Pathways Algebra II MODULE 1 FUNCTIONS AND CONSTANT

Pathways Algebra II MODULE 1 FUNCTIONS AND CONSTANT RATE OF CHANGE Part 2: Investigations #6 -10 Investigation #6: Introduction to Systems of Equations Investigation #7: Solving Systems of Equations Using the Substitution Method Investigation #8: The Elimination Method Investigation #9: Possibilities for Solutions to a System of Linear Functions Investigation #10: Three Variable Linear Systems © 2017 CARLSON & O’BRYAN 1

Pathways Algebra II Investigation 6 INTRODUCTION TO SYSTEMS OFEQUATIONS © 2017 CARLSON & O’BRYAN

Pathways Algebra II For Exercises #1 -3, use the following context. Tim and Joni left their home in different cars headed for Denver along the same route. Tim left at 2: 00 pm and traveled at a constant rate of 60 miles per hour. Joni left at 3: 00 pm and traveled at a constant rate of 72 miles per hour. © 2017 CARLSON & O’BRYAN Inv 1. 6 3

Pathways Algebra II 1. a. Fill in the following table. Clock time: 2: 00 pm 3: 00 pm 4: 30 pm 6: 45 pm 7: 30 pm 8: 00 pm 8: 15 pm Number Tim's Joni's of hours distance since Tim since Joni from home hit the (in miles) road © 2017 CARLSON & O’BRYAN Distance between them (in miles) Inv 1. 6 4

Pathways Algebra II 1. a. Fill in the following table. Clock time: 2: 00 pm 3: 00 pm 4: 30 pm 6: 45 pm 7: 30 pm 8: 00 pm 8: 15 pm Number Tim's Joni's Distance of hours distance between since Tim since Joni from home them (in hit the (in miles) road 0 N/A 0 unknown 1 0 60 0 60 2. 5 150 108 42 4. 75 3. 75 285 270 15 5. 5 4. 5 330 324 6 6 5 360 0 6. 25 5. 25 375 378 3* © 2017 CARLSON & O’BRYAN Inv 1. 6 5

Pathways Algebra II b. Define variable(s) as needed and define a function g to represent the distance Tim has traveled in terms of the number of hours t that have passed since 2: 00 pm. Let M represent Tim’s distance from home (in miles) g(t) = 60 t where M = g(t) c. Define variable(s) as needed and define a function h to represent the distance Joni has traveled in terms of the number of hours t that have passed since 2: 00 pm. Let J represent Joni’s distance from home (in miles) h(t) = 72(t – 1) where J = h(t) © 2017 CARLSON & O’BRYAN Inv 1. 6 6

Pathways Algebra II 2. Circle any of the following statements that are true. For each statement you circle, explain what the statement means in the context of this situation. a. g(2. 5) < h(2. 5) g(2. 5) = h(2. 5) g(2. 5) > h(2. 5) The circled statement says that Tim’s distance from home in miles 2. 5 hours since 2: 00 p. m. (at 4: 30 p. m. ) is greater than Joni’s distance from home in miles 2. 5 hours since 2: 00 p. m. (at 4: 30 p. m. ). b. g(6) < h(6) g(6) = h(6) g(6) > h(6) The circled statement says that Tim’s distance from home in miles 6 hours since 2: 00 p. m. (at 8: 00 p. m. ) is the same as Joni’s distance from home in miles 6 hours since 2: 00 p. m. (at 8: 00 p. m. ). © 2017 CARLSON & O’BRYAN Inv 1. 6 7

Pathways Algebra II 2. Circle any of the following statements that are true. For each statement you circle, explain what the statement means in the context of this situation. c. g(6. 25) < h(6. 25) g(6. 25) = h(6. 25) g(6. 25) > h(6. 25) The circled statement says that Tim’s distance from home in miles 6. 25 hours since 2: 00 p. m. (at 8: 15 p. m. ) is less than Joni’s distance from home in miles 6. 25 hours since 2: 00 p. m. (at 8: 15 p. m. ). © 2017 CARLSON & O’BRYAN Inv 1. 6 8

Pathways Algebra II 3. a. Graph g and h on the given axes. © 2017 CARLSON & O’BRYAN Since we don’t have information about Joni’s distance from home prior to 3: 00 p. m. , the graph of h begins at (1, 0). Inv 1. 6 10

Pathways Algebra II b. Use your graph to determine whether each of the following statements is true or false. i) g(4) < h(4) For part (i), we determine which function has a greater output value when the input value is 4. We find that g(4) > h(4), so the given statement is false. See graph. © 2017 CARLSON & O’BRYAN Inv 1. 6 11

Pathways Algebra II b. Use your graph to determine whether each of the following statements is true or false. ii) h(5. 25) < g(5. 25) For part (ii), we determine which function has a greater output value when the input value is 5. 25. We find that g(5. 25) > h(5. 25), so the given statement is true. See graph. © 2017 CARLSON & O’BRYAN Inv 1. 6 12

Pathways Algebra II b. Use your graph to determine whether each of the following statements is true or false. iii) g(6) = h(6) For part (iii) we determine if the output value is the same in both functions when the input value is 6. g(6) = h(6) so the statement is true. © 2017 CARLSON & O’BRYAN Inv 1. 6 13

