain your answer. 3. Graph the function rules y = 8 x and y = 5 x on the coordinate plane. Preparing for upcoming lessons 4. Does the graph have an intersection point? If so, state the intersection point. If not, explain why there is no intersection point. Focus skill: Slope and geometric connections 5. Graph the function rule y = - 2x + 7 on the coordinate grid. 6. Use the graph you made in question 5 to solve the equation 9 = - 2x + 7 for x. Answer with supporting work:

51 Topic 16: Formulating and solving systems 155 Lesson 16.9 Modeling with systems of inequalities 16.9 OPENER Desmond needs to schedule the lawn mowing jobs each week so that he and Shelly can do all of the mowing and edging that is required. What are some issues that Desmond needs to consider? 16.9 CORE ACTIVITY Based on their previous experience, Desmond and Shelly come up with the following time estimates: Standard- sized interior lot: 1 hour to mow and a half hour to edge Larger corner lot: 2 hours to mow and 45 minutes to edge Desmond can spend at most 30 hours a week mowing lawns. Shelly can only spend at most 12 hours per week edging. 1. How can Desmond model this information as a system of two linear inequalities? Standard- sized yards Large yards Constraints Number of Yards Number of mowing hours per yard Number of edging hours per yard System of inequalities: 2. Determine whether the following combinations of sizes of lawns represent feasible numbers of lawns to mow and edge in any one week, given the time constraints represented by the system of inequalities. Explain your conclusions in terms of the amount of time Desmond and Shelly will work in each case. a. 10 standard- size interior lawns and 8 large corner lawns b. 6 standard- size interior lawns and 15 large corner lawns c. 6 standard- size interior lawns and 12 large corner lawns d. 18 standard- size lawns and 6 large corner lawns

52 156 Unit 6 Systems of linear equations and inequalities 3. Sketch the graph of x + 2y Sketch the graph of x + y Sketch the graph of the system x + 2y x + y Desmond makes two observations. a. First, Desmond notices that because neither variable can be negative, his problem is really modeled by a system of four inequalities, not just two inequalities. Write these two new inequalities, along with the original two inequalities, to show the complete system of four inequalities to which Desmond is referring. 1 3 b. Desmond also notices that the point (10,8) is below both of the lines, x + 2y 30 and x + y 12. He tells Shelly 2 4 that the location of this point agrees with the observation he made earlier about the mowing schedule that point represents. Can you explain what Desmond means?

53 Topic 16: Formulating and solving systems Complete the statements to explain the other points representing combinations of sizes of lawns. Use the answer choices provided. Shelly above both below Desmond neither on The point (6,15) falls both lines. This corresponds to Desmond nor Shelly having adequate time required for mowing and edging 6 interior lots and 15 corner lots. The point (6,12) falls both lines. This corrresponds to Desmond and Shelly having exactly the time required for mowing and edging 6 interior lots and 12 corner lots. The point (18,6) falls the line x + 2y = 30 and the line 1! 2 x y = 12. This corresponds to having exactly the time required to mow, but having less than the time required to edge, for a weekly schedule of 18 interior lots and 6 corner lots. 8. Desmond and Shelly decide to allocate more time per week to their respective tasks. Desmond increases his mowing time to a maximum of 36 hours per week. Shelly increases her edging time to a maximum of 15 hours per week. Write a system of inequalities that represents the new constraints. Graph the system of inequalities, shading the solution set. Are the mowing schedules represented by the points (6,15) and (18,6) now feasible?

54 158 Unit 6 Systems of linear equations and inequalities 16.9 REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions. Problems we did well on: Skills and/or concepts that are addressed in these problems: Problems we did not do well on: Skills and/or concepts that are addressed in these problems: Addressing areas of incomplete understanding Use this page and notebook paper to take notes and re- work particular online assessment problems that your class identifies. Problem # Work for problem: Problem # Work for problem: Problem # Work for problem:

55 Topic 16: Formulating and solving systems 159 HOMEWORK 16.9 Notes or additional instructions based on whole- class discussion of homework assignment: Next class period, you will take a mid- unit assessment. One good study strategy to prepare for tests is to review the important topics, skills, and ideas you have learned. Here is a list of some of the important skills and ideas that you have worked on in this topic. Use this list to help you review these skills and concepts, especially by looking at your course materials. Another good study strategy to prepare for tests is to re- work problems that you did in class. Some specific activities to re- work are listed. Important skills and concepts from the topic: Identifying the variables and conditions in a situation and writing a system of equations Understanding the meaning of a solution of a system of equations and verifying a solution Solving a system of equations using informal methods (number sense, logical thinking, guess and check Solving a system of equations using two tables; using tables by generating and testing combinations Review: graphing review, rewriting equations in y= form Solving a system of equations using graphs by hand Solving a system of equations using graphs on the graphing calculator Writing and solving a system of inequalities by graphing by hand Part I: Study for the mid- unit assessment by reviewing the key topic ideas listed above. Part II: Complete the online More practice for the topic Formulating and solving systems. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study. Part III: Complete Staying Sharp 16.9.

56 160 Unit 6 Systems of linear equations and inequalities STAYING SHARP Find the value of each of the following expressions. Show your supporting work in each case. 2. If there are 15 boys in a class of 25 students, what is the ratio of girls to boys in the class? Practicing algebra skills & concepts a b Answer: Preparing for upcoming lessons 3. Solve for x: 1! 4 x = 5! 4 x 1 Answer with supporting work: 4. Express the following equation in terms of a: F = m a Answer with supporting work: 5. Melinda takes readings of the depth of water in a pond every day for 1 week. Here are her results in meters: 6. Using this chart, what is the total time, in hours, of biking for the five different legs of the trip? Reviewing pre- algebra ideas 6.5, 6.8, 7.2, 6.8, 6.6, 6.3, 6.4 What are the mean, median, and mode water levels for the week, rounded to the nearest tenth? Answer with supporting work: Answer with supporting work: