Solving Sstems of Equations 11 th Grade 7-Da Unit Plan Tools Used: TI-83 graphing calculator (teacher) Casio graphing calculator (teacher) TV connection for Casio (teacher) Set of Casio calculators (students) B: Nicole M. McCo 1
Objectives of Unit: The students will recognize the properties of sstems of equations The students will discover four different methods to solving sstems of equations The students will be able to choose the easiest method when solving a sstem Standards addressed throughout lesson: NCTM Standards Numbers & Operations Algebra Problem Solving Communication Connections Representation NYS Ke Ideas Ke Idea 2- Number and Numeration Ke Idea 3 Operations Ke Idea 4 Modeling/Multiple Representation Ke Idea 7 Patterns/Functions Textbook Information: Publisher Scott Foresman Addison Wesle Title The Universit of Chicago School Mathematics Project Authors Senk, Thompson, Viktora, Usiskin, Abbel, Levin, Weinhold, Rubenstein, Jaskowiak, Flanders, Jakucn, Pillsbur Chapter/pages Chapter 5/pages 279-311 Copright 1998 2
Unit Overview Da 1 Solving Sstems of Equations Graphicall Da 2 Solving Sstems of Equations Algebraicall Using Substitution Da 3 Solving Sstems of Equations Algebraicall Using Linear Combinations Da 4 Review of Linear Combinations; Preview to Matrices Da 5 Solving Sstems of Equations Using Matrices on the Casio Da 6 Solving Sstems of Equations using an method word problems Da 7 More word problems and quiz 3
Da 1 Lesson: Solving sstems of equations graphicall Objectives o Students will recognize properties of sstems of equations o Students will estimate solutions to sstems b graphing Extra Materials o None Opening Activit Student Bellwork: The students have 5 minutes to complete the following problem. The students will then pass the paper to a different student and the will correct each others papers. The teacher will collect the papers after we correct them as a class. Main Activit Have the students graph the following on separate graphs (provided b teacher): 3x - 5 2x 2 x 4 This will determine who knows how to use their graphing calculator and who needs more help/practice. As a class, we will discuss how to find the solution to a sstem of linear equations using the graphs of the lines. NOTE This method can onl be used if solving for 1 or 2 variables due to the limited powers (no 3-D capabilities on the graphing calculators) This will be mentioned to the students earl in the lesson. The students will find the solution(s) to the following sstems while working in groups. Then, each group will present one sstem and show how the arrived at the solution(s). Ï 1 Ï x - 7 Ï 3x + 13 x + 1 1. 2. 3. 2 Ó -2x + 5 Ó x + 1 Óx - 2-2 Ï9x - 12 4. Ó 9x + 6 Ï + 3 7x 5. Ó 6x + 2 Ï 5x 6. Ó -3x + 1 Closing Activit Have the students write a brief sentence or two answering this question: In example 4, there was no solution because the lines were parallel. Give a situation where there would be an infinite amount of solutions to a sstem of linear equations. Think about the parallel lines example. Homework Page 283 # 10-12, 15 4
Da 2 Lesson: Solving sstems of equations algebraicall using substitution Objectives o Students will solve 2x2 and 3x3 sstems using substitution o Students will recognize properties of sstems of equations Extra Materials o Class set of Lesson Master 5-3 B Opening Activit Student Bellwork: The students have 5 minutes to complete the following problem. The students will then pass the paper to a different student and the will correct each others papers. The teacher will collect the papers after we correct them as a class. Have students solve the following: If -1, solve the following equations for x: 1. x -2 + 4 2. 4x + 3 9 2 3. 