Pathways Algebra II c. For what values of t is Tim ahead of Joni? That is, solve the inequality g(t) > h(t) for t. From the table of values and the graph we can say that g(t) > h(t) when 0 ≤ t < 6. We can also solve the inequality. Since we don’t have information about their distances from home prior to 2: 00 p. m. , we should restrict the interval to 0 ≤ t < 6. © 2017 CARLSON & O’BRYAN Inv 1. 6 14

Pathways Algebra II 4. Kim and Allison are both in a 100 -mile bike race. At the halfway checkpoint Kim trails Allison by 10 miles. Kim still has a lot of energy left and rides at a constant speed of 15 miles per hour for the rest of the race. Allison, on the other hand, is getting tired and is riding at a constant rate of 11 miles per hour for the rest of the race. a. Make a drawing of this situation and define variables to represent relevant varying quantities. © 2017 CARLSON & O’BRYAN Inv 1. 6 15

Pathways Algebra II a. Make a drawing of this situation and define variables to represent relevant varying quantities. • Let K represent Kim’s distance from the halfway point of the race in miles • Let A represent Allison’s distance from the halfway point of the race in miles • Let t represent the elapsed time since Kim passed the halfway point of the race in hours. © 2017 CARLSON & O’BRYAN Inv 1. 6 16

Pathways Algebra II 4. Kim and Allison are both in a 100 -mile bike race. At the halfway checkpoint Kim trails Allison by 10 miles. Kim still has a lot of energy left and rides at a constant speed of 15 miles per hour for the rest of the race. Allison, on the other hand, is getting tired and is riding at a constant rate of 11 miles per hour for the rest of the race. b. Define a function to represent Kim’s distance beyond the halfway checkpoint in terms of the number of hours since she passed the checkpoint. © 2017 CARLSON & O’BRYAN Inv 1. 6 17

Pathways Algebra II 4. Kim and Allison are both in a 100 -mile bike race. At the halfway checkpoint Kim trails Allison by 10 miles. Kim still has a lot of energy left and rides at a constant speed of 15 miles per hour for the rest of the race. Allison, on the other hand, is getting tired and is riding at a constant rate of 11 miles per hour for the rest of the race. c. Define a function to represent Allison’s distance beyond the halfway checkpoint in terms of the number of hours since Kim passed the halfway checkpoint. (Hint: Keep in mind that when Kim passes the halfway checkpoint, Allison is 10 miles beyond the halfway checkpoint. ) © 2017 CARLSON & O’BRYAN Inv 1. 6 18

Pathways Algebra II d. Graph both functions on the same axes. © 2017 CARLSON

Pathways Algebra II e. What does the point of intersection of the graphs in part (d) represent? It represents the point where the number of hours since Kim passed the checkpoint and the distance from the halfway checkpoint of the race (in miles) is the same for both riders. Push students to make sure that they know it represents the point where the output values are the same for both functions for the same value of the input. f. Use function notation to represent the meaning of the intersection point of the functions’ graphs. f (t) = g(t) for the same input t g. Who finished the race first – Kim or Allison? How do you know? Kim. The graph shows us that Kim reaches the finish line (a distance of 50 miles from the halfway checkpoint) when Allison’s distance from the checkpoint is less than 50 miles. © 2017 CARLSON & O’BRYAN Inv 1. 6 21

Pathways Algebra II For both contexts in this investigation we defined at least two functions in two variables. Notice that the input quantities in both functions were the same and that the output quantities were measured in the same units. When this is the case we have created a system. Systems A system involves two or more functions that share the same input and output quantities. Note that it is possible for either the input or output quantities to be slightly different as long as they are measured in the same units. © 2017 CARLSON & O’BRYAN Inv 1. 6 22

Pathways Algebra II At some point during each exploration we were asked to determine the input value that results in both functions having the same output value. (Put another way, for two functions f and g we found the ordered pair (x, y) that was a solution to both y = f (x) and y = g(x). ) Ordered pairs that make each function relationship true are called solutions to the system. Solution to a System An ordered pair (x, y) is a solution to a system if it represents an input-output pair that is true for each function in the system. © 2017 CARLSON & O’BRYAN Inv 1. 6 23

Pathways Algebra II 5. a. Write down the solution(s) to the system in the first context (Joni and Tim driving to Denver) then demonstrate or explain how this ordered pair fits the definition of being a solution to the system. (6, 360). When t = 6 (6 hours since 2: 00 p. m. , or at 8: 00 p. m. ), both Tim and Joni are 360 miles from home. Since (6, 360) satisfies both of the original functions g(t) = 60 t and h(t) = 72(t – 1), it is a solution to the system. © 2017 CARLSON & O’BRYAN Inv 1. 6 24

Pathways Algebra II b. Give the solution(s) to the system in the second context (Kim and Allison in a bike race) then demonstrate or explain how this ordered pair fits the definition of being a solution to the system. (2. 5, 37. 5). When t = 2. 5 (2. 5 hours since Kim passed the halfway mark checkpoint), both Kim and Allison are 37. 5 miles beyond the checkpoint (they’ve both completed 87. 5 miles of the race). Since (2. 5, 37. 5) satisfies both of the original functions f (t) = 15 t and g(t) = 11 t + 10, it is a solution to the system. © 2017 CARLSON & O’BRYAN Inv 1. 6 25

