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MATH 130 MATH 130 Finite Mathematics Mathematics Finite 41 2-Math 130 Course Guide Self-paced study. Anytime. Anywhere! Math 130 Finite Mathematics University of Idaho 3 Semester-Hour Credits Prepared by: Judith Terrio, MS Instructor of Mathematics University of Idaho RV: 2/2012 2-Math 130 Copyright Independent Study in Idaho/Idaho State Board of Education 2-Math 130 Table of Contents Welcome! ........................................................................................................................................................ 1 Policies and Procedures .................................................................................................................................. 1 Course Description .......................................................................................................................................... 1 Course Materials ............................................................................................................................................. 1 Course Delivery ............................................................................................................................................... 2 Course Introduction ........................................................................................................................................ 2 Course Objectives ........................................................................................................................................... 2 Lessons ............................................................................................................................................................ 3 Exams .............................................................................................................................................................. 4 Grading............................................................................................................................................................ 4 About the Course Developer .......................................................................................................................... 5 Contacting Your Instructor.............................................................................................................................. 5 Disability Support Services .............................................................................................................................. 5 Assignment Submission Log ............................................................................................................................ 6 Lesson 1: Lesson 2: Lesson 3: Linear Functions [Chapter 1, sections 1, 2, and 3] ...................................................................... 7 Linear Systems [Chapter 2, sections 1 and 2] .............................................................................. 10 Matrices [Chapter 2, sections 3, 4, 5] .......................................................................................... 16 Exam 1 Information: Covers Lessons 1-3 ...............................................................................................19 Lesson 4: Lesson 5: Lesson 6: Matrix Equations [Chapter 2, section 6] ...................................................................................... 21 Linear Programming: Geometric Approach [Chapter 3, sections 1, 2, 3, and 4]......................... 26 Linear Programming: The Simplex Method [Chapter 4, sections 1, 2, and 3] ............................. 33 Exam 2 Information: Covers Lessons 4-6 ...............................................................................................40 Lesson 7: Lesson 8: Lesson 9: Lesson 10: Sets and Counting I [Chapter 6, sections 1, 2, and 3] .................................................................. 42 Sets and Counting II [Chapter 6, sections 4, 5, and 6] ................................................................. 46 Probability I [Chapter 7, sections 1, 2, and 3].............................................................................. 50 Probability II [Chapter 7, sections 4, 5, and 6]............................................................................. 53 Exam 3 Information: Covers Lessons 7-10 .............................................................................................56 Lesson 11: Self-study: Markov Chains [Chapter 7, section 7] ....................................................................... 