8 x + Answers: 1. x 6 2. x 3 3. x 7 Main Activit Students will learn how to solve sstems of equations using the substitution method and Lesson Master 5-3 B. Together we will discuss how to use substitution to solve for the variables. The teacher will put the following example on the board: 5 2 + 3 4 + 1 5 Ask the students how to change this from two different equations to one equation that is equivalent. Answer: 2 + 3 4 + 1; because the both equal 5, the must both be equal to each other. Discuss when and wh we use substitution. - It is used when we have more than 1 unknown (variable) and one of the equations can be solved for 1 variable. Then we substitute that variable s value in terms of the other variable(s) into the other equation. Discuss how to use substitution. - This is when we, as a class, will work through the odd numbered problems on the Lesson Master 5-3 B. The students will solve 2x2 and 3x3 sstems using substitution. 5
Closing Activit On a half sheet of paper students will comment on the substitution method for solving sstems of equations. The will include: - When and wh we use substitution - How to use substitution - If the prefer graphing the two lines or solving algebraicall and wh This will be the Ticket out of Class and each student needs to hand the teacher a paper to exit the room. (NOTE: No late passes will be allowed if student refuses!) Homework Finish worksheet 6
7 Lesson Master 5-3B In 1-8, use substitution to solve the sstem. Then check. 1. Ó Ï + - - 5 2 7 x x 2. Ó Ï + + 1 13 3 x x 3. Ó Ï - - 11 6 21 1 2 3 n m n m 4. Ó Ï - x x 4 4 5. Ó Ï - + 990 1.5 7.5 78.1.25 x x 6. Ó Ï - + - - + a c a b c b a 4 3 26 3 6 4 7. Ó Ï + + - + 1 1 10 x x z z x 8. Ó Ï - - + 2 2 1 2 1 x x
Lesson Master 5-3B (Answer Ke) In 1-8, use substitution to solve the sstem. Then check. 1. Ï x - 7 Ó -2x + 5 2. Ï 3x + 13 Ó x + 1 x4; -3 x-6; -5 3. Ï3m - 2n 1 Ó21m - 6n 11 4. Ïx 4 Óx -4 m2/3; n1/2 x-4; 1 x4; -1 5. Ï.25x +.1 78 Ó7.5-1.5x 990 6. Ï4a + 6b - 3c -26 b a + 3 Óc -4a x240; 180 a-2; b1; c8 7. Ïx + z 10 z -x + 1 Ó x + 1 8. Ï 1 x + 1 2 Óx - 2-2 x3; 4; z-2 x-3; -4; z2 infinitel man solutions 8
Da 3 Lesson: Solving sstems of equations algebraicall using linear combinations Objectives o Students will solve 2x2 and 3x3 sstems using linear combinations o Students will recognize properties of sstems of equations Extra Materials o Class set of Lesson Master 5-4 B Opening Activit Student Bellwork: The students have 5 minutes to complete the following problem. The students will then pass the paper to a different student and the will correct each others papers. The teacher will collect the papers after we correct them as a class. Have students solve the following: 2x + 3 4x + 3 29 Answers: x 2 7 Main Activit Students will learn how to solve sstems of equations using the linear combinations method and Lesson Master 5-4 B. Together we will discuss how to use linear combinations to solve for the variables. The teacher will put this example on the board: 4x 6 3 2x + 12-6 Ask the following questions: 1. How would we get the coefficients of the x s to be the same number with opposite signs? Answer: Multipl the 2x b -2 2. How would we get the coefficients of the s to be the same number with opposite signs? Answer: Multipl the 6 b 2 Discuss when and wh we use linear combinations. - It is used when we are not able to solve for one variable easil (or at all) and/or one equations variable is a multiple of the others. Discuss how to use linear combinations. - This is when, as a class, we will do the problems together on Lesson Master 5-4 B. Students will solve 2x2 and 3x3 sstems using linear combinations. Closing Activit On a half sheet of paper students will discuss which algebraic method of solving the sstems of equations the would use to solve the following problems and wh. 1. 6x + 12 5 (substitution) 2. x + 9 (linear combinations) 2 10x 2x 2 9