Pathways Algebra II c. When you graph a system, how can you identify the solution(s)? When we graph the functions making up the system we can look for any points where the graphs intersect. Since a point on the graph of a function represents an input and corresponding output value for the function, any points where two or more graphs intersect represent input-output pairs that are common to all of the functions, and are thus solutions to the system. d. If you have a table of values for two functions in a system, how can you identify the solution(s)? The solutions will appear in the tables where the same input value produces the same output value for each function (the rows in the tables will look identical). © 2017 CARLSON & O’BRYAN Inv 1. 6 26

Pathways Algebra II 6. One of your classmates said the following. “When you have a system, the solution is the only thing that matters. If you know the solution, then nothing else is important. ” How do you respond to this comment? © 2017 CARLSON & O’BRYAN Inv 1. 6 27

Pathways Algebra II Investigation 7 SOLVING SYSTEMS OFEQUATIONS USING THE SUBSTITUTION METHOD © 2017

Pathways Algebra II When we have two functions, such as f (x) = 3 x – 4 and g(x) = –x + 28, and we want to find the solution (that is, the ordered pair where the input and output of each function are identical), we set up the equation f (x) = g(x) and solve it for x. This is one example of a technique called the substitution method. The name substitution comes from the fact that in the second step of the solution method we substitute the rule that determines the functions’ outputs in place of f (x) and g(x). f (x) = g(x) 3 x – 4 = –x + 28 To solve the system, we first solve 3 x – 4 = –x + 28 for x. © 2017 CARLSON & O’BRYAN Inv 1. 7 29

Pathways Algebra II 1. Complete the solution process to show that x = 8

Pathways Algebra II Now we can substitute 8 in for x in either function. Since we solved the system, the corresponding output values for f and g will be the same. We have found that (8, 20) is the solution of this system. If we graph f and g as originally defined, the solution will be the point where the graphs of f and g intersect. The x-value of the ordered pair (8) is the input value where the output values of f and g are equal (20). © 2017 CARLSON & O’BRYAN Inv 1. 7 31

Pathways Algebra II 2. Graph the system using a calculator or other graphing software

Pathways Algebra II In Exercises #3 -4, solve the given system. Graph the functions

Pathways Algebra II For Exercises #5 -6, use the following context. Edward, James, and Luis decided to start living healthier lives – several weeks ago they began a regular program of diet and exercise. To help motivate themselves, they each contributed $100 towards a winner’s prize. Whoever weighs the least after 12 weeks gets to keep all of the money. Over the course of the 12 weeks, each person lost weight at approximately a constant rate. Edward (who initially weighed 220 pounds) lost 2. 3 pounds per week, James (who initially weighed 242 pounds) lost 4. 6 pounds per week, and Luis (who initially weighed 231 pounds) lost 2. 5 pounds per week. © 2017 CARLSON & O’BRYAN Inv 1. 7 35

Pathways Algebra II 5. a. Complete the following table showing each contestant’s weight at different points throughout the competition. Show the expression used to calculate each weight as well as the exact values. Number of weeks since Edward’s weight competition (in pounds) began James’s weight Luis’s weight (in pounds) 2 3 8 © 2017 CARLSON & O’BRYAN Inv 1. 7 36

Pathways Algebra II 5. a. Complete the following table showing each contestant’s weight at different points throughout the competition. Show the expression used to calculate each weight as well as the exact values. Number of weeks since Edward’s weight competition (in pounds) began 220 – 2. 3(2) = 2 215. 4 220 – 2. 3(3) = 3 213. 1 220 – 2. 3(8) = 8 201. 6 James’s weight Luis’s weight (in pounds) 242 – 4. 6(2) = 231 – 2. 5(2) = 232. 8 226 242 – 4. 6(3) = 231 – 2. 5(3) = 228. 2 223. 5 242 – 4. 6(8) = 231 – 2. 5(8) = 205. 2 211 © 2017 CARLSON & O’BRYAN Inv 1. 7 37

Pathways Algebra II b. If t represents the number of weeks since the competition began, define the function rules that model each person’s weight in pounds. Edward: f (t) = James: g(t) = Luis: h(t) = © 2017 CARLSON & O’BRYAN Inv 1. 7 38

Pathways Algebra II c. Explain what it means to solve each of the following equations for t in this context. (In other words, if you solve each of the following equations, what would the solution tell you? ) i. f (t) = g(t) Solving this equation for t determines the number of weeks since the competition began when Edward and James weighed the same amount. ii. g(t) = h(t) Solving this equation for t determines the number of weeks since the competition began when James and Luis weighed the same amount. iii. f (t) = h(t) Solving this equation for t determines the number of weeks since the competition began when Edward and Luis weighed the same amount. © 2017 CARLSON & O’BRYAN Inv 1. 7 39

Pathways Algebra II d. Solve each of the equations in part (c) for t. *NOTE: The value of t such that f (t) = h(t) is outside the domain of this competition. We do not have information about each contestant’s weight outside of the 12 -week competition. Therefore, there is “no solution” to the equation f (t) = h(t) for this context. © 2017 CARLSON & O’BRYAN Inv 1. 7 40

Pathways Algebra II e. What would it mean if there was a “common solution” to the entire system of three equations in two variables? Is there a “common solution”? It would mean that there is a value of t such that the weight of all three contestants is the same. This does not happen in this context. © 2017 CARLSON & O’BRYAN Inv 1. 7 41