58 Final Exam Information: Covers Lessons 1-11 ........................................................................................60 Lesson Submission Form ......................................................................................................................73 2-Math 130 Math 130: Finite Mathematics 3 Semester-Hour Credits: U-Idaho Welcome! Whether you are a new or returning student, welcome to the Independent Study in Idaho (ISI) program. Below, you will find information pertinent to your course including the course description, course materials, course objectives, as well as information about lessons, exams, and grading. Policies and Procedures Important! As you read this section, you will see the following icon: Use this icon to direct yourself to essential ISI information. Students are responsible for following ISI’s policies. Refer to ISI’s website at www.uidaho.edu/isi, select About ISI, Policies for the most current policies, procedures and course information. The Registration Confirmation Letter sent to you upon registration provides your course instructor’s contact information and lesson guidelines. If you have any questions or concerns, please contact the ISI office for clarification before beginning your course. Course Description Systems of linear equations and inequalities, matrices, linear programming, and probability. Prerequisite: Sufficient score on SAT, ACT, or COMPASS Math Test; or Math 108 with a C or better. Required test scores can be found here: www.uidaho.edu/registrar/registration/placement/math.htm. U-Idaho students: May be used as core credit in J-3-c. 10 graded lessons, 1 self-study lesson, 4 proctored exams Course Materials Required Course Materials Rolf, Howard L. Finite Mathematics, Enhanced Edition (with Enhanced WebAssign with eBook for One Term Math and Science Printed Access Card). 7th ed. Belmont, CA: Thomson Brooks/Cole, 2010. ISBN-10: 0-538-49732-7. ISBN-13: 978-0-538-49732-9. Recommended Course Materials Rolf, Howard L. Student Solutions Manual for Rolf’s Finite Mathematics. 7th ed. Belmont, CA: Thomson Brooks/Cole, 2007. ISBN-10: 0-495-11843-5. ISBN-13: 978-0495118435. If you are unsure if your background is sufficient, or if it has been a long time since you have done any formal mathematics, I suggest you look at the Appendices, starting on page 777 in Finite Mathematics, Enhanced Edition (with Enhanced WebAssign with eBook for One Term Math and Science Printed Access Card). The answers to the odd numbered problems are in the back of the text. Work some of these from each of the four sections. Also, the material in Chapter 1 “Functions and Lines”, sections 1 and 2 and Chapter 2 “Linear Systems”, section 1, should mostly be review from algebra. If it seems unfamiliar to you, then you should consider taking the algebra course, Math 108 [Intermediate Algebra], before proceeding with this course. 1 2-Math 130 Independent Study in Idaho course materials are available for purchase at the VandalStore (University of Idaho bookstore). Your Registration Confirmation Letter contains the VandalStore’s contact information. Independent Study in Idaho courses are updated and revised periodically. Ordering course materials from the VandalStore at the time of registration allows you to purchase the correct edition(s) of textbooks, course guides, and supplemental materials. Contact the VandalStore directly for questions regarding course materials that you have ordered. If purchasing textbooks from another source, refer to the ISBN(s) for the textbook(s) listed for this course to ensure that you obtain the correct edition(s). Course Delivery This course is available online. An electronic course guide is accessible through BbLearn at no additional cost. Refer to your Registration Confirmation Letter for instructions on how to access BbLearn. Course Introduction The goal of the first two-thirds of this course is to teach you to solve a particular type of application, called linear programming. These problems occur in numerous settings in the “real” world, as you will see from the variety of applications we will do. We will be studying two methods for solving this type of problem: a graphical method and a matrix method. The first two chapters lay the groundwork for linear programming, and then the graphical approach is presented in Chapter 3 “Linear Programming,” and the matrix method in Chapter 4 “Linear Programming: The Simplex Method.” The last one-third of this course is devoted to studying probability. Probability lays the groundwork for statistical methods, which most students now study, regardless of their majors. Chapters 6 “Sets and Counting” and 7 “Probability” will discuss counting and probabilities. Please note that the textbook is set up in the same manner as most of the math textbooks you have used in the past. The notation “Section 2.3” means Chapter 2, Section 3. The exercise sets are at the end of each section and there is a set of review exercises at the end of every chapter. Also, the answers to the odd numbered problems are all in the back to the textbook beginning on page 807. Course Objectives Upon successful completion of this course, the student will be able to: Solve systems of linear equations by the Elimination Method. Solve systems of linear equations by the Gauss-Jordan Method. Solve systems of linear equations using Matrix Inversion. Solve applications of linear equations. Solve linear programming problems by graphing. Solve linear programming problems by the Simplex Method. Use sets and Venn diagrams to solve counting problems. Use permutations and combinations to solve counting problems. Find probabilities for equally likely events and compound events. Use conditional probabilities and independence to solve probability problems. Use Bayes’ Rule to determine probabilities. Solve Markov Chain problems. 