3. 2x + 3 + z 13 (linear combinations) 4. 3x (substitution) 5x 2 4z 7 x 48 4x + 5 + 3z 25 This will be the Ticket out of Class and each student needs to hand the teacher a paper to exit the room. (NOTE: No late passes will be allowed if student refuses!) Homework Finish worksheet 5-4 B 10
Lesson Master 5-4B In 1-8, use linear combinations to solve the sstem. Then check. 1. Ï4x + -12 Ó2x + 2-15 2. Ï4x + 3 2.6 Ó5x - 2 2.1 3. Ï2a + b - 5c -21 a + 2b - 2c -15 Óa - 4b + c 18 4. Ï8m - 2n -16 Ó2m -.5n -4 5. 2 2 Ï 12x - 5 523 2 2 Ó 6x + 2 482 6. Ï4x + 5-14 Ó8x + 10-20 7. Ï1 x - -8 4 1 x + 4 14 Ó2 8. Ïd + 9e - f 13 3d + e + 2 f -7 Ó2d + 2e + 2 f -6 11
Lesson Master 5-4B Answer Ke In 1-8, use linear combinations to solve the sstem. Then check. 1. Ï4x + -12 Ó2x + 2-15 2. Ï4x + 3 2.6 Ó5x - 2 2.1 x-1.5; -6 x.5;.2 3. Ï2a + b - 5c -21 a + 2b - 2c -15 Óa - 4b + c 18 4. Ï8m - 2n -16 Ó2m -.5n -4 a-1; b-4; c3 infinitel man solutions 5. 2 2 Ï 12x - 5 523 2 2 Ó 6x + 2 482 6. Ï4x + 5-14 Ó8x + 10-20 x8; 7 x8; -7 x-8; 7 x-8; -7 no solution 7. Ï1 x - -8 4 1 x + 4 14 Ó2 8. Ïd + 9e - f 13 3d + e + 2 f -7 Ó2d + 2e + 2 f -6 x-12; 5 d0; e1; f-4 12
Da 4 Lesson: Review of Linear Combinations and Solving sstems of equations algebraicall using matrices on graphing calculator Objectives o Students will solve 2x2 and 3x3 sstems using matrices on graphing calculator o Students will recognize properties of sstems of equations Extra Materials o Students worksheets from previous class da (Lesson Master 5-4 B) Opening Activit Student Bellwork: The students have 5 minutes to complete the following problem. The students will then pass the paper to a different student and the will correct each others papers. The teacher will collect the papers after we correct them as a class. Have students solve the following: Solve the following sstem of equations using linear combinations: 2x 4 Answer: x 4 -x + 3 8 4 Main Activit Students will review how to use the linear combination method to solve sstems of equations. As a class, we will go over the homework problems that caused the students difficult. After all homework questions have been answered and the students have done a 3x3 sstem, we will quickl set up the matrix method on the calculators. STEPS: Turn calculator on to the main menu. Go to EQUA and hit EXE. This is a simultaneous process, so hit F1 for simultaneous. Next, we need to enter the number of unknowns. For a 3x3 we will have 3 unknowns (variables). Enter in all of the coefficients and the constants as prompted on the screen. After all the numbers are entered, hit F1 to solve. This will give the answer matrix to the 3x3 sstem we entered. Ask the students which wa the prefer. I guarantee it will be the matrix method. Closing Activit Have the students tr another sstem from their homework on the calculator. Have them do as man as time will allow. Discuss with them quickl that this will not however be accepted as full credit work on an exam. There is some work involved and we will discuss that tomorrow. 13
Da 5 Lesson: Solving sstems of equations algebraicall using matrices on the calculator and wh it works Objectives o Students will solve 2x2 and 3x3 sstems using matrices on the calculator o Students will recognize properties of sstems of equations o Students will understand the steps involved to solve sstems using the matrix method Extra Materials o Class set of worksheets Opening Activit Student Bellwork: The students have 5 minutes to complete the following problem. The students will then pass the paper to a different student and the will correct each others papers. The teacher will collect the papers after we correct them as a class. Main Activit Have students solve the following