Pathways Algebra II 6. a. Graph the system involving three functions, then plot on your graph the coordinate points representing your solutions to part (c) of Exercise #5. © 2017 CARLSON & O’BRYAN Inv 1. 7 42

Pathways Algebra II 6. a. Graph the system involving three functions, then plot on your graph the coordinate points representing your solutions to part (c) of Exercise #5. © 2017 CARLSON & O’BRYAN Inv 1. 7 43

Pathways Algebra II b. Who won the competition? How do you know? James won. His weight was less than Edward’s weight and Luis weight after 12 weeks. c. Provide the interval (if any) over which each of the following is true during the 12 -week competition. i. Edward weighed less than James ii. James weighed less than Luis iii. Luis weighed less than James © 2017 CARLSON & O’BRYAN Inv 1. 7 44

Pathways Algebra II 7. Comment on the following statement: “Solutions to a system are important and useful, but the solution to a system is not always the ‘answer’ to a question involving a system. ” © 2017 CARLSON & O’BRYAN Inv 1. 7 45

Pathways Algebra II In Exercises #8 -9, solve the given system. (Remember to find the output value of the solution as well as the input value. ) Graph each system with a graphing calculator or similar software to verify your solution. 8. © 2017 CARLSON & O’BRYAN Inv 1. 7 46

Pathways Algebra II In Exercises #8 -9, solve the given system. (Remember to find the output value of the solution as well as the input value. ) Graph each system with a graphing calculator or similar software to verify your solution. 9. © 2017 CARLSON & O’BRYAN Inv 1. 7 47

Pathways Algebra II Sometimes you might encounter systems written in different formats. For example, instead of a system such as f (x) = 2 x – 7 and g(x) = – 3 x – 10, where we can solve the system by setting up the equation f (x) = g(x) and solving for x, we may be given a system such as x = y + 2 and 2 x + 8 y = 14 in which the formulas are not both explicitly defining the output value in terms of the input value. The following example shows the process. Remember that before we can even begin we must make the assumption that the values of x and y are identical in both formulas. In other words, we must assume that a solution exists in order to initiate the process. © 2017 CARLSON & O’BRYAN Inv 1. 7 48

Pathways Algebra II Since x = y + 2, we can write From here

Pathways Algebra II We now use y = 1 to find the corresponding value

Pathways Algebra II In Exercises #10 -11 use substitution to solve the given system. Check your solution by substituting the answer back into each initial formula. 10. © 2017 CARLSON & O’BRYAN Inv 1. 7 51

Pathways Algebra II In Exercises #10 -11 use substitution to solve the given system. Check your solution by substituting the answer back into each initial formula. 11. © 2017 CARLSON & O’BRYAN Inv 1. 7 52

Pathways Algebra II Investigation 8 THE ELIMINATION METHOD © 2017 CARLSON & O’BRYAN Inv

Pathways Algebra II Some linear relationships are best represented in standard form, or in the form ax + by = c where a, b, and c are constants (also called a linear combination). This form is most common when neither variable quantity can easily be identified as an input or output value, but instead their values combine to form some constant. For example, a high school drama club sold both adult and student tickets to their play. Adult tickets cost $5 each and student tickets cost $3 each. If the total amount of money collected for tickets was $1, 141, write a relationship between the number of tickets sold and the total income. Let x represent the number of adult tickets sold. Let y represent the number of student tickets sold. Then… © 2017 CARLSON & O’BRYAN Inv 1. 8 54

Pathways Algebra II 1. Fill in the blanks above and fill in the descriptions of what each expression represents. In this investigation we will develop and practice techniques for solving systems involving linear relationships written in standard form. © 2017 CARLSON & O’BRYAN Inv 1. 8 55

Pathways Algebra II 1. Fill in the blanks above and fill in the descriptions of what each expression represents. In this investigation we will develop and practice techniques for solving systems involving linear relationships written in standard form. © 2017 CARLSON & O’BRYAN Inv 1. 8 56

Pathways Algebra II 2. John and Amy both went to the store to buy snacks (chips and soda) for a party. John paid $12 (before tax) and bought four 2 liters of soda and two bags of chips. Amy paid $9 (before tax) and bought four 2 -liters and one bag of chips. (Note: You may assume that they each paid the same price per item. ) a. How much does a bag of chips cost? How do you know? A bag of chips costs $3 because the only difference in what they purchased was one bag of chips and the difference in their bill was $3. So the additional $3 John paid must be the price of one bag of chips. © 2017 CARLSON & O’BRYAN Inv 1. 8 57

Pathways Algebra II 2. John and Amy both went to the store to buy snacks (chips and soda) for a party. John paid $12 (before tax) and bought four 2 liters of soda and two bags of chips. Amy paid $9 (before tax) and bought four 2 -liters and one bag of chips. (Note: You may assume that they each paid the same price per item. ) b. How much does a 2 -liter bottle of soda cost? If a bag of chips costs $3, then Amy paid $6 for four 2 -liter bottles of soda. So each 2 -liter bottle of soda costs ¼ of $6, or $1. 50. © 2017 CARLSON & O’BRYAN Inv 1. 8 58

