Linear Systems and Matrices

Transcription

1 Linear Systems and Matrices A lot of work in science from biology to criminology is done using computers, and matrices make lighter work of complex data..1 A Tale of Two Systems Solving Systems of Two Equations We Can Work It Out Solving Systems of Three Equations Algebraically Step Inside the Matrix Introduction to Matrices and Matrix Operations Another Tool in the Toolbox Solving Matrix Equations _A_TX_Ch0_ indd 87 14/03/14 :18 PM

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3 A Tale of Two Systems Solving Systems of Two Equations.1 Learning Goals In this lesson you will: Solve systems of two linear equations. Solve systems of equations involving one linear and one quadratic equation. Your body is an amazing collection of different systems. Your cardiovascular system pumps blood throughout your body, your skeletal system provides shape and support, and your nervous system controls communication between your senses and your brain. Your skin, including your hair and fingernails, is a system all by itself the integumentary system and it protects all of your body s other systems. You also have a digestive system, endocrine system, excretory system, immune system, muscular system, reproductive system, and respiratory system. Why do we call these systems systems? What do you think makes up a system? 89

4 Problem 1 Systematic Déjà Vu You would like to take a taxi to the airport. There are two local taxi companies. Friendly s Cab Company charges $.60 plus $0.0 per one-sixth of a mile driven. Anderson Taxi, Inc. charges $5.00 plus $0.10 per one-sixth of a mile driven. 1. Formulate a system of two linear equations in two variables to represent this problem situation. Be sure to define your variables.. Graph the system of equations y Cost (dollars) Distance Driven (in one-sixth miles) x 3. Estimate the solution to the system of equations. Justify your reasoning. Remember, the solution to a system of linear equations occurs when the values of the variables satisfy all of the linear equations. 90 Chapter Linear Systems and Matrices

5 ?4. Steven and Juan were asked to use an algebraic method to solve the system of two linear equations in two variables. Juan Steven I solved the system of equations using substitution. y 5 0.0x 1.60 y x x x x 5.40 x 5 4 Substitute x 5 4 into one of the linear equations. y 5 0.0x 1.60 y 5 0.0(4) 1.60 y y The solution to the system is (4, 7.4). I solved the system of equations using linear combinations. y 5 0.0x 1.60 y x x 1 y x y x x 5.40 x 5 4 Substitute x 5 4 into one of the linear equations. y x y (4) y y The solution to the system is (4, 7.4). Whose method is correct? Justify your reasoning. 5. What does the solution mean in terms of the problem situation?.1 Solving Systems of Two Equations 91

6 6. Suppose that Anderson Taxi, Inc. decides to increase its fare to $0.0 per one-sixth mile driven. Write a new system of equations to reflect the increased fare. When will the cost of using the two taxi companies be equal? Explain your reasoning. 7. Think about the graphs of different systems of two linear equations. a. Describe the different ways in which the two graphs can intersect, and provide a sketch of each case. b. How does this relate to the number of solutions to a system of two linear equations? 9 Chapter Linear Systems and Matrices

7 Problem Now Let s Kick It Up a Notch! A system of equations can also involve non-linear equations, such as quadratic equations. Luckily, the methods for solving a system of non-linear equations are similar to methods for solving a system of linear equations. 1. Consider the system of a linear equation and a quadratic equation: x 1 y 5 8 x y 5 5. a. Use substitution to write a new equation that can be used to solve this system. b. Solve the resulting equation for x. c. Calculate the corresponding value(s) for y. Determine the solution(s) to the system of equations..1 Solving Systems of Two Equations 93

8 d. Graph and label each equation of the system and identify the point(s) of intersection. y x e. What do you notice about the solutions that you determined algebraically and graphically?. Think about the graphs of a linear equation and a quadratic equation. a. Describe the different ways in which the two graphs can intersect, and provide a sketch of each case. b. How does this relate to the number of solutions to a system of one linear and one quadratic equation? 94 Chapter Linear Systems and Matrices

9 3. Can a system of a linear equation and a quadratic equation ever have infinitely many solutions? Explain your reasoning. 4. Solve each system of two equations in two variables algebraically. Then verify the solution graphically. a. y 5 x 1x 1 8 y 5 3x 1 y x Solving Systems of Two Equations 95

