Unit 4
Contents
1. It will look like this on your handheld RAD AUTO REAL a Press the key The handheld will display the result RAD AUTO REAL a You have eliminated the y variable Sct 2 y 16 5x 2 y 8 8 24 P Subtracting two equations works the same way as adding two equations The key is to remember that you must enclose each equation in brackets when you type it into the handheld How to Multiply to Find an Equivalent Equation Consider the following linear system 2x 5y 7 6x 3 If we immediately add or subtract the equations neither the x or the y variable is eliminated Instead we must multiply one of the equations by an integer so that the coefficients match Then if we subtract the equations a variable will be eliminated Let s multiply the first equation by 3 Remember that all terms on both sides of the equation must be multiplied by 3 so that the equation stays balanced Here is how to do this on a handheld First be certain that you are on a Calculator page If you need help with this see the Getting Started section from earlier in this manual Press the following keys OP DOOODDOOOOODD D What you have typed should look like this RAD AUTO REAL a Press the key The handheld will display the result RAD AUTO REAL Now we have a system that we 6x 15 21 can begin to solve by subtracting the equations 3 2 3 5 3 7 6 15 21 TIPS4RM Grade 10 Applied Unit 4 Linear Systems2. Number of Books How can the flat fee and the cost per book be used to draw the graph What is the point of intersection of the two lines What does it represent Under what conditions is it best to go with Descartes Publishing A Under what conditions is it best to go with School Memories TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 12 4 1 2 What s My Equation Continued Problem D The school is putting on the play Algebra The Musical Adult tickets were sold at a cost of 8 and student tickets were sold at a cost of 5 A total of 220 tickets were sold to the premiere and a total of 1460 was collected from ticket sales How many adult and student tickets were sold to the premiere of the musical To solve the question complete the table of values and the graph Let x represent the of student tickets sold Let y represent the of adult tickets sold Algebra The Musical D x y x y x 0 0 40 40 E 80 80 2 120 120 5 160 160 5 200 200 t 2 z 20 40 60 80 100 120 140 160 180 200 Number of Student Tickets 1 What is the approximate point of intersection of the two lines What does it represent 2 Does the rest of the graph other than the POI give us any information about the number of tickets sold TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 13 4 1 3 Meaning of the Po
3. Linear Systems July 2008 4 34 4 5 2 An Elimination Introduction You know that two integers can be added or subtracted 5 15 _7 _6 12 9 In the same way equations be added or subtracted 2y 19 10 20 80 5x 2y 5 10x 15 25 8 24 55 Notice that by adding the equations in the first linear system the y variable was eliminated there were Oy which makes it possible to solve for x By subtracting the equations in the second linear system the x variable was eliminated there were which makes it possible to solve for y 1 Work in pairs to consider the following linear systems Decide what operation addition or subtraction would result in the elimination of a variable You may use CAS on the handheld to help you decide 9x y 4 3x 50 7x 338 14x 1 12x y 115 9x 366 18x 454 19x 2y 102 17x 8y 323 12x 5y 316 19x 2y 50 6x 8y 114 9x 235 7X l6y 441 5x 3y 188 15x 2y 409 7 17 476 6x 11 344 2 What needs to be true about a linear system so that a variable is eliminated when the equations are added or subtracted TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 35 4 5 3 Solving a Linear System by Elimination 1 How would you begin solving this linear system Addition or Subtraction 5x 4y 7 3x 4y 17 2 Solve
4. 4 49 4 7 5 3 Ways Continued The student council is providing lunch and music for the grade 10 class They have two quotes from Lunch Express and Let s Do Lunch The costs for each were given as follow Lunch Express If 100 students attend it will cost 1 000 If 200 students attend it will cost 1 500 Let s Do Lunch If 50 students attend it will cost 700 If 150 students attend it will cost 1 350 Solve the system using the three different methods Equations for the companies Lunch Express Let s Do Lunch Lunch Express Let s Do Lunch Graphing Method Substitution Method 50 100 150 200 250 300 Point of intersection Point of intersection TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 50 4 7 5 3 Ways Continued Elimination Method Point of intersection The student council has 1 800 in their budget for the lunch They prefer Let s Do Lunch what is the greatest number of grade 10 students they can have at the lunch What does the ordered pair 25 750 mean on the Lunch Express line TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 51 4 6 Unit Summary Page Unit Name Using a graphic organizer of your choice create a unit summary TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 52 4 R Reflecting on My Learning 3 2 1 3 Thi