Pathways Algebra II c. Write a formula to represent the statement “John bought four 2 liters of soda and two bags of chips and paid a total of $12. ” Define any necessary variables. Let d represent the cost (in dollars) of purchasing one 2 -liter bottle of soda and let c represent the cost of purchasing one bag of chips. 4 d + 2 c = 12 d. Write a formula to represent the statement “Amy bought four 2 liters of soda and one bags of chips and paid a total of $9. ” Define any necessary variables (if they haven’t been defined already in part (c)). 4 d + c = 9 © 2017 CARLSON & O’BRYAN Inv 1. 8 59

Pathways Algebra II 3. Jessica likes to exercise. Each morning she goes to the track at the high school and spends some time jogging and some time running. One morning, she walked 2 miles and jogged 3 miles, which took her 44 minutes. The next morning, she walked 3 miles and jogged 3 miles, which took her 54 minutes. (Note: Assume that she always runs a mile at the same speed and always walks a mile at the same speed regardless of the day. ) a. What are the unknown quantities in this situation? Define variables to represent the values of these unknown quantities. We don’t know how long it takes Jessica to walk one mile and we don’t know how long it takes Jessica to jog one mile. Let w represent the number of minutes it takes Jessica to walk one mile and let j represent the number of minutes it takes Jessica to jog one mile. © 2017 CARLSON & O’BRYAN Inv 1. 8 60

Pathways Algebra II b. Write formulas to represent the relationship between the total amount of time she exercised and the exercises she performed each of these two days. 2 w + 3 j = 44 and 3 w + 3 j = 54 © 2017 CARLSON & O’BRYAN Inv 1. 8 61

Pathways Algebra II c. How long does it takes Jessica to run a mile and to walk a mile? The only difference between the two days of exercising is that she walked an additional mile on the second day and spent an additional 10 minutes exercising. This tells us that it takes her 10 minutes to walk a mile. Also, on the second day she spent a total of 30 minutes walking and 24 minutes jogging. Jogging 3 miles in 24 minutes means it takes her 8 minutes to jog a mile. We can double-check these conclusions on the first day of exercising to make sure that it makes sense that it takes 10 minutes to walk a mile and 8 minutes to jog a mile. © 2017 CARLSON & O’BRYAN Inv 1. 8 62

Pathways Algebra II 4. Coach Gonzalez is a P. E. teacher who had to replace some baseballs and footballs this year that were lost or damaged. One time he went to the store and bought 10 baseballs and 3 footballs for $44 (before tax). Another time, he bought 6 baseballs and 3 footballs for $36 (before tax). (Note: Assume that he paid the same price per item both shopping trips. ) a. How much does a baseball cost? How much does a football cost? Explain your reasoning. The difference in his shopping trips was 4 baseballs (he purchased the same number of footballs each trip). Since the cost of the items purchased differed by $8, and this is entirely due to the 4 extra baseballs purchased on his first trip, each baseball must cost $2. Therefore, in his first shopping trip he spent $20 on baseballs, and thus $24 on three footballs. Each football costs $8. © 2017 CARLSON & O’BRYAN Inv 1. 8 63

Pathways Algebra II b. Write two formulas representing the relationship between the items her purchased and the total cost for each shopping trip. Define variables as necessary. Let b represent the cost of each baseball (in dollars) and t represent the cost of each football (in dollars). Then 10 b + 3 t = 44 and 6 b + 3 t = 36. © 2017 CARLSON & O’BRYAN Inv 1. 8 64

Pathways Algebra II c. Use the elimination method to solve the system. Explain how

Pathways Algebra II Exercises #5 -6 demonstrate the basic reasoning behind the Elimination Method

Pathways Algebra II In Exercises #7 -8, do the following. a) Use the Elimination Method to solve the system. b) Check your answer by graphing the given functions. 7. © 2017 CARLSON & O’BRYAN Inv 1. 8 68

Pathways Algebra II In Exercises #7 -8, do the following. a) Use the Elimination Method to solve the system. b) Check your answer by graphing the given functions. 7. © 2017 CARLSON & O’BRYAN Inv 1. 8 69

Pathways Algebra II In Exercises #7 -8, do the following. a) Use the Elimination Method to solve the system. b) Check your answer by graphing the given functions. 8. © 2017 CARLSON & O’BRYAN Inv 1. 8 70

Pathways Algebra II In Exercises #7 -8, do the following. a) Use the Elimination Method to solve the system. b) Check your answer by graphing the given functions. 8. © 2017 CARLSON & O’BRYAN Inv 1. 8 71

Pathways Algebra II 9. Coach Gonzalez has to replace more than just baseballs and footballs. This year he also had to replace some dodge balls and softballs. The first time he went to the store he bought 2 dodge balls and 2 softballs and paid $23. 50 (before tax). The second time he bought 4 dodge balls and 9 softballs and paid $63. 25 (before tax). a. Without writing any formulas or equations, can you determine the price of a dodge ball and a softball? If so, explain your method. © 2017 CARLSON & O’BRYAN Inv 1. 8 72