sstem using the matrix method: 2x 2 + 4z 7 Answer: x -33-4x + 2 3z 14-126.5 x + 4-12z 1 z -45 Have the students learn how the matrix method works without actuall showing them how to use a matrix. We will as a class go through the steps to solving a sstem using the matrix method. Step 1: Change sstem of equations to matrix equation: Ï2x 2 + 4z 7-4x + 2 3z 14 Óx + 4-12z 1 This sstem becomes the following matrix coefficient matrix constant matrix È 2-2 4 Èx È 7 Í Í - 4 2-3 Í Í Í Í 14 ÍÎ 1 4-12 ÍÎ z ÍÎ 1 variable matrix Step 2: Label the matrices: È 2 Í Í - 4 ÍÎ 1-2 4 Èx È 7 2-3 Í Í Í Í 14 4-12 ÍÎ z ÍÎ 1 A B 14
Step 3: Multipl [ A ] -1 x [ A ] so that all we have on the left side of the equation is the variable because we have an equation, we need to multipl [ A ] -1 x [ B ] on the right side of the equation. matrix. And, So we now have: [ A ] -1 x [ A] Step 4: [ ] -1 A x [ ] Èx Í Í ÍÎ z Èx Í Í ÍÎ z [ A ] -1 x [ B] A cancel out to become the identit matrix (essentiall 1) and so we are now [ A ] -1 x [ B ], where [ A ] -1 x [ ] B is the answer matrix. left with: This is enough work to show for full credit. Now, the students will put numbers into the calculator to get the answer matrix. The will work on the worksheets in pairs and solve for the variables, showing all of their work. Closing Activit Explain in our own words wh [ A ] -1 x [ ] A cancels out (becomes 1). Give an example using numbers. 1 1 Example Answer: 2 1, where is the inverse and 1 is the identit. 2 2 Homework Finish worksheet 15
Name Date Solving Sstems of Equations using Matrices Directions: Solve the following sstems of equations using the matrix method on our calculator. 1) 3x + 4z -17 1) Number of unknowns 4x + 3-5z 4 x + 6 + 2z -6 2) m + n + p + q 7 2) Number of unknowns -2m + 4n p + 3q 1 4m 2n + 4p + q 4 -m + 2n 3p 2q 8 3) s + t u 5 3) Number of unknowns 2s - 5t + 3u 10 -s + 6t 7u 2 4) a + 2b + 3c + 4d + 5e 6 4) Number of unknowns -a 3b 2c 5d 4e 12 4a + 7b 7c + 8d e -2-3a + 2b + 8c 2e 14 6a 5b 2c + d 4e 0 5) 2h j + 4k 2m 23 5) Number of unknowns 4h + 2j k + 3m -1 h 5j + 8k 4m 19-3h + j 2k -6 16
Name Answer Ke Date Solving Sstems of Equations using Matrices Directions: Solve the following sstems of equations using the matrix method on our calculator. 1) 3x + 4z -17 1) Number of unknowns 3 4x + 3-5z 4 x + 6 + 2z -6 x -2.2731;.2098; z -2.4927 2) m + n + p + q 7 2) Number of unknowns 4-2m + 4n p + 3q 1 4m 2n + 4p + q 4 -m + 2n 3p 2q 8 m -3; n 9; p 11; q -10 3) s + t u 5 3) Number of unknowns 3 2s - 5t + 3u 10 -s + 6t 7u 2 s 4 1/3; t -1 2/3; u -2 1/3 4) a + 2b + 3c + 4d + 5e 6 4) Number of unknowns 5 -a 3b 2c 5d 4e 12 4a + 7b 7c + 8d e -2-3a + 2b + 8c 2e 14 6a 5b 2c + d 4e 0 a 32.4846; b 25.9596; c 8.4963; d -31.2455; e 4.2179 5) 2h j + 4k 2m 23 5) Number of unknowns 4 4h + 2j k + 3m -1 h 5j + 8k 4m 19-3h + j 2k -6 h 2.4285; j 6.5714; k 2.6428; m -7.0714 17
Da 6 Lesson: Given word problems, change into sstems of equations and solve for the variables Objectives o Students will change word problems into sstems of equations and solve for variables o Students will solve 2x2 and 3x3 sstems using matrices on the calculator o Students will recognize properties of sstems of equations Extra Materials o Class set of word problem worksheets Opening Activit Student Bellwork: The students have 5 minutes to complete the following problem. The students will then pass the paper to a different student and the will correct each others papers. The teacher will collect the papers after we correct them as a class. Main Activit Have students solve the following sstem using whichever method the choose: -3x + 4-2 Answers: x -2.8 -x + 2 6 1.6 Given word problems, students will change into a sstem of equations and then solve for the variables. We will do some examples as a class, and some examples will be done as pairs. Students can use an method that we have learned to solve the sstems. Closing Activit Given this sstem, tr to write a word problem that would make sense. You don t need to solve the sstem, just write a word problem. Homework x + 10 7x 3 25.60 Finish worksheet on word problems. 18