10 b. y 5 x 3x 4 y 5 x 8 y x c. y 5 x 1 4x 1 3 y 5 4x 1 y x Chapter Linear Systems and Matrices

11 5. Simon and his sister are playing a guessing game. Simon tells his sister that he is thinking of two positive numbers. The first number minus the second number is 15. The square of the first number minus 0 times the second number is equal to 300. a. Formulate a system of one linear and one quadratic equation to represent the two numbers. Be sure to define your variables. b. Solve the system of equations. c. What are the two numbers that Simon is thinking of? Explain your reasoning..1 Solving Systems of Two Equations 97

12 Problem 3 Say Cheese! A photographer specializes in taking senior portraits. She records the amount of profit she earns for each senior portrait package that she sells. Number of Portraits Sold Profit (dollars) Use quadratic regression to write an equation to model the photographer s profit. Be sure to define your variables.. Each month, the photographer must keep track of her costs and revenue. Her costs consist of a fixed amount of $1400, which includes rent, utilities, and workers salaries, as well as $30 per package to print the portraits. Write a linear equation to model the photographer s costs. 3. Formulate a system of one linear and one quadratic equation to represent this problem situation. 98 Chapter Linear Systems and Matrices

13 4. Graph the system of equations y x 5. Solve the system of equations. 6. Explain the solution(s) in terms of the problem situation..1 Solving Systems of Two Equations 99

14 7. Determine the amount of portrait packages that the photographer needs to sell in order to make a profit. Is this solution reasonable in terms of the problem situation? Due to the rising costs of running a business, the photographer anticipates fixed costs in the next year to be $400 per month, whereas the costs to print each portrait will increase to $35 per portrait. 8. Formulate a new system of equations to reflect the changes in the cost. 9. Graph the system y x 10. Determine the solution(s) to the system of equations. 11. Explain the solution(s) in terms of the problem situation. Are the solutions reasonable? Be prepared to share your solutions and methods. 100 Chapter Linear Systems and Matrices

15 We Can Work It Out Solving Systems of Three Equations Algebraically. Learning Goals Key Term In this lesson, you will: Formulate and solve systems of three linear equations in three variables by using substitution. Formulate and solve systems of three linear equations in three variables by using Gaussian elimination. Gaussian elimination Although the method you ll learn about in this lesson Gaussian elimination was named after a mathematician who lived in the late 18th and early 19th centuries, there is evidence of its use as early as the nd century in China. And the method also appears in notes by Isaac Newton in the 17th century. Ironically, although the method is named after him, Gauss simply contributed a handy notation for it. 101

16 Problem 1 Yeah, That s the Ticket! As a fundraising event, your club sold tickets to a special viewing of a new summer movie. The fundraiser was successful: you sold all 800 seats in the school s auditorium! You sold tickets at three different prices: $.50 for children under 1 years old, $3.50 for youth between 1 and 18 years old, and $5.00 for adults. The total amount of money taken in was $937.50, and there were 4 times as many youth tickets as children s tickets sold. How many of each type of ticket were sold? 1. Formulate a system of three linear equations in three variables to represent this situation. Be sure to define your variables. You can solve systems of three linear equations in three variables by using substitution. First, solve one equation for a variable and then substitute that expression into the other two equations. This reduces the system of three equations in three variables to a system of two equations in two variables, which you can then solve using any method.. The equation t 5 4c is already solved for a variable. Substitute 4c for t in the other two equations to create a system of two equations in two variables. Once the system is in two equations, you can use graphing, substitution, or linear combinations to solve it. 10 Chapter Linear Systems and Matrices

17 3. Solve the resulting system. Explain what your solution represents in terms of the problem situation.. Solving Systems of Three Equations Algebraically 103