Pathways Algebra II a. Without writing any formulas or equations, can you determine the price of a dodge ball and a softball? If so, explain your method. We can imagine that if we take his first shopping trip and pretend that he purchased twice as many dodge balls and softballs that his total bill (before tax) will exactly double. We now have a hypothetical first shopping trip where he purchased 4 dodge balls and 4 softballs for $47, and the second shopping trip where he purchased 4 dodge balls and 9 softballs for $63. 25. The difference in these shopping trips is 5 softballs and $16. 25. So each softball costs $3. 25. In his second shopping trip he thus spent $29. 25 on softballs, so he must have spent $34 on 4 dodge balls, making the cost of each dodge ball $8. 50. © 2017 CARLSON & O’BRYAN Inv 1. 8 73

Pathways Algebra II b. Write two formulas showing the relationship between the items he purchased and the total cost for of each shopping trip. Define variables as necessary. Let d represent the price of a dodge ball (in dollars) and let b represent the price of a softball (in dollars). Then 2 d + 2 b = 23. 50 and 4 d + 9 b = 63. 25. © 2017 CARLSON & O’BRYAN Inv 1. 8 74

Pathways Algebra II b. Write two formulas showing the relationship between the items he purchased and the total cost for of each shopping trip. Define variables as necessary. © 2017 CARLSON & O’BRYAN Inv 1. 8 75

Pathways Algebra II 10. Student Council sold spirit beads and stuffed animals of their mascot to raise money for charity. They sold a total of 320 items. Spirit beads sold for $2 each and stuffed animals for $15 each. The total amount of revenue raised was $2135. a. Write a formula to represent the total number of items sold based on the total number of spirit beads and stuffed animals sold. Define variables as necessary. Let b represent the number of spirit beads sold and let a represent the number of stuffed animals sold. Then b + a = 320. © 2017 CARLSON & O’BRYAN Inv 1. 8 76

Pathways Algebra II b. Write a formula to represent the relationship between the total

Pathways Algebra II c. How many of each item did Student Council sell? Student

Pathways Algebra II In Exercises #11 -12, use the elimination method to solve the

Pathways Algebra II In Exercises #13 -14, do the following. a) Use the elimination method to solve the system. b) Check your answer by graphing the given functions. 13. © 2017 CARLSON & O’BRYAN Inv 1. 8 81

Pathways Algebra II In Exercises #13 -14, do the following. a) Use the elimination method to solve the system. b) Check your answer by graphing the given functions. 13. © 2017 CARLSON & O’BRYAN Inv 1. 8 82

Pathways Algebra II In Exercises #13 -14, do the following. a) Use the elimination method to solve the system. b) Check your answer by graphing the given functions. 14. © 2017 CARLSON & O’BRYAN Inv 1. 8 83

Pathways Algebra II In Exercises #13 -14, do the following. a) Use the elimination method to solve the system. b) Check your answer by graphing the given functions. 14. © 2017 CARLSON & O’BRYAN Inv 1. 8 84

Pathways Algebra II 15. The Othello Theatre sold 420 tickets (made up of regular admission tickets and senior discount tickets) for one performance of its most recent show and took in $4760 for that night. Tickets regularly cost $12 each, but senior citizens could purchase a discounted ticket costing $8. How many regular tickets and how many senior citizen tickets were sold? Let c represent the number of regular-priced tickets sold and let t represent the number of discounted tickets sold. Then we get the formulas c + t = 420 and 12 c + 8 t = 4760. Make sure students can explain what each of these formulas (and their constituent parts) represent. © 2017 CARLSON & O’BRYAN Inv 1. 8 85

Pathways Algebra II © 2017 CARLSON & O’BRYAN Inv 1. 8 86

Pathways Algebra II 350 regular-priced tickets were sold and 70 discounted ticket were sold

Pathways Algebra II Investigation 9 POSSIBILITIES FORSOLUTIONS TO A SYSTEM OFLINEAR FUNCTIONS © 2017

Pathways Algebra II So far in this module we have focused on systems that have exactly one solution. However, a system of equations can have either: i) exactly one solution, ii) no solution, or iii) infinitely many solutions. © 2017 CARLSON & O’BRYAN Inv 1. 9 89

Pathways Algebra II 1. Make a rough sketch of the graphs of systems that

Pathways Algebra II 2. Consider the system consisting of x + y = 12 and 2 x + 2 y = 24. a. Graph this system. What do observe? The graphs of both functions are identical. © 2017 CARLSON & O’BRYAN Inv 1. 9 94

Pathways Algebra II 2. Consider the system consisting of x + y = 12 and 2 x + 2 y = 24. b. What ordered pair(s) (x, y) are solutions to the system? Every point on the graph of x + y = 12 is also a solution to 2 x + 2 y = 24. This system has infinitely many solutions. © 2017 CARLSON & O’BRYAN Inv 1. 9 95

Pathways Algebra II 2. Consider the system consisting of x + y = 12 and 2 x + 2 y = 24. c. Use either the substitution method or the elimination method and attempt to solve the system. What do you observe? Our solution method seems to break down. Instead of being able to find a value of one of the variables, we are left with a true statement of equality with no variables present. This happens because they share every possible solution, so there is no single value of either variable needed for the other variable to have equal value in both relationships. It is always true. © 2017 CARLSON & O’BRYAN Inv 1. 9 96

Pathways Algebra II The systems in Exercises #3 -4 have infinitely many solutions. For each, do the following. a) Graph the system to verify that there are infinitely many solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 3. © 2017 CARLSON & O’BRYAN Inv 1. 9 97