5. August 2008 4 6 4 1 1 Working on Commission Nahid works at Euclid s Electronics She is paid a salary of 200 per week plus a commission of 5 of her sales during the week The equation P 0 055 200 represents Nahid s pay for the week where P represents the total pay for the week and s represents her total sales If Nahid earned 290 in a week use the Euclid s Electronics Wages i hd determine how 500 ELLE Use your handheld to help you solve Refer to the user manual if you need to review how to solve using the handheld Week s Pay 500 1001 1500 2000 2500 Total Value of Sales Nahid is offered another job at Fermat s Footwear where the pay is a salary of 100 per week and 1096 commission on all sales The graph below represents the Pay vs Sales for this job Which of the following equations do you think represents pay for one week at Fermat s Footwear Fermat s Footwear Wages 500 i b 0 105 100 LIT c P 2100s 10 a P 0 01s 100 d P 0 05s 200 Provide a reason or justify why you selected the equation that you chose Refer back to the equation for Euclid s Electronics for hints Week s Pay 500 1001 1500 2000 2500 Total Value of Sales TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 7 4 1 1 Working on Commission Continued Which One is Better Nahid needs help determinin
6. 2 1 53 4 RLS Reflecting on Learning Skills 54 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 2 Nspire CAS Handheld Manual Getting Started When you turn on the handheld press You will be asked whether you want to save the Do you want to save this document document Select No To do this use the large circular navpad to move to the right then press the Yes in the middle of the navpad 1 Add Calculator Next select 1 Add Calculator To do this press the button again 2 Add Graphs amp Geometry 3 Add Lists amp Spreadsheet You are now ready use CAS on the handheld 5 Add Data amp Statistics Some Helpful Shortcuts If you make a mistake at any point that you want to undo press C If you undo something that you want back again press How to Solve for a Variable Example One Say that you wish to solve the equation 6x 2y 16 for the variable y To do this first be certain that you are on a Calculator page If you need help with this see the Getting Started section above First type in the equation that you want to solve Use the number pad and the green letter keys the operations are located on the right and the equals sign is in the top left corner of the keypad When you have typed in the equation press the key found in the bottom right corner RAD AUTO REAL Li The
7. month which is the best plan for her Explain 2 Which plan is cheaper if you only plan to buy 4 songs per month How do you know from the graph 3 Which cell phone plan would you choose and why Relate your answer back to the POI TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 17 4 2 3 Where Do We Meet For each of the following situations find the point of intersection and describe the meaning of this point Describe which company or service you would choose under what circumstances A template has been provided for the first situations Lawn Service 1074 7 a 80 Point of intersection 70 Interpretation of the point 60 z 50 If the job lasts less than hours 8 40 A choose If the job lasts more 30 w I b ee than hours choose l If the job lasts hours 20 choose either company and the cost is 10 2 12 3 4 5 6 78 9 10 Time hours Car Rental Companies Point of intersection Interpretation of the point If the kilometers driven is less than choose If the kilometers driven is more than choose If the kilometers driven is choose either company the cost is 10 20 30 40 50 60 70 80 90 100 Kilometers Driven TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 18 4 2 3 Where Do We Meet Continued For each of the following situations find the point of intersection and describe the meaning
8. of this point Refer back to the template provided for the first situations Cost Snowboard Rental Time hours Point of intersection Interpretation of the point Cost Long Distance Plans LLLLLLLLAZALI L LLL LL LLLI LL LL LLLA LX T3 ri Pt LUE E LLLI tal 17 4 E tt Yi i tt tt PE a a i a 10 15 20 25 30 35 40 45 50 Talking Time hours Point of intersection Interpretation of the point TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 2 4 Does This Line Cross From the list of relations below determine which lines cross through the point 2 3 You may use the graph to assist you 1 y 22x43 2 1 3 2 7 4 y 2 3 b x22 6 2 Questions 1 Which of the lines passes through the point 2 3 2 Is there another way to determine if the line passes through the point other than graphing Explain 3 Without graphing how can you quickly determine if a horizontal or vertical line passes through a point 4 Other than the point 2 3 what are the other points of intersection on your graph 5 Is it possible for two lines to have more than one point of intersection with each other Discuss this with your partner TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 20 4 2 5 Is this Accurate 1 Find the point of intersection Solve the system using gra