2 2-Math 130 Lessons Each lesson includes the following components: • reading assignments • overview • self-study practice problems • graded assignments Reading mathematics takes practice, time, and effort. You should spend a lot of time reading the text. Read very slowly, thinking about each sentence. Read with a pencil in hand so you can write in the margins and work through the given examples. Often you will need to fill in a step or two. Authors sometimes leave out a few steps of arithmetic or algebra in order to keep the number of pages down to a manageable size. If you have any questions on the reading, you can always contact the instructor. I hope that you will find the textbook to be very readable. I chose it because students seem to like it and find the author to be highly understandable. There are two features that I particularly like that I would like you to look at right now. In the margins of some pages, the author sometimes writes little “cautions” (see p. 20) or “notes” (see p. 23). These are bits of information that I would have included in the overview, but don’t need to because this author is very thorough. So, pay attention to these tidbits of information. The second really nice feature, especially for an independent learner, is the direct link that the author provides between examples and exercises. Please look at the end of example 12 on page 23 in the textbook. Note where it says, in blue, “Now you are ready to Work Exercise 92, pg. 30.” Now turn to that exercise and notice how it refers you to the example. I hope you will find this to be a very helpful feature. And, while we are looking at the textbook, notice the very large number of examples that are in each section. Quite frankly, I love this textbook for a course like this. I hope you appreciate it and find that it eases your learning. In the overview section of each lesson, I point out material that is especially important, any topics in the reading that you may skip, some tips of my own and, sometimes, I give other examples that I think you will find helpful. The self-study practice problems are intended to help develop your understanding of the basic concepts and to increase your competence with the algebraic and arithmetic manipulations needed to solve problems. There is an old saying: “Math is not a spectator sport.” I can read and understand the rules of basketball or directions on how to drive a car. However, unless I train my muscles and practice, I would not be a very competent basketball team member, nor would you want to ride in a car with me driving! Practice is the key to succeeding in a math class. It is a lot of work; that is true. However, the rewards come when each problem gets a little bit easier and then each lesson gets a little bit easier as you exercise your math brain muscles. There are a lot of false starts in mathematics. That’s okay. The key is to be aware of this and to not get frustrated. Take a break, go for a walk and clear your head. This is what we all have to do; it is the nature of math problem solving and indeed, of all problem solving in life. These practice problems are the same ones that I assign to my on-campus students. The answers to the self-study practice problems are in the back of the textbook and in the Student’s Solution Manual for 3 2-Math 130 Rolf’s Finite Mathematics. Do not turn these problems in, but do not neglect them. Feel free to contact the grader if you need help on any of these. Graded assignments take the place of in-class quizzes. They give both of us a way to access which concepts you need more practice on before taking an exam. Equally important is that you learn to write mathematics correctly. These assignments need to be done neatly and completely. You are going to be graded on the thoroughness of your work as well as its accuracy. Study Hints: • Complete all reading assignments. Complete all the self-study problems before doing the graded problems. • Set a schedule allowing for course completion one month prior to your personal deadline. An Assignment Submission Log is provided for this purpose. • Web pages and URL links in the World Wide Web are continuously changing. Contact your instructor if you find a broken webpage or URL. Keep a copy of every lesson submitted. Submit the entire lesson at one time (partial or incomplete assignments will receive a zero). Refer to your Registration Confirmation Letter for further details on your instructor’s lesson guidelines and requirements. Also refer to the ISI Policies and Procedures for essential ISI policies on submitting lessons to your instructor. Exams • • You must wait for grades and comments on lessons prior to taking subsequent exams. For your instructor’s exam guidelines, refer to your Registration Confirmation Letter. In this course guide, on the Exam Information pages, there are test outlines containing textbook exercises that are similar to what you will be tested on. You should do all of those problems to prepare for the test. You may find that you need to do them more than once. Keep track of the problems you had to look up in the textbook, in this guide, or on your past homework before you