Worksheet on Word Problems 1. Five ards of fabric and three spools of thread cost $40.12. Two ards of the same fabric and ten spools of the same thread cost $23.88. Find the cost of a ard of fabric and the cost of a spool of thread. Fabric Thread 2. Half a watermelon and a half pound of cherries cost $3.09. A whole watermelon and two pounds of cherries cost $8.16. a. Write a sstem of equations that can be used to find the cost of each tpe of fruit. b. Solve the sstem to find the cost of each tpe of fruit. Watermelon Cherries 3. Two apples and six plums provide 300 calories. Three apples and five plums provide 350 calories. How man calories are provided b five apples and eight plums? 19
4. At Wet Pets, a starter aquarium kit costs $15 plus $.60 per fish. At Gills and Frills, the same kit is $13 plus $.80 per fish. a. Give an equation for the cost c of f fish at each store. Wet Pets Gills and Frills b. For what number of fish is the cost the same at the two stores? 5. For the Summer Rock Festival, there is one price for students, one for adults, and another for senior citizens. The Rueda famil bought 3 student tickets and 2 adult tickets for $104. The Cosentinos bought 5 student tickets, 1 adult ticket, and 2 senior citizen tickets for $155. The Cragins bought 2 of each for $126. a. Write a sstem of equations that can be used to find the cost of each ticket b. Solve the sstem to find the cost of each ticket. Students Adults Senior Citizens 6. A biccle, three triccles, and a uniccle cost $561. Seven biccles and a triccle cost $906. Five uniccles, two biccles, and seven triccles cost $1758. a. Set up a sstem of equations that can be used to find the cost of each item. b. Solve the sstem to find the cost of each tpe of ccle. Biccle Triccle Uniccle 20
Worksheet on Word Problems Answer Ke 1. Five ards of fabric and three spools of thread cost $40.12. Two ards of the same fabric and ten spools of the same thread cost $23.88. Find the cost of a ard of fabric and the cost of a spool of thread. Fabric $7.49 Thread $.89 5f + 3t 40.12 2f + 10t 23.88 2. Half a watermelon and a half pound of cherries cost $3.09. A whole watermelon and two pounds of cherries cost $8.16. a. Write a sstem of equations that can be used to find the cost of each tpe of fruit..5w +.5c 3.09 w + 2c 8.16 b. Solve the sstem to find the cost of each tpe of fruit. Watermelon $4.20 Cherries $1.98 per lb. 3. Two apples and six plums provide 300 calories. Three apples and five plums provide 350 calories. How man calories are provided b five apples and eight plums? 575 calories 2a + 6p 300 3a + 5p 350 a 75 p 25 5a + 8p? 5(75) + 8(25) 575 21
4. At Wet Pets, a starter aquarium kit costs $15 plus $.60 per fish. At Gills and Frills, the same kit is $13 plus $.80 per fish. a. Give an equation for the cost c of f fish at each store. Wet Pets C 15 +.60f Gills and Frills C 13 +.80f b. For what number of fish is the cost the same at the two stores? 10 fish 5. For the Summer Rock Festival, there is one price for students, one for adults, and another for senior citizens. The Rueda famil bought 3 student tickets and 2 adult tickets for $104. The Cosentinos bought 5 student tickets, 1 adult ticket, and 2 senior citizen tickets for $155. The Cragins bought 2 of each for $126. a. Write a sstem of equations that can be used to find the cost of each ticket 3s + 2a 104 5s + a + 2c 155 2s + 2a + 2c 126 b. Solve the sstem to find the cost of each ticket. Students $18 Adults $25 Senior Citizens $20 6. A biccle, three triccles, and a uniccle cost $561. Seven biccles and a triccle cost $906. Five uniccles, two biccles, and seven triccles cost $1758. a. Set up a sstem of equations that can be used to find the cost of each item. b + 3t + u 561 7b + t 906 2b + 7t + 5u 1758 b. Solve the sstem to find the cost of each tpe of ccle. Biccle $117 Triccle $87 Uniccle $183 22