9. system into y mx b form then graph For example here is how the first equation can be re arranged 4 2y 12 2y 4 12 2 4 12 _ 2 2 2 2X 6 tT tT ete oT BERR EEE LT TTT TT ee ee te eT TT Tt Ltt tT Tt Pe T SATT eee BERR EEE Eli eee eee 4 What you notice about the slope each equation that you re arranged What do you notice about how the two lines visually relate to each other Is there a solution to this linear system TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 42 4 6 6 Here s To The Crazy Ones Continued 5 8 Solve the following linear system by elimination 6x 2y 2 12x 4y 4 Did you encounter any results that are unusual Explain what is different compared to questions you have already solved Re arrange each equation from the linear system into y mx b form then graph BERR EEE eee LT TTT TT Tt et ee TT BERR EEE eee eee eee SR TT TTT tT TT What do you notice about the slope and y intercept in each equation that you re arranged What do you notice about how the two lines visually relate to each other Is there a solution to this linear system If so how many TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 43 4 7 3 Which Method For Sy
10. the system 3 In your own words describe what you must do to solve a linear system by elimination TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 36 4 6 1 What s the difference 2 6 Original equation The new equation is Multiply the equation by the constant Complete the table of values for your equation and plot the values on the grid provided an ptt tT tt ttt _ SRR Ree Pee _ SRR Ree Pees SRR Pee pt tT ttt tt eee TT eee eee BRR eRe PERE pt tt tt tt eiii 4 37 TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 6 2 Elimination Preparation 1 Consider the following linear system 3k 15m 15 k apod If you add or subtract the equations will a variable be eliminated Explain 2 What could be done to create the conditions necessary for elimination TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 38 4 6 3 Algebra the Musical Redux Recall that in the first lesson of this unit you solved the following problem by graphing The school is putting on the play Algebra The Musical Adult tickets were sold at a cost of 8 and student tickets were sold at a cost of 5 A total of 220 tickets were sold to the premiere and a total of 1460 was collected from ticket sales How many adult and student tickets were sold to the premiere of the musical If x represents the number
11. 5 per kilometre driven Which company should Yasser choose to rent the car from Problem B Equations The school council is trying to determine where to hold the athletic banquet The Algebra Ballroom charges an 800 flat fee and 60 per person The Geometry Hall charges a 1000 flat fee and 55 per person Which location should the school council select for the athletic banquet Problem C Equations The yearbook club is considering two different companies to print the yearbook The Descartes Publishing Company charges a flat fee of 475 plus 4 50 per book School Memories charges a flat fee of 550 plus 4 25 per book Which company should the yearbook club select to print this year s yearbook Problem D Equations The school is putting on the play Algebra The Musical Adult tickets were sold at a cost of 8 and student tickets were sold at a cost of 5 A total of 220 tickets were sold to the premiere and a total of 1460 was collected from ticket sales How many adult and student tickets were sold to the premiere of the musical EQUATIONS 1 4 50 475x 2 60 800x y 3 y 1000 55x 4 45 0 15x 5 1000 55 6 45 0 15 7T 220 8 5 8y 220 9 y 2 4 25 550 10 y 550x 4 25 11 y 800 60x 12 1460 13 0 25 35 14 4 50 475 15 y 2 85x 0 25 16 5 8 1460 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 9 4 1 2 What s My Equ
12. 8 4 15 4 2 1 A Visual Cell Phone Problem Two cell phone companies charge a monthly flat fee plus an additional cost for each minute of time used The graph below shows the Time vs Cost relationship for one month Cell Phone Costs Talk More We Talk Cost 200 400 600 800 1000 1200 Time minutes 1 What is the Point of Intersection POI and what is the meaning of the POI in relation to the cell phone plans 2 Under what conditions is it best to use the Talk More cell phone plan 3 Under what conditions is it best to use the We Talk cell phone plan 4 How does the graph help you to determine which cell phone plan is the most appropriate at any given time TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 16 4 2 2 Music is My Best Friend iTones and Music Mine are two online music providers Each company charges a monthly membership fee and then a per song download rate iTones charges 10 per month and 1 per song C n 10 Music Mine charges 7 per month and 1 50 per song C 1 5n 7 Where C represents the total cost for one month and n represents the number of songs purchased Create a table of values showing the total charges for up to 8 songs purchased Graph the lines on the same graph below iTones Music Mine Online Music Purchasing N Cc n 0 0 1 1 Cost Number of Songs 1 If Lulu plans to purchase 7 songs this
13. IPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 33 4 5 1 A Trip to Jim Hortons It s summer vacation Ah sweet freedom The only problem is that you re the designated coffee gopher at the office where you have a summer job On Monday you were sent out to Jimmies to pick up five small coffees and seven extra large coffees You remember that the total cost was 14 95 including tax On Tuesday you were sent out to get three small coffees and seven medium coffees You recall that the total came to 12 75 with tax Its Wednesday morning and your coffee crazy co workers are calling for their cup joe Unfortunately since the morning fix hasn t arrived yet no one can remember how much a small or extra large coffee costs including yourself You need to find out how much each size costs to collect the correct amount of money for the Wednesday coffee run 1 Let s be the number of small coffees ordered on a single day Let e be the number of extra large coffees ordered on a single day As a class can we decide on an equation to represent the purchases made on Monday and an equation to represent the purchases made on Tuesday Monday s equation Tuesday s equation 2 Now we have a linear system Take a few minutes to solve the linear system using substitution in the space below Then pair with another student to discuss your solution 3 What problems if any did you encounter TIPS4RM Grade 10 Applied Unit 4