could solve the problems. Do those over again until you can do every problem without any references. Refer to Grading for specific information on lesson/exam points and percentages. Proctor Selection/Scheduling Exams All exams require a proctor. Refer to the ISI Policies and Procedures for guidelines on how to choose a proctor and schedule exams. Complete the Proctor Information Form and send it to the ISI office at least two weeks prior to scheduling your first exam. 4 2-Math 130 Grading The course grade will be based upon the following considerations: Assignments (10 @ 10 points each) Unit Exams (3 @ 100 points each) Comprehensive Final Exam Total 100 points 300 points 100 points 500 points A letter grade will be assigned at the end of the course based on the standard 10 percent scale. A = 500 – 450 points B = 449 – 400 points C = 399 – 350 points D = 349 – 300 points F = 299 points or lower Self-study Lessons Lesson 11 is the only lesson for which there is no graded assignment. As with the self-study exercises listed in the other lessons, the answers to lesson 11’s problems are in the back of the textbook. The final course grade is issued after all lessons and exams have been graded. Refer to the ISI Policies and Procedures for information about confidentiality of student grades, course completion, time considerations, and requesting a transcript. About the Course Developer Judith Terrio got her math ability from her father and her love of people from her mother. She grew up and went to college on the east coast. Judi obtained her secondary teaching certificate and taught a few years in Colorado before moving to beautiful Idaho. In north Idaho, Judi taught high school for many years before earning a master’s degree in mathematics and subsequently becoming an instructor at the University of Idaho. She thinks she is one of the most fortunate people alive because her passion and her job are one in the same. She loves teaching! She is also a cat whisperer. Contacting Your Instructor You will receive specific course requirements and instructor contact information in your Registration Confirmation Letter, which you will receive upon registration. Please copy the ISI office at [email protected] when corresponding and submitting lessons by email to your instructor. Disability Support Services Refer to the ISI Policies and Procedures for information on Disability Support Services (DSS). 5 2-Math 130 Assignment Submission Log Send the completed Proctor Information Form to the ISI office at least two weeks prior to taking your first exam. Lesson Projected Date for Completion Date Submitted 1 2 3 It is time to make arrangements with your proctor to take Exam 1. Exam 1 4 5 6 It is time to make arrangements with your proctor to take Exam 2. Exam 2 7 8 9 10 It is time to make arrangements with your proctor to take Exam 3. Exam 3 It is time to make arrangements with your proctor to take the Final Exam. Exam 4 6 Grade Received Cumulative Point Totals 2-Math 130 Lesson 1 Linear Functions Reading Assignment Preface “To the Student,” pp. xi–xii Chapter 1, Functions and Lines, Sections 1.1, 1.2, and 1.3 Overview As mentioned in the introduction, the goal of the first two-thirds of this course is to teach you to solve a particular type of application, called linear programming. One of the very nice things about these problems is that the variables are only to the first power and are only added or subtracted to one another (i.e., no square roots, no xy or x/y in the equations). Sounds great, doesn’t it? This means that we will be concerned mostly with lines. In actuality, applications may involve many, many variables, but fortunately we have technology to lend a helping hand in solving those more complex problems. The technological methods used are just an extension of the methods used with lines or with three or four variables. So, we start with the study of functions, particularly linear functions, as the foundation for this course. Section 1.1 Functions Please pay special attention to how variables are defined when using mathematics to describe an application. For instance, let’s say I’m going to the store to buy bananas. To define a variable by only saying, “let x = bananas,” is not a sufficient definition. Is x the number of individual bananas, the number of bunches, or the number of pounds of bananas? You must include a quantifier (like “number of” or “pounds of”) when defining a variable. Section 1.2 Lines This section provides you with an opportunity to review and refresh your algebra skills concerning lines. Here is a list of formulas that you will need. Slope of a line between two given points (x1, y1) and (x2, y2) is m y y2 y1 , provided x2 ≠x1 x x2 x1 Note that this is said “delta y over delta x” or “change in y over change in x.” Δ is the capital Greek letter “delta” and “change” means the difference in two numbers (like the change you get when you make a cash purchase.) Slope-Intercept Form Point-Slope Form Standard Form Horizontal Line Vertical Line y = mx + b, where m is the slope and the point (0, b) is the y-intercept y – y1 = m(x –x1), where m is the slope and (x1, y1) is a known point Ax + By = C, where A is a whole number greater than or equal to 0 y = b [m = 0, y-intercept is (0, b), there is no x-intercept] x = a [slope is undefined, there is no y-intercept, x-intercept is (a, 0)] *There are basically two methods for sketching a line: (1) Using a point on the line and its slope (see textbook examples 4 [pp. 17-8], 9 [p. 21], and 10 [pp. 21-2]) (2) Using the line’s x- and y- intercepts 7 2-Math 130 The second method is the one that we will use most often, so it is important for you to understand how to find these two points. Example: Given 2x 3 y 12. A point on the x-axis has a y -coordinate of zero, A point on the y -axis has a x-coordinate of zero, so find the x-intercept by setting y 0. so find the y -intercept by setting x 0. 