Pathways Algebra II The systems in Exercises #3 -4 have infinitely many solutions. For each, do the following. a) Graph the system to verify that there are infinitely many solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 3. © 2017 CARLSON & O’BRYAN Inv 1. 9 98

Pathways Algebra II The systems in Exercises #3 -4 have infinitely many solutions. For each, do the following. a) Graph the system to verify that there are infinitely many solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 4. © 2017 CARLSON & O’BRYAN Inv 1. 9 99

Pathways Algebra II The systems in Exercises #3 -4 have infinitely many solutions. For each, do the following. a) Graph the system to verify that there are infinitely many solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 4. © 2017 CARLSON & O’BRYAN Inv 1. 9 100

Pathways Algebra II 5. Consider the system consisting of f (x) = 3 x + 4 and g(x) = 3 x – 1 where y = f (x) and y = g(x). a. Graph this system. What do you observe? © 2017 CARLSON & O’BRYAN Inv 1. 9 101

Pathways Algebra II 5. Consider the system consisting of f (x) = 3 x + 4 and g(x) = 3 x – 1 where y = f (x) and y = g(x). a. Graph this system. What do you observe? The lines are parallel. They do not intersect. [Another helpful observation is that for all values of x, the difference in y values is always 5 (the difference in value of the vertical intercepts). ] © 2017 CARLSON & O’BRYAN Inv 1. 9 102

Pathways Algebra II b. What ordered pair(s) (x, y) are solutions to the system? This system has no solution. c. Use either the substitution method or the elimination method and attempt to solve the system. What do you observe? Our solution method seems to break down. Instead of being able to find a value of one of the variables, we are left with a false statement of equality with no variables present. This is because no values of one variable can produce the same value for the other variable in both relationships. © 2017 CARLSON & O’BRYAN Inv 1. 9 103

Pathways Algebra II The systems in Exercises #6 -7 have no solution. For each, do the following. a) Graph the system to verify that there are no solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 6. © 2017 CARLSON & O’BRYAN Inv 1. 9 104

Pathways Algebra II The systems in Exercises #6 -7 have no solution. For each, do the following. a) Graph the system to verify that there are no solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 6. © 2017 CARLSON & O’BRYAN Inv 1. 9 105

Pathways Algebra II The systems in Exercises #6 -7 have no solution. For each, do the following. a) Graph the system to verify that there are no solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 7. © 2017 CARLSON & O’BRYAN Inv 1. 9 106

Pathways Algebra II The systems in Exercises #6 -7 have no solution. For each, do the following. a) Graph the system to verify that there are no solutions. b) Use either the substitution method or elimination method to attempt to solve the system. 7. © 2017 CARLSON & O’BRYAN Inv 1. 9 107

Pathways Algebra II 8. In a 100 -meter race, Kacie got out of the starting blocks before her main competitor. A photo was taken and shows Kacie at the 30 meter mark at exactly the same time that her competitor, Molly, passed the 29 -meter mark. Kacie and Molly held the same speed of 10. 8 meters per second for the remainder of the race. a. Define a function f to represent Kacie’s number of meters from the starting line as a function of the number of seconds n since the picture was taken. f (n) = 10. 8 n + 30 where k = f (n), with k representing Kacie’s distance from the starting line in meters. © 2017 CARLSON & O’BRYAN Inv 1. 9 108

Pathways Algebra II 8. In a 100 -meter race, Kacie got out of the starting blocks before her main competitor. A photo was taken and shows Kacie at the 30 meter mark at exactly the same time that her competitor, Molly, passed the 29 -meter mark. Kacie and Molly held the same speed of 10. 8 meters per second for the remainder of the race. b. Define a function g to represent Molly’s number of meters from the starting line as a function of the number of seconds n since the picture was taken. g(n) = 10. 8 n + 29 where m = g(n), with m representing Molly’s distance from the starting line in meters. © 2017 CARLSON & O’BRYAN Inv 1. 9 109

Pathways Algebra II 8. In a 100 -meter race, Kacie got out of the starting blocks before her main competitor. A photo was taken and shows Kacie at the 30 meter mark at exactly the same time that her competitor, Molly, passed the 29 -meter mark. Kacie and Molly held the same speed of 10. 8 meters per second for the remainder of the race. c. Will Molly ever catch Kacie? Explain. No. Explanations may vary. The two runners are traveling at the same speed for the rest of the race, so they will remain 1 meter apart for the remainder of the race. © 2017 CARLSON & O’BRYAN Inv 1. 9 110

Pathways Algebra II 8. In a 100 -meter race, Kacie got out of the starting blocks before her main competitor. A photo was taken and shows Kacie at the 30 meter mark at exactly the same time that her competitor, Molly, passed the 29 -meter mark. Kacie and Molly held the same speed of 10. 8 meters per second for the remainder of the race. d. T or F: This system has no solutions. Explain. True. There is no value of n (number of seconds since the picture was taken) such that Kacie’s distance from the starting line and Molly’s distance from the starting line are identical. © 2017 CARLSON & O’BRYAN Inv 1. 9 111