14. TIPSARM Targeted Implementation and Planning Supports for Revised Mathematics Grades 7 8 9 Applied 10 Applied Course Grade 10 Applied Mathematics MFM2P Unit 4 Linear Systems Quantity Dufferin Peel Jt set REVISED Catholic District zi School Board ALY August 2008 Unit 4 Linear Systems Section Activity Page Nspire CAS Manual 3 4 1 1 Working on Commission 7 4 1 2 What s My Equation 9 4 1 3 Meaning of he Point of Intersection 14 4 2 1 A Visual Cell Phone Problem 16 4 2 2 Music is My Best Friend 17 4 2 3 Where Do We Meet 18 4 2 4 Does This Line Cross 20 4 2 5 Is This Accurate 21 4 3 1 What s My POI 22 4 3 2 A Better Way 23 4 3 3 Putting the Pieces Together 24 4 3 4 The Sub Way 25 4 3 5 What s My Equation Part 2 26 4 4 1 The Lowdown on Downloads 28 4 4 2 Putting the Process Together 29 4 4 3 An Interesting Problem 31 4 4 4 The Sub Steps 33 4 5 1 A Trip to Jim Hortons 34 4 5 2 An Elimination Introduction 35 4 5 3 Solving a Linear System by Elimination 36 4 6 1 What s the Difference 37 4 6 2 Elimination Preparation 38 4 6 3 Algebra the Musical Redux 39 4 6 4 Two for You 40 4 6 5 Help an Absent Friend 41 4 6 6 Here s to the Crazy Ones 42 4 7 3 Which Method 44 4 7 4 The Frayer Model 47 4 7 5 3 Ways 48 4 S Unit Summary 52 4 R Reflecting on My Learning 3
15. ation Continued Problem A Yasser is renting a car Zeno Car Rental charges 45 for the rental of the car and 0 10 per kilometre driven Erdos Car Rental charges 35 for the rental of the same car and 0 25 per kilometre driven Which company should Yasser choose to rent the car from To solve the question complete the table of values and the graph Zeno vs Erdos Zeno Erdos Distance Cost Distance Cost km km 0 0 10 10 20 20 30 30 amp 40 40 50 50 8 60 60 70 70 80 80 90 90 100 100 10 20 30 40 50 60 70 80 90 100 Kilometers Driven 1 How can the car rental cost and the cost per kilometre be used to draw the graph 2 What is the point of intersection of the two lines What does it represent 3 Under what conditions is it best to rent from Zeno Car Rental 4 Under what conditions is it best to rent from Erdos Car Rental TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 10 4 1 2 What s My Equation Continued Problem B The school council is trying to determine where to hold the athletic banquet The Algebra Ballroom charges an 800 flat fee and 60 per person The Geometry Hall charges a 1000 flat fee and 55 per person Which location should the school council select for the athletic banquet To solve the question complete the table of values and the graph Al
16. e rental of the car and 0 15 per kilometre driven Erdos Car Rental charges 35 for the rental of the same car and 0 25 per kilometre driven For what distance do the two rental companies charge the same amount Problem B The school council is trying to determine where to hold the athletic banquet The Algebra Ballroom charges an 800 flat fee and 60 per person The Geometry Hall charges a 1000 flat fee and 55 per person For what amount of guests do the two banquet halls charge the same amount Problem C The yearbook club is considering two different companies to print the yearbook The Descartes Publishing Company charges a flat fee of 475 plus 4 50 per book School Memories charges a flat fee of 550 plus 4 25 per book For what amount of books do the two companies charge the same amount Equations 45 0 15 35 0 25x Equations y 60x 800 55x 1000 Equations y 475 4 50 550 4 25 am solving problem TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 26 4 3 5 What s My Equation Part 2 continued 1 Looking at your problem how can you tell from the equation which company is cheaper before the point of intersection where the costs are equal 2 Looking at your problem how can you tell from the equation which company is cheaper after the point of intersection where the costs are equal 3 15 this true for all problems 4 No
17. e systems Each system below has only one equation given Assign each system to one person in the group and create a second equation that will give the solution 1 2 when solved System A System B System C Equation 1 x 6y 11 Equation 2 Equation 1 14x 9y 4 Equation 2 Equation 1 9x 7 5 Equation 2 7 Share your equation with everyone in the group and copy down the equations from the rest of the group Review the equations created and make sure they follow the rule that you noted in 5 You will solve these three systems as a part of your home activity TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 32 4 4 4 The Sub Steps Solving using the method of substitution requires five steps The steps are given below in the text boxes Discuss with your partner what you think the correct order is for the steps and then write the steps in the space provided Solve the system in the chart as model of solving by substitution Check using your handheld State the point of intersection Solve the resulting equation Substitute the isolated expression into Substitute your solution into an original the other equation equation to solve for the other variable Isolate for a variable The easiest variable to isolate for has a coefficient of 1 Example Solve Steps for Solving by Substitution 4x y 2x 3y 10 T