2 x 3(0) 12 2 x 12 x6 2(0) 3 y 12 - 3 y 12 y 4 To sketch the line, connect the two points (6, 0) and (0, -4) with a line which continues in both directions. *When given two points, there are two ways to find the equation of a line: (1) Using the slope-intercept form (which is the method you are probably the most familiar with) (2) Using the point-slope form The second method is the one that we will use most often. It is actually less work and less prone to errors. Example: Find an equation of the line passing through 3, 1 and 2, 9 . y 1 9 10 2 x 3 2 5 Then substitute the slope and either point into the form y y1 m( x x1 ) First find the line's slope: m y 1 2( x 3) y 1 2 x 6 y 2 x 5 y 9 2( x 2) y 9 2 x 4 y 2 x 5 Section 1.3 Linear Models This section deals with the use of math to model real-world situations. The major focus in this section is on the cost, revenue, and profit functions. If x = number of items sold, then: The cost function C(x) = ax + b; where C(x) is the total cost for producing x items, a = unit cost, the cost for producing each item, which includes such things as material and labor costs; and b = fixed costs, which don’t depend on the number of items made, such as factory rent, heating and electricity, office staff salaries and insurance, etc. The revenue function R(x) = px; where R(x) is the gross amount of money received for selling x items and, p = unit selling price The profit function P(x) = R(x) – C(x); the amount of money received from selling x items minus the cost of producing those x items. 8 2-Math 130 The break-even point is the number of units needed to be sold so that P(x) = 0 or R(x) = C(x). Example: Textbook exercise #40, p. 52. There are three values given in this problem: Fixed costs of $28,000, unit costs of $48/each, and selling price of $62/each. So, if we let x = the number of TV stands made, then the cost function is C(x) = 48x + 28000 and the revenue function is R(x) = 62x. To find the break-even point, equate the revenue function and the cost function: R ( x) C ( x) 62 x 48 x 28000 14 x 28000 x 2000 So, 2000 TV stands must be sold to break even. Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. As I mentioned above, it is crucial for you to practice, to do math, and not just read about it. The answers to the self-study practice problems are in the back of the textbook and in the Student Solution Manual for Rolf’s Finite Mathematics. Section 1.1 (p. 6): Section 1.2 (p. 28): Section 1.3 (p. 50) #1, 9, 13–23 odd #7–43 every other odd, 45–83 odd, 87, 89, 93, 95, 99, 103, 109 #1–29 odd, 37, 43, 51 Graded Assignment #1 Before beginning the first written assignment, refer to your Registration Confirmation Letter for your instructor’s lesson requirements. If emailing lessons to your instructor, please copy the ISI office at [email protected] Show all your work in a neat and organized manner, or you will receive no credit. Good luck and have some fun! Treat yourself when you finish the lesson. Your labors deserve a reward. (16 problems) Section 1.1 (p. 6): Section 1.2 (p. 28): Section 1.3 (p. 50): #10, 18 #14, 16, 18, 54, 56, 64, 66, 92, 96 #14, 16, 18, 30, 48 9 2-Math 130 Lesson 2 Linear Systems Reading Assignment Chapter 2, Linear Systems, Sections 2.1 and 2.2 Overview In this chapter we will investigate systems of lines and systems of planes. First, we will look at geometric and algebraic methods for finding the relationship between lines in a plane or planes in space. Then we will look at a method, which may be new to you, that uses something called an augmented matrix. Section 2.1 Algebraic Elimination In this section, we will review, from algebra, the solving of systems of two lines in the same plane. Also, we will look at some applications for systems of lines. To solve a system of equations means to find any points in common. If I have two lines in the same plane, then there are three possible arrangements. The lines intersect in one point and have one unique solution The lines are parallel and have no solutions. The equations are the same line and have an infinite number of solutions. The solution is a point (x1, x2). There is no solution. The solution is the set of all points on the line (k, mk+b). *There are two basic ways to solve a system of two linear equations: (1) by substitution, or (2) by elimination You most likely used both of these methods in algebra. In this class, the elimination method is the method we will use most often. 