18 4. Allison and Emily were asked to solve this system of three linear equations using substitution. Analyze their methods. x 1 y 1 z 5 x 3y 1 z x 1 3y z 5 5 Allison x 1 y 1 z 5 x 5 y z x 3y 1 z x 1 3y z 5 5 ( y z) 3y 1 z ( y z) 1 3y z y z 3y 1 z y 4z 1 3y z y y 5z 5 5 5y 5 10 y 1 5z y 5 10 Then I solved the system y 1 5z Emily 4x 1 3y z 5 5 4x 1 3y 5 5 z x 3y 1 z 5 14 x 1 y 1 z 5 x 3y 1 (4x 1 3y 5) 5 14 x 1 y 1 (4x 1 3y 5) 5 x 3y 1 8x 1 6y x 1 4y x 1 3y x 1 4y x 1 3y x 1 3y 5 4 Then I solved the system 5x 1 4y Chapter Linear Systems and Matrices

19 a. Describe the similarities and differences in their methods. b. Demonstrate that Allison s method and Emily s method will both yield the same solution. Explain your reasoning. c. Could this system have been solved using a different substitution? How do you select which variable to solve for? Explain your reasoning.. Solving Systems of Three Equations Algebraically 105

20 5. During the movie fundraising event, the concession stand at the auditorium sells popcorn, fruit, and drinks. The price of a box of popcorn is $0.50 more than the price of a piece of fruit, and the price of two drinks is $0.50 less than the price of three pieces of fruit. You order two boxes of popcorn, one piece of fruit, and three drinks and the total comes to $7.75. a. Formulate a system of three linear equations in three variables to represent this situation. Be sure to define your variables. b. Calculate the price of each item. Use substitution to solve the system of three linear equations in three variables. 106 Chapter Linear Systems and Matrices

21 Problem Operation: Elimination Gaussian elimination is an algorithm for solving linear systems of equations, named after the mathematician Carl Friedrich Gauss. It involves using linear combinations of the equations in the system to isolate one variable per equation. The goal of Gaussian elimination is to eliminate all variables except one in each equation. To do this, you should look for a variable that has a coefficient of 1. Multiply the equation by a constant, and add it to another equation to eliminate a variable. When using this method, there are three rules: Swap the positions of the two rows. Multiply an equation by a nonzero constant. Add one equation to the multiple of another. x 1 y 1 3z 5 3 Consider the system x 1 3y 1 z 5 5 3x 1 y 1 z 5 4 The second equation has a variable, x, with a coefficient of 1. You can use the second equation to eliminate the x in the other two equations. 1. Multiply the second equation (x 1 3y 1 z 5 5) x 6y 4z 5 10 by, and add the result 1x 1 y 1 3z 5 3 to the first equation. 4y z 5 7. Replace the first equation in the system with the new equation. 4y z 5 7 x 1 3y 1 z 5 5 3x 1 y 1 z Multiply the second 3(x 1 3y 1 z 5 5) 3x 9y 6z 5 15 equation by 3, and 13x 1 y 1 z 5 4 add the result to the 8y 5z 5 11 third equation. 4. Replace the third equation in the system with the new equation. 4y z 5 7 x 1 3y 1 z 5 5 8y 5z 5 11 You have now isolated the x variable to the first equation. Continue in this same manner to isolate y and z.. Solving Systems of Three Equations Algebraically 107

22 1. Continue to solve the system of three linear equations in three variables from the worked example using Gaussian elimination. a. Swap the first and second equations so that the isolated x variable is on top. x 1 3y 1 z 5 5 4y z 5 7 8y 5z 5 11 Remember, look for variables with a coefficient of 1 or variables that are multiples of each other to help you decide which variables to eliminate. b. Multiply the second equation by, and add the result to the third equation. Replace the third equation. c. Multiply the third equation by 1 3. d. Add the third equation to the second equation. Replace the second equation. e. Multiply the second equation by 1 4. f. Multiply the second equation by 3 and add the result to the first equation. Replace the first equation. g. Multiply the third equation by and add the result to the first equation. Replace the first equation. 108 Chapter Linear Systems and Matrices

23 . Jamie claims that the system in Question 1 could have been solved more efficiently by starting with a different approach. Jamie I multiplied the second equation by -5 and added it to the third equation. x 1 3y 1 z 5 5 4y z 5 7 8y 5z 511 5(4y z 5 7) 0y 1 5z y 5z y 5 4 a. How is Jamie s method different from the method used in Question 1?. Solving Systems of Three Equations Algebraically 109

24 b. Complete Jamie s method. Describe your steps. c. How does this solution compare to the solution you got in Question 1? d. Compare the two methods. What can you determine about the order in which equations are combined when using Gaussian elimination? 110 Chapter Linear Systems and Matrices