18. er pad and the green letter keys the operations x are located on the right and the equals sign is in the top left corner of the keypad When you have typed in the equation press the key found in the bottom right corner RAD AUTO REAL a The top of your screen will look something like this 6 2 18 6 2 18 z Now decide how you would start solving for y Perhaps you ve decided that subtracting 6x from both RAD AUTO REAL sides of the equation is a good start Wonderful To 6202 18 6 2 18 do this immediately press 627469 Notice that the handheld automatically inserts Ans What is this Ans stands for the last answer you found If you now RAD AUTO REAL Ll press the key the handheld will subtract 6x from A 6 x 2 y 18 the left side and the right side of 6x 2y 18 You will see this result 6 2 18 6 6 xt 2 y 18 2 18 6 Continue solving the equation You probably see that to finally isolate the y variable it is necessary to divide the equation by 2 on both sides Again just start typing the operation you want to perform Press lt 2 62 The handheld will insert Ans for you Press to calculate the result As you can see the handheld reports that RAD AUTO REAL y 3 x 3 2 18 6 2 18 Is this the result you expected 29186 Your teacher will discuss this with your c
19. fee What is the hourly fee HINT Start by writing let statements to define the variables you will use For example Let a represent the amount of the annual fee Let h represent the amount of the hourly fee 2 This past summer you ran a food booth at a local festival You sold hotdogs for 1 each and samosas for 2 50 each From 205 purchases you made 400 in total To help plan purchases for next year s festival you d like to know how many hotdogs and samosas were sold Unfortunately you forgot to keep track of this when selling the food Can you determine how many hotdogs and samosas were sold NOTE Assume one hotdog or one samosa per purchase TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 40 4 6 5 Help an Absent Friend Consider the following linear system 2X 1 7 How would you solve it Write words a description of the steps you would take To help you understand what to write pretend for a moment that you are writing the instructions for a friend who is not in class today What steps would you need to describe TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 41 4 6 6 Here s The Crazy Ones 1 Solve the following linear system by elimination 4 2y 12 8x 32 2 Did you encounter any results that are unusual Explain what is different compared to questions you have already solved 3 Re arrange each equation from the linear
20. g which job she should keep She decides to look at them as a system of equations when she creates a graph comparing the two equations at the same time Analyze the graph and complete the questions below Euclid s Electronics Fermat s Footwear 500 1000 1500 2000 2500 Total Value of Sales 1 Where the two lines cross is called the point on intersection or the solution to the system At what coordinates do the two lines cross 2 What does this coordinate represent in terms of Nahid s sales and pay for the week 3 If Nahid usually makes 1500 worth of sales per week which job should she take Explain 4 How does the graph help Nahid determine which is the better job 5 What does the point 1000 250 represent in the graph TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 8 4 1 2 What s My Equation You are given four problems below Each problem will require two equations to solve it The equations that are needed to solve each problem appear at the bottom of the handout Match the equations with the problems and compare your answers with another student Note There more equations than problems and all the equations use x for the independent variable and y for the dependent variable Problem A Equations Yasser is renting a car Zeno Car Rental charges 45 for the rental of the car and 0 15 per kilometre driven Erdos Car Rental charges 35 for the rental of the same car and 0 2
21. gebra Ballroom Geometry Hall Algebra Ballroom vs Geometry Hall Number Number of Cost of Cost People People 0 0 10 10 30 30 40 40 9 50 50 60 60 70 70 80 80 90 90 100 100 0 20 30 40 50 60 70 80 90 100 Number of People 1 How can the flat fee and the per person cost be used to draw the graph 2 What is the point of intersection of the two lines What does it represent 3 Under what conditions is it best to go with Algebra Ballroom 4 Under what conditions is it best to go with Geometry Hall TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 11 4 1 2 What s My Equation Continued Problem C The yearbook club is considering two different companies to print the yearbook The Descartes Publishing Company charges a flat fee of 475 plus 4 50 per book School Memories charges a flat fee of 550 plus 4 25 per book Which company should the yearbook club select to print this year s yearbook To solve the question complete the table of values and the graph Descartes School Memories Descartes vs School Memories Number Number 3000 Hime cost TTT TTT 0 0 50 uw LE LL LA 100 100 2100 LLL LLL 150 150 9 100 a 200 8 LL tt tt tt HRE 350 350 800 400 400 B00 450 450 300 309 309 50 100 150 200 250 300 350 400 450 500