3x 4 y 5 Example: Given 2 x 3 y 4 We need to get the coefficients in front of either x or y to be the same, except opposite signs. That way we can add the two equations together and one of the variables drops out. Since the signs in front of the y’s are already opposite, I will multiply the first equation by three and the second equation by four. In the next box, notice how I indicate what I am doing above and below the arrow: 10 2-Math 130 3 x 4 y 5 3 R1 9 x 12 y 15 add 1 17 x 0 1 x 4 R2 17 2x 3y 4 8 x 12 y 16 To get the corresponding y -coordinate, substitute the x-value into either of the given equations. multiply by 17 3( 171 ) 4 y 5 3 28 y 85 28 y 88 y 88 22 28 7 The solution to this system is the point ( 171 , 722 ). The textbook, Section 2.1, examples 7(pp. 72-3) and 8 (pp. 73-4) show you what happens when you use the elimination method on a system of parallel equations or a system of two identical lines. Demand, Supply, and Equilibrium In Section 1.3, when we considered the revenue function R(x) = px (p. 41), we assumed that the unit price was a constant amount, independent of the number of items sold. Actually, price can depend on the amount that consumers want or the price can depend on the amount that is available. When x = number of items in demand price usually rises with demand. When x = number of items produced, price usually lessens as more units are made. P = ax + b is called the demand equation P = cx + d is called the supply equation x x The equilibrium point is the point (x, p) which satisfies both of these equations. Example: Textbook exercise #46, p. 78. Let x be the number of CD players. For the demand equation, we are given two points (10, 130) and (25, 100). Finding the equation of the line, using the slope and one of the points (see Lesson 1, Section 1.2 Lines) we find the demand equation to be p = -2x + 150. Similarly, for the supply equation, we are given two points (40, 130) and (10, 100). Finding the equation of the line, using the slope and one of the points we find the supply equation to be p = x + 90. In this case it is easiest to solve for the equilibrium point using substitution -2x + 150 = x + 90 -3x = -60 x = 20 and substituting 20 into either the demand or the supply equations gives p = 110 So, the equilibrium quantity is 20 CD players at a price of $110 each. 11 2-Math 130 Section 2.2 Augmented Matrices and the Gauss-Jordan Method The textbook starts this section by algebraically solving systems of three equations with three variables. I am not going to require you to solve these in this way. The method we are going to use starts on p. 86. It is known as the Gauss-Jordan Method (after two mathematicians who developed it) and uses a structure called an augmented matrix. Please read pp. 86–92 before proceeding. Suppose we are trying to solve the following system of equations to find the point or points, if any, that all three equations have in common. x – 2y + z = 7 3x + y – z = 2 2x + 3y + 2z = 7 1 2 1 7 1 0 0 a followingthena set of rules 0 1 0 b change into 3 1 1 2 2 3 2 7 0 0 1 c The first step is to write the system as an augmented matrix. Once you have all 1’s in the diagonals and zeros everywhere else to the left of the vertical line, you now have a new system of equations: x + 0y + 0 z = a 0x + y + 0 z = b 0x + 0y + z = c x=a y = b The solution point is (a, b, c). z=c or, simply So, the heart of the Gauss-Jordan Method is how to change the first augmented matrix into diagonal form. The following are the three legal rules and the notation we use to show what has been done between any two matrices: R1 ↔ R 2 c R 1 → R1 c R 1 + d R2 → R1 Swap any two rows, in this case, rows 1 and 2. Replace any row with a nonzero multiple of itself. Replace any row with the sum of a multiple of itself and a multiple of another row. So, how does one start? There is an order to follow or else you might find yourself undoing some of the zeros that you got in one step while performing another step. There are three possible cases for the first crucial step. 1. If the first row, first column is not a 1, but there is a 1 in the first column, then use the first rule to swap two rows and get the one in the a11 position. 2. If there are no 1’s in the first column use a nonzero a11 (swapping rows if necessary) with rule 1 two to multiply row one by a . 11 3. However, if this results in messy fractions, you can work with any nonzero number in the first position. Getting ones makes the other steps easier, but it is not a crucial step. Concentrate on getting the zeros. Get all the zeros in the first column and then the zeros in the second column and finally the last column. For example, solve the following system using the Gauss-Jordan method: 12 2-Math 130 0 2 2 y 3z 7 3 7 set-up 3 6 12 3 3 x 6 y 12 z 3 5 x 2 y 2 z 7 5 2 2 7 Notice that we do not have any 1’s in the first column. However, if we multiply the second row by ⅓ and swap the first and second rows, we will have a 1 in the top left corner and no fractions in that row. 0 2 0 2 3 7 1 3 7 1 2 4 1 3 R2 R2 R1 R2 1 2 4 1 0 2 3 7 3 6 12 3 5 2 5 2 5 2 2 7 2 7 2 7 If you can do so correctly, you may perform two operations at once, but no more than two. Here is the same matrix reduced by doing two steps in one. 0 1 2 3 7 1 2 4 1 3 R2 R2 2 3 7 R1 R2 0 3 6 12 3 5 2 5 2 2 7 2 7 R R 1 2 or one could do and get the same result. 