Pathways Algebra II 8. In a 100 -meter race, Kacie got out of the starting blocks before her main competitor. A photo was taken and shows Kacie at the 30 meter mark at exactly the same time that her competitor, Molly, passed the 29 -meter mark. Kacie and Molly held the same speed of 10. 8 meters per second for the remainder of the race. e. Modify the situation above so that the system has infinitely many solutions. “A photo was taken and shows Kacie at the 30 meter mark at exactly the same time that her competitor, Molly, is at the 30 meter mark. Kacie and Molly held the same speed of 10. 8 meters per second for the remainder of the race. ” © 2017 CARLSON & O’BRYAN Inv 1. 9 112

Pathways Algebra II 9. Determine if each of the given systems of linear equations has one solution, no solutions, or infinitely many solutions. Justify your answer for each system. No solution. The functions have a. the same rate of change and different vertical intercepts. b. No solution. The functions have the same rate of change and different vertical intercepts. © 2017 CARLSON & O’BRYAN Inv 1. 9 113

Pathways Algebra II c. d. Infinitely many solutions. If we multiply both sides of 2. 5 x + 4. 5 y = 12 by 2, we get 5 x + 9 y = 24, which is identical to the other function formula in the system. Any ordered pair for 2. 5 x + 4. 5 y = 12 is also an ordered pair for 5 x + 9 y = 24. One solution. It’s clear that functions have different rates of change, and thus there must be a single solution (represented by the intersection point on the graph). © 2017 CARLSON & O’BRYAN Inv 1. 9 114

Pathways Algebra II e. f. No solution. When we solve 4. 5 x – 7. 5 y = 6 for y we get y = 0. 6 x – 0. 8, or y = (3/5)x – (4/5). The functions have the same rate of change and different vertical intercepts. One solution. When we solve 0. 5 x + 1. 2 y = 4. 8 for y we get y = –(5/12)x + 4. The functions have different rates of change, and thus there must be a single solution (represented by the intersection point on the graph). © 2017 CARLSON & O’BRYAN Inv 1. 9 115

Pathways Algebra II 10. A system of equations with no solution is said to be inconsistent and a system of equations with infinitely many solution is said to be dependent. Which of the systems in Exercise #9 are a. inconsistent? From Exercise #9, parts (a), (b), and (e) are inconsistent. b. dependent? From Exercise #9, part (c) is dependent. © 2017 CARLSON & O’BRYAN Inv 1. 9 116

Pathways Algebra II Investigation 10 THREE VARIABLELINEAR SYSTEMS © 2017 CARLSON & O’BRYAN Inv

Pathways Algebra II Sometimes a system includes more than three variables. For example, a florist purchased three different types of flowers to decorate for a wedding. The following table shows the costs the florist paid to buy each type of flower and what the florist charged the customer. Roses Carnations Daisies Cost to Florist (per dozen) $5 $3 $4 Charge to Customer (per dozen) $12 $9 $10 If the florist bought a total of 40 dozen flowers, paid a total of $170, and charged the customer a total of $428, then we can determine exactly how many of each type of flower was purchased by writing and solving a system with three different equations. Let x be the number of roses purchased (in dozens). Let y be the number of carnations purchased (in dozens). Let z be the number of daisies purchased (in dozens). © 2017 CARLSON & O’BRYAN Inv 1. 10 118

Pathways Algebra II © 2017 CARLSON & O’BRYAN Inv 1. 10 119

Pathways Algebra II To solve the system by hand, we create two pairs of equations with the goal of eliminating the same variable on both pairs. Exercise #1 is completed for you. First Pair: Second Pair: © 2017 CARLSON & O’BRYAN Inv 1. 10 120

Pathways Algebra II 1. Use the elimination method to create an equation in terms

Pathways Algebra II 2. Use the elimination method to create an equation in terms

Pathways Algebra II We now take the pair of equations involving only x, y, and constant values and solve the system. Once we have the solution values for x and y, substitute them back into one of the original equations to find the solution value for z. 3. Write down the two equations derived in Exercises #1 and #2 and finish solving the system. The florist provided 18 dozen roses, 8 dozen carnations, and 14 dozen daisies for the wedding. © 2017 CARLSON & O’BRYAN Inv 1. 10 123

Pathways Algebra II 4. Write and solve a system of three equations to answer the following question: The florist provided flowers for another wedding and discovered that flower prices had changed. Roses Carnations Daisies Cost to Florist Charge to Customer (per dozen) $6 $14 $4 $10 $5 $12 If the florist bought a total of 45 dozen flowers, paid a total of $219, and charged the customer a total of $528, how many of each type of flower was purchased? © 2017 CARLSON & O’BRYAN Inv 1. 10 124

Pathways Algebra II First Pair: © 2017 CARLSON & O’BRYAN Inv 1. 10 125

Pathways Algebra II Second Pair: © 2017 CARLSON & O’BRYAN Inv 1. 10 126

Pathways Algebra II The florist provided 16 dozen roses, 22 dozen carnations, and 7

Pathways Algebra II In Exercises #5 -8, solve the system of equations. Note that when you have a system with three variables, you must have at least three equations in order to solve the system. 6. 5. x = 6, y = – 3, and z = 2 x = – 1, y = 3, and z = 4 © 2017 CARLSON & O’BRYAN Inv 1. 10 128

Pathways Algebra II In Exercises #5 -8, solve the system of equations. Note that when you have a system with three variables, you must have at least three equations in order to solve the system. 8. 7. x = 2, y = – 1, and z = 4 x = 2, y = 7, and z = – 5 © 2017 CARLSON & O’BRYAN Inv 1. 10 129