22. he pieces out and glue them in the correct order on a separate piece of paper Use the handheld to help you Hint There are 10 steps in the complete solution 2 lt TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 29 TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 30 4 4 3 An Interesting Problem Consider the following three systems of equations 2y 4 5 6 2x 4 7x 10 13 5 2 1 x 5 1 In order to solve these systems by substitution we need to first isolate one variable in one equation Circle the variable in each system that would require just one step to isolate Compare your choice with a neighbour 2 Using your handheld to help you isolate the variable you selected in 1 for each system 3 Assign each system to one person in your group and solve the system assigned to you in the space below Use your handheld to help solve and check am solving system 4 Compare your solution to the rest of the group What do you notice TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 31 4 4 3 An Interesting Problem Continued 5 The chart below lists out only the numbers that appear in each system What do you notice about the numbers in each system System A System B System C 1 2 9 4 5 6 2 1 4 7 10 13 5 3 13 1 3 5 6 Let s see if this works with mor
23. int of Intersection 1 Your family wants to rent a car for a weekend trip Cars R Us charges 60 00 per weekend for a midsize car plus 0 20 per km Travel With Us charges 0 50 per km Renting a Car a Graph both options on the grid and determine the number of kilometres where both companies will cost the same amount 200 b Explain what this means for your weekend trip 150 Cost 100 50 100 200 300 400 Number of kms 2 Anthony and Anne are bicycling at a Provincial Park Anthony travels at the rate of 10 km hr and begins 2 km from the park entrance Anne begins at the park entrance and travels at a rate of 15 km hr They both travel at a constant rate towards the Outdoor Education Centre Bicycling in the Park Graph both routes on the grid and determine the meaning of the point of intersection Distance From Park Entrance km Time hours TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 14 4 1 3 Meaning of the Point of Intersection continued 3 Foracar wash fundraiser Team A washes 2 cars per hour starting a 7 00 a m Team B begins washing cars at 9 00 a m and washes 3 cars per hour Graph the car washing progress of each team on the grid and determine the meaning of the point of intersection as well as the meaning of the points before and after the point of intersection Fund Raiser Number of Cars Hours Elapsed TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 200
24. iversaries Nick and Heather are getting married in September and they have two catering companies to choose from Frugal Gourmet Cookie s Catering Cesaer Salad Chicken Picatta Roasted Potatoes Steamed Vegetables Sherbert Coffee or Tea Minestrone Soup Mixed green salad Prime Rib Garlic Mashed Potatoes Asparagus Apple Pie Coffee or Tea The cost C for the different menu options includes the cost of the hall rental and price per person n Cookie s Catering C 40n 500 Frugal Gourmet 45 350 Cookie s Catering Frugal Gourmet Solve the system using the graphing method 10 20 30 40 50 60 70 80 90 100 Point of intersection TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 48 4 7 5 3 Ways Continued Solve the system using substitution Point of intersection Solve the system using elimination Point of intersection What does the point of intersection mean in this catering problem You used 3 different methods to solve the system what did you notice about the points of intersection Does this surprise you Nick and Heather have invited 80 people to their wedding How much will it cost for each menu Heather prefers the Frugal Gourmet menu to Cookie s Catering How much more will she pay for her preference TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008
25. lass 2 y 18 6 x 3 3 After the discussion use the space below to write your 2 own explanation of what this means TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 4 Nspire CAS Handheld Manual Continued How to Check a Solution to a Linear System Say that you have solved the following linear system 4 2y 24 8x 18 and you believe the solution is 3 3 This would be tedious to check by pencil and paper but it is quick to check with the handheld Here is how to do it First be certain that you are on a Calculator page If you need help with this see the Getting Started section from earlier in this manual You are going to press the following keys to check the solution against the first equation POPOVOOVOOOOYOODO OUOOUOE This means Check this equation such that x z and y 3 Don t forget to press the key lower right corner of keypad before and after the ANO Here is what it will look like on your handheld screen RAD AUTO REAL Li To have the handheld check the solution press the 9 key If the solution is correct the handheld will I IDE return the result true If the solution is notcorrect the handheld will report false You should find that the handheld reports that E is a correct solution for the first equation Remember it is necessary to check the solution in the second equation as well It works the same way as before bu