25 There is no one correct order in which to perform the steps to Gaussian elimination. The idea is to keep performing linear combinations until each equation contains a different isolated variable.? of how to start the problem. 3. Ethan, Jackson, and Olivia are each asked to use Gaussian elimination to solve a system of three linear equations in three variables, but they each have a different idea x 1 y 1 z 5 3 3x y 1 z 5 11 x 1 y z 5 8 Ethan I would add equation 1 to equation 3, and replace equation 3. This would eliminate the x variable. x 1 y 1 z 5 3 x 1 y z 5 8 y 1 z 5 5 x 1 y 1 z 5 3 3x y 1 z 5 11 y 1 z 5 5 Jackson I would add equation 1 to equation and replace equation. This would eliminate the y variable. x 1 y 1 z 5 3 3x y 1 z x 1 4z 5 14 x 1 y z 5 3 5x 1 4z 5 14 x 1 y z 5 8 Olivia I would subtract equation from equation 1 and replace equation. This would eliminate the z variable. x 1 y 1 z 5 3 3x 1 y z 5 11 x 1 y 5 8 x 1 y 1 z 5 3 x 1 y 5 8 x 1 y z 5 8 a. Describe the similarities and differences among all three methods. b. Whose method is correct? Explain your reasoning. c. Is there a different way to begin the problem than the ones that were mentioned? Justify your answer.. Solving Systems of Three Equations Algebraically 111

26 4. Solve the system of three linear equations in three variables from Question 3 using Gaussian elimination. 11 Chapter Linear Systems and Matrices

27 5. Colleen has 1 coins in her pocket. The mix of quarters, nickels, and dimes add up to two dollars, and she has three times as many quarters as nickels. a. Formulate a system of three linear equations in three variables to represent this problem situation. Be sure to define your variables. Don t forget the three rules of Gaussian elimination! Swap, add, or multiply by a scalar.. Solving Systems of Three Equations Algebraically 113

28 b. How many of each coin does Colleen have in her pocket? Solve this system using Gaussian elimination. 114 Chapter Linear Systems and Matrices

29 Problem 3 Have It Your Way You can use either substitution or Gaussian elimination to solve systems of three linear equations in three variables. 1. List an advantage and a disadvantage to the substitution method.. List an advantage and a disadvantage to the Gaussian elimination method. 3. For each system, determine whether you would prefer to use substitution or Gaussian elimination. Justify your reasoning. Perform the first step. a. 5x 1 3y 1 z 5 3 4x 3y 5 4 x 3y 1 4z 5 14 b. x 1 y 1 z 5 7 3x 1 4y z 5 10 x y 5 7. Solving Systems of Three Equations Algebraically 115

30 4. Solve the system using whichever method you prefer. x 1 3y 4z 5 16 y 1 z 5 1 3x 4z 5 9 Be prepared to share your methods and solutions. 116 Chapter Linear Systems and Matrices

31 Step Inside the Matrix Introduction to Matrices and Matrix Operations.3 Learning Goals In this lesson you will: Determine the dimensions of a matrix. Identify specific matrix elements. Perform matrix operations such as addition, subtraction, and multiplication. Key Terms matrix (matrices) dimensions square matrix matrix element scalar multiplication matrix multiplication In computer programming, matrices are widely used, although depending on the language you write in, they can be called arrays or lists. Regardless, every modern computer programming language comes ready with a way to implement matrices. Programming with matrices (or arrays or lists) allows operations to be applied across a large collection of values, which provides for increased speed in processing and conciseness in writing programs. These types of operations are essential when dealing with large amounts of data. Most every computer software you use from social media to operating systems employs matrices to get work done. 117