Da 7 Lesson: Given word problems, change into sstems of equations and solve for the variables Quiz on Sections 5-2 to 5-4 Objectives o Students will change word problems into sstems of equations and solve for variables o Students will solve 2x2 and 3x3 sstems using matrices on the calculator o Students will recognize properties of sstems of equations Extra Materials o Class set of quizzes Opening Activit Main Activit Quiz on Sections 5-2, 5-3, 5-4, 5-6 and including word problems As a class, we will review how to change a word problem into a sstem of equations. We will go over in detail the previous night s homework sheet answering an and all questions that ma arise. The students will first share their answers on the board: o One student will put the sstem of equations on the board o Another student will solve the problem for the necessar information This method will be repeated for each homework problem. Closing Activit Have students change this word problem into a sstem of 2 equations and hand in before exiting the classroom: At the zoo, Ja bought 3 slices of vegetable pizza and 1 small lemonade for $5.40. Terri paid $4.80 for 2 slices of vegetable pizza and 2 small lemonades. Answer: 3P + 1L 5.40 2P + 2L 4.80 23
Name Chapter Quiz Date Advanced Algebra In questions 1 & 2, solve the sstem graphicall or algebraicall. Show all work. 1. Ï12s - 8t 56 Ó5s + 3t - 2 1. 2. Ï19x + 4-7 Ó 3x + 6 2. 3. Consider the sstem graphed below. How man solutions does the sstem have? 3. In questions 4 & 5, refer the the following situation: At Federal Rent-A-Car, the cost of a one-da rental of a midsize car is $45 plus $0.27 per mile driven. At Read Rentals, the cost is $27 per da plus $0.36 per mile driven. 4. Let x the number of miles driven and 4. the cost of a one-da rental with x miles driven. Set up a sstem of two equations to describe this situation. 5. a. For what number of miles driven will 5. a. the cost of a one-da rental be the same at Federal Rent-A-Car and at Read Rental. b. What is the cost for this number of b. miles driven? *Extra Credit* For what value of k does Justif our answer. Ïkx + 2 12 Ó9x + 2 8 have no solution? 24
Name Answer Ke Date Chapter Quiz Advanced Algebra In questions 1 & 2, solve the sstem graphicall or algebraicall. Show all work. 1. Ï12s - 8t 56 Ó5s + 3t - 2 1. s 2; t -4 2. Ï19x + 4-7 Ó 3x + 6 2. x -1; 3 3. Consider the sstem graphed below. How man solutions does the sstem have? 3. 2 solutions In questions 4 & 5, refer the the following situation: At Federal Rent-A-Car, the cost of a one-da rental of a midsize car is $45 plus $0.27 per mile driven. At Read Rentals, the cost is $27 per da plus $0.36 per mile driven. 4. Let x the number of miles driven and 4. 45+.27x the cost of a one-da rental with x miles driven. Set up a sstem of two 27+.36x equations to describe this situation. 5. a. For what number of miles driven will 5. a. 200 miles the cost of a one-da rental be the same at Federal Rent-A-Car and at Read Rental. b. What is the cost for this number of b. $99.00 miles driven? *Extra Credit* For what value of k does Justif our answer. Ïkx + 2 12 Ó9x + 2 8 have no solution? When k 9, there is no solution because when ou perform linear combinations on the sstem ou get that 04 and this will never be true, therefore the lines are parallel, and no solution exists. 25