26. ngs know well from this unit 2 Things need explained more 1 Question still have TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 53 4 RLS Reflecting on Learning Skills Students should be aware of the importance that these skills have on your performance After receiving your marked assessment answer the following questions Be honest with yourself Good Learning Skills will help you now in other courses and in the future E Always Sometimes S Need Improvement N Never Organization e EGSN came prepared for class with all materials e EGSN My work is submitted on time e EGSN keep my notebook organized I 5 e zzzzzzo attempt all of my homework use my class time efficiently limit my talking to the math topic on hand am on time If am away ask someone what missed complete the work from the day that missed e gt 5 x e mmmms 9 mrimmns3 ITI TI TI ITI mm am an active participant in pairs group work co operate with others within my group respect the opinions of others nnng lt QDODS oooooo zzz e e participate in class discussion lessons When have difficulty seek extra help After resolve my difficulties reattempt the problem review the daily lesson ideas concepts e O O0 2 2 2 2 o ndependen
27. of student tickets sold and y represents the number of adult tickets sold then the equations that model this problem are from cost of tickets 5x 8y 1460 from number of tickets sold x y 220 You probably remember that this problem took a while to solve by graphing and the answer you found was not necessarily very accurate since you read the point of intersection off of the graph You will work with a partner now to solve this problem using the method of elimination 1 Since you have been asked to eliminate the x or y variable circle one first what will be the first step you take to create the conditions necessary for elimination 2 Solve the linear system now You may use CAS to help find the answer and check the solution Use the space below for rough work 3 Does it matter which variable is eliminated first That is does it change the final answer 4 Think back to when you solved this problem by graphing Do you find the method of elimination easier or harder Explain TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 39 4 6 4 Two for You Try solving the following questions using the method of elimination You may use CAS on the handhelds to help solve the questions and check your solutions 1 Afitness club charges an annual fee and an hourly fee In a single year member A worked out for 76 hours and paid 277 in total Member B worked out for 49 hours and paid 223 in total What is the annual
28. phical method a 2 1 b y x 2 3 2 y 2x 7 Point of intersection is Point of intersection is TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 21 4 3 1 What s my POI Each one of you will solve one of the systems of equations given below e Once you have solved the system you were assigned trade with your partner and check their solution Share your feedback with your partner e Once you have shared your feedback and are confident in the solutions to the systems post your point of intersection under the appropriate heading on the class list alia 2 7 di 3 4 101 8 6 pT ols Point of Intersection Point of Intersection TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 22 4 3 2 A Better Way Solve the following systems of equations algebraically Use your CAS handheld to solve and check if needed 1 Equation 1 y 22 10 and Equation2 12 Point of intersection ___ 2 Equation 1 3x 2y 33 and Equation2 2x x 7 Point of intersection ___ In each of the systems you solved above which equation did you choose to solve first Why did you select that equation in each case TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 23 4 3 3 Putting the Pieces Together System 1 2 Zand y 3x 4 Record
29. stem A determine if you can solve the system using each of the three methods you have learned and if you can then solve 2x 3y 10 4x 5y 2 Justify why you can or cannot solve using this method Graphing Justification Substitution Justification Elimination Justification TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 7 3 Which Method Continued For System B determine if you can solve the system using each of the three methods you have learned and if you can then solve 2 5 4 Graphing Justify why you can or cannot Justification solve using this method Substitution Elimination Justification Justification TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 7 3 Which Method Continued For System C determine if you can solve the system using each of the three methods you have learned and if you can then solve pese y 4x 5 Graphing Justify why you can or cannot Justification solve using this method Substitution Elimination Justification Justification TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 7 4 The Frayer Model Facts Characteristics Non Examples TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 47 4 7 5 3 Ways Two catering companies provide food and the banquet hall for weddings proms and ann