32 Problem 1 If the Shoe Fits Systems of equations can become cumbersome to solve by hand, particularly when the system contains three or more variables. There is another method that you can use to solve systems of linear equations using technology. It involves a mathematical object called a matrix. A matrix (plural matrices) is an array of numbers composed of rows and columns. A matrix is usually designated by a capital letter. The dimensions of a matrix are its number of rows and its number of columns. A square matrix is a matrix that has an equal number of rows and columns. Rows are horizontal, while columns are vertical. This matrix is an n 3 m matrix, because it has n rows and m columns. a 11 a 1 a 1m a 1 a a m a n1 a n a nm Each number in the matrix is known as a matrix element. Each matrix element is labeled using the notation a i j, where a i j means the number in the ith row and the jth column. The Fleet Feet shoe store in your town just got a shipment of Phantoms, a top-selling running shoe. They are available in sizes 9 through 13 and in widths C, D, E, and EE. The manager of the store keeps track of the inventory according to the number of pairs of shoes in each size and width using the following matrix: A 5 Use matrix A to answer each question. C D E EE Describe what each row and each column represents in terms of the problem situation. 118 Chapter Linear Systems and Matrices

33 . What are the dimensions of this matrix? 3. Determine the number that is in each location. Describe the meaning of the matrix element in terms of the problem situation. a. The element a 34 b. The element a 45 c. The element a 1 4. The Fleet Feet shoe store in a neighboring town carries the same types of shoes and represents its inventory using matrix B. B 5 C D E EE 0 a. What are the dimensions of matrix B? b. What matrix element represents the number of size 10E shoes? Write your answer in matrix notation. c. How is b 4 different from b 4?.3 Introduction to Matrices and Matrix Operations 119

34 5. Coach Tirone, a local track coach, wants every player on her team to wear Phantoms. She calls the managers of both Fleet Feet branches and requests a copy of their inventory report. a. Determine the total number of Phantoms available at both branches of Fleet Feet for each shoe size. Represent your findings as matrix C. b. Describe the method you used to calculate each matrix element for matrix C. 6. Suppose Coach Tirone has a record of each team member s shoe size, represented by matrix D. D 5 C D E EE a. Calculate the number of Phantoms that Fleet Feet would have left in their inventory at both stores after the entire track team buys their shoes. Record your answer as matrix E. b. Describe the method you used to calculate each matrix element for matrix D. 10 Chapter Linear Systems and Matrices

35 7. Coach Tirone also inquires about the Mirage running shoes, which are only available through the Fleet Feet website. The available stock is represented by matrix F. F C D E EE EEE Are there enough Mirages available to outfit the entire team? Explain your reasoning in terms of the problem situation. 8. Coach Tirone decides that every member of the team needs to own two pairs of shoes to prevent overuse. a. Determine how many shoes the track team needs to fulfill Coach Tirone s request. b. Describe the method you used to calculate each matrix element. Multiplying each element of a matrix by a constant is called scalar multiplication. The resulting matrix is designated by a capital letter with a constant coefficient, such as D. You can use technology to help you perform basic matrix operations such as addition, subtraction, and multiplication. 9. Consider the matrices A and B Calculate A 1 B..3 Introduction to Matrices and Matrix Operations 11

36 You can use technology to add matrices. Step 1: Using a graphing calculator, open the matrix menu by pressing nd and then the x 1 key. NAMES MATH EDIT 1:[A] :[B] 3:[C] 4:[D] 5:[E] 6:[F] 7 T [G] Step : Enter the dimensions of matrix A and each matrix element. Then return to the home screen. MATRIX[A] *3 [1-5 4 ] [0-7 ] 1,1=1 Step 3: Repeat the process to enter matrix B. MATRIX[B] *3 [3-1 ] [ ] 1,1=3 Step 4: Open the matrix menu by pressing nd and then the x 1 key. Select matrix A and press enter. Type the 1 key, and then repeat the process to select matrix B. Press enter. [A]+[B] J j 10. Compare the answer you calculated in Question 9 with the answer returned by your calculator. How do they compare? 1 Chapter Linear Systems and Matrices

37 11. Calculate A B. Then use technology to verify your answer. 1. Answer each question about the properties of matrices for addition and subtraction. Then, use the given matrices to support your claim with an example or counterexample. A B D E C a. Describe the dimensions of matrices that can be added or subtracted. Describe the dimensions of matrices that cannot be added or subtracted. Explain your reasoning..3 Introduction to Matrices and Matrix Operations 13

38 b. Determine whether matrix addition and subtraction is commutative. 14 Chapter Linear Systems and Matrices

39 c. Determine whether matrix addition and subtraction is associative..3 Introduction to Matrices and Matrix Operations 15