30. t you can save RAD AUTO REAL 0 some typing Use the circular navpad and press up twice The first command you entered is highlighted Now press the key The handheld copies the command down to the line you are working on You can now use the navpad to move left until the cursor is behind the symbol It looks like this Press the key to erase the first equation then type the second equation Then press amp As you can see 3 3 is a correct solution for the 2 9 8 xx 6 y iB and y 3 second equation as well Since the solution is correct for both equations from the linear system we know it must be the right answer TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 5 Nspire CAS Handheld Manual Continued How to Add or Subtract Two Equations The elimination method for solving a linear system involves adding or subtracting the given equations Say that you are considering the following linear system 3x 2y 16 5x 2y 8 You probably agree that if we add these two equations the y variable will be eliminated Here is how to do this on a handheld First be certain that you are on a Calculator page If you need help with this see the Getting Started section from earlier in this manual When you type in the equations be certain to enclose them in brackets Remember that we decided to add the equations Here are the keys you should press OVOOOVOVOM COO POOPOUOG
31. the line that results once you have substituted and then solve the equation using your CAS handheld Resulting equation and my solution Don t forget to solve for both variables Check your solution using the CAS handheld Compare your solution to someone else in the class System 2 2 0 and x 12 Record the line that results once you have substituted and then solve the equation using your CAS handheld Resulting equation and my solution Don t forget to solve for both variables Check your solution using the CAS handheld Compare your solution to someone else in the class TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 24 4 3 4 The Sub Way e n groups of three have each person in the group solve one of the systems below e Use your CAS handheld to help you solve and check your system e Share your solutions with each person in the group System A System B 4 24 5 12 13 y 2 4and 5x y 4 0 System C Challenge y 5x CHALLENGE Plot each of the POI s from Systems A B and C and find the equation of the line that connects the three points Equation of Line TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 25 4 3 5 What s My Equation Part 2 Problem A Yasser is renting a car Zeno Car Rental charges 45 for th
32. tly attempt the work on my own try before seeking help If have difficulties ask others but stay on task am committed to tasks at hand know all the different ways available my school where can seek extra o mmmmz O OO lt D o 2 Yes tried my best What will do differently in the next unit to improve TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 54
33. top of your screen will look something like this 6x 2y 16 TTE Now decide how you would start in solving for y RAD AUTO REAL 0 Perhaps you ve decided that subtracting 6x from both sides of the equation is a good start Wonderful To 6 2 16 6 x 2 y 16 do this immediately press Notice that the Pc handheld automatically inserts Ans What is this Ans stands for the last answer you found If you now RAD AUTO REAL press the lt key the handheld will subtract from SD 6 2 16 2 the left side and the right side of 6x 2y 16 You eee Will see this result 6 2 16 2 y 16 6 x 1 gt Continue solving the equation You probably see that to finally isolate the y variable it is necessary to divide the equation by 2 on both sides Again just start 2 16 6 3 8 typing the operation you want to perform Press 2 DAL The handheld will insert Ans for you Press to calculate the result As you can see the handheld reports that y 3x 8 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 3 Nspire CAS Handheld Manual Continued How to Solve for a Variable Example Two Say that you wish to solve the equation 6x 2y 18 for the variable y To do this first be certain that you are on a Calculator page If you need help with this see the Getting Started section above First type in the equation that you want to solve Use the numb
34. w that you ve solved the problems using two different methods which method do you prefer Why 5 When do you think solving by substitution would be preferable to solving by graphing TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 27 4 4 1 The Lowdown on Downloads Let s return to our music downloading problem Here s the problem again iTones and Music Mine are two online music providers Each company charges a monthly membership fee and then a per song download rate iTones charges 10 per month and 1 per song 10 Music Mine charges 7 per month and 1 50 per song 1 5 7 1 Find the number of songs that would need to download where the costs are the same for the two music providers 2 Paris solved the problem and then made the following conclusion If you download less than 6 songs per month than choose iTones since the cost per song is less If you download more than 6 songs per month than choose Music Mine since the fixed cost is less If you download exactly 6 songs per month choose either Is the conclusion that Paris made correct If not underline the part s of her conclusion that are incorrect and then rewrite it so that it is correct TIPS4RM Grade 10 Applied Unit 4 Linear Systems July 2008 4 28 4 4 2 Putting the Pieces Together Continued Solve the system y 10 2x and x 2y 10 The steps to the solution to this system are given in the pieces below Cut t
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