40 Problem Go For the Gold Coach Tirone s team is at their first track meet. The finishing places and number of medals earned for four of her sprinters are displayed in matrix A. Points are awarded according to the scoring matrix B. A 5 Lauren Kerri Meaghan Erin 3 1st nd 3rd 4th 5th st nd B 5 3rd 4th 5th Points Determine each sprinter s score for the meet.. Describe the method you used to calculate each sprinter s score. 3. Record the sprinters overall scores as a matrix. Label the rows and columns. 16 Chapter Linear Systems and Matrices

41 When you determined the matrix representing the overall scores for each sprinter, you actually performed a process called matrix multiplication. In matrix multiplication, an element a pq of the product matrix is determined by multiplying each element in row p of the first matrix by an element from column q in the second matrix and calculating the sum of the products. In order to multiply matrices, the number of columns in the first matrix must be the same as the number of rows in the second matrix. Lauren Kerri Meaghan Erin Sprinter s Matrix Scoring Matrix Results Matrix 1st nd 3rd 4th 5th st nd 3rd 4th 5th Points Record the dimensions for each of the matrices in the table. 5 Lauren Kerri Meaghan Erin Points Dimensions of Sprinter s Matrix Dimensions of Scoring Matrix Dimensions of Results Matrix Notice that the number of columns in the sprinter matrix is equal to the number rows in the scoring matrix. 5. Josh made an observation about the dimensions of the product matrix. Josh sprinters by medals 3 medals by points 5 sprinters 3 points results Write a generalization about the dimensions of the product matrix, based on Josh s statement..3 Introduction to Matrices and Matrix Operations 17

42 3 6. Consider the matrices A and B a. Is the product matrix AB defined? Explain your reasoning. b. Predict the dimensions of the product matrix AB. Explain your reasoning. To multiply matrices, start with the first row of the first matrix and the first column of the second matrix. Multiply each corresponding element, and then calculate the sum of the products. Because it is the product of the first row in matrix A and the first column in matrix B, the product will go in the first row, first column of the product matrix AB (3) 1 0(0) 1 (5)(1) 1 3() Repeat the process to calculate the remaining elements of the product matrix. 18 Chapter Linear Systems and Matrices

43 7. Complete the matrix multiplication from the Worked Example. Calculate each element of the product matrix. a. second row, first column b. first row, second column c. second row, second column d. Write the final product matrix. e. Verify your answer using technology. 8. Would the dimensions of the product matrix BA be the same as product matrix AB? Justify your reasoning..3 Introduction to Matrices and Matrix Operations 19

44 9. Consider the matrices D, E, and F. Calculate each product. Use technology to verify your answers. 4 D E 5 a. FD F b. DF c. ED 130 Chapter Linear Systems and Matrices

45 10. Describe Andrew s error. Then, multiply the matrices correctly. Andrew Answer each question about the properties of matrices for multiplication. Explain your reasoning. Then, use the given matrices to support your claim with an example or counterexample. Use the matrices R, S, and T to answer each question. 6 1 R 5 [ 1 5], S , T a. Describe the dimensions of matrices that can be multiplied. Describe the dimensions of matrices that cannot be multiplied. Explain your reasoning..3 Introduction to Matrices and Matrix Operations 131

46 b. Determine whether matrix multiplication is commutative. 13 Chapter Linear Systems and Matrices

47 ?c. Sammy makes the following claim. Sammy Matrix multiplication is associative, as long as each of the product matrices is defined. (R 3 S) 3 T ( ) (1 3 ) R 3 (S 3 T) ( ) (3 3 3) Determine if Sammy s claim is accurate. Justify your reasoning..3 Introduction to Matrices and Matrix Operations 133

48 Talk the Talk 1. Describe the properties of matrices in words for each matrix operation. Then provide an example. Required Matrix Dimensions Matrix Addition Associative? Commutative? 134 Chapter Linear Systems and Matrices

49 Required Matrix Dimensions Matrix Subtraction Associative? Commutative?.3 Introduction to Matrices and Matrix Operations 135

50 Required Matrix Dimensions Matrix Multiplication Associative? Commutative? Be prepared to share your solutions and methods. 136 Chapter Linear Systems and Matrices