1 R R1 3 1 The next step is to get all zeros in the first column under the one. Since the second row already has a zero, we next get a zero where the five is by using the 1 in row one. Notice that all the new values in row three are now divisible by 2 (or -2). Sometimes it will make your work easier to reduce a row so you have smaller numbers to work with. You can do this in a separate step or in the same step if you notice it at the time. 1 2 4 1 1 2 5 R1 R3 R3 2 0 2 3 7 0 5 2 2 7 0 12 4 1 3 7 22 2 1 2 4 1 1 2 4 1 5 R1 R3 R3 3 7 1 0 2 3 7 0 2 R3 R3 2 5 2 2 7 0 6 11 1 For this problem, we now have a choice as to how to proceed depending on your competence with 1 fractions. After getting column one to look like 0 , the next step is to get a 1 in the second column, 0 second row. We would do this by using rule two: ½ R2 → R2. For this matrix, we would now have to work with fractions in the second row. Halves aren’t too messy to work with, but we can do something else to avoid the fractions at this stage and that is to use rule three. Please recall the third step, which I wrote immediately prior to this example. Concentrate on the zeros; we can get the 1’s at the end. So, to avoid fractions, proceed as follows, similar to the elimination method for a 2 x 2 system. *Notice that when getting the zeros in column two, we multiply the second row to get the number needed for our zeros. This element, the second row and second column is called the pivot element. 13 2-Math 130 1 2 4 1 1 0 7 8 1 1 0 7 8 1R2 R1 R1 2 R3 R3 3 7 3 7 0 2 3 7 3 R2 R3 R3 0 2 0 2 0 6 11 1 0 0 20 20 0 0 1 1 1 0 7 8 1 0 0 1 1 1 0 0 1 7 R3 R1 R1 2 R2 R2 0 1 0 2 3 R3 R2 R2 0 2 3 7 0 2 0 4 0 0 1 1 0 0 1 1 0 0 1 1 You can easily check your answer by substituting the values into all three of the original equations to see if they are satisfied. Now we get the zeros in the third column by pivoting on the 1 in the third row, third column. The solution is now determined to be x = -1, y = 2, and z = 1. 2 y 3z 7 2(2) 3(1) 7 x 1, y 2, z 1 3(1) 6(2) 12(1) 3 3x 6 y 12 z 3 5 x 2 y 2 z 7 5(1) 2(2) 2(1) 7 Section 2.2 Applications You may find it helpful for some of the applications to use a table to organize the given data. Example: Textbook exercise #54, pp. 100 Let A = the price ($) of a medium T-shirt Let B = the price ($) of a large T-shirt Let C = the price ($) of an extra-large T-shirt Organizing the given information into a table with the variables horizontally across the top makes translation into the system of equations straightforward. Alpha Beta Beta Gamma Pi Beta A B C 3 8 13 16 9 0 6 11 14 Total Cost ($) 357.50 326.00 453.00 14 3 A 8 B 13C 357.50 16 A 9 B 326.00 6 A 11B 14C 453.00 2-Math 130 Example: Textbook exercise #62, p. 101 Let x = number of minutes jogging Let y = number of minutes playing handball Let z = number of minutes cycling Sometimes using a table doesn’t to capture all of given information. In this exercise, only the information about calories fits into table format. The other information needs to be translated into an equation as you learned in algebra class. x y z Total Calories 13 11 7 640 The total number of minutes is one hour, so And minutes jogging = minutes cycling So, x = z or 13 x 11 y 7 z 640 (calories) x y z 60 (minutes) x z 0 Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 2.1 (p. 77): Section 2.2 (p. 98): #17 – 33 odd, 41 – 53 odd, 59 #11, 21, 23, 27, 29, 37, 39, 41, 45, 53, 55, 57, 59, 63 Graded Assignment #2 Show all your work in a neat and organized manner or you will receive no credit. (10 problems) Section 2.1 (p. 77): Section 2.2 (p. 98): #20, 24, 48, 54 #40, 42, 46, 60, 64, 70 [set up, but do not solve #64] 15 2-Math 130 Lesson 3 Matrices Reading Assignment Chapter 2, Linear Systems, Sections 2.3, 2.4, and 2.5 Overview Section 2.3 Gauss-Jordan Method for General Systems of Equations Recall that back in the beginning of Lesson 2 (in textbook Section 2.1 Algebraic Elimination and this course guide) we found that there are three possibilities for solutions of a 2 x 2 system of equations. In this lesson, we will extend that idea to systems of equations and variables larger than 2 x 2. We want to investigate what happens when using the Gauss-Jordan Method in these cases. The textbook has 11 examples that are done out in complete detail, so I will not fill in all the details below. You might try for yourself to see if you can. If you haven’t done so yet, please read pages 108 through 110 before proceeding. It’s not always going to be possible, in these cases, to reduce the augmented matrix into the diagonal form. 1 0 0 a 0 1 0 b This form is actually a particular case of what is called the reduced echelon form. 0 0 1 c The basic structure for the reduced echelon form is to put a nonzero number in the first row, first column, just as before and then get every other number in the column to be zeros. Then locate the first nonzero number in the second row and make all the others in that row a zero. Note that the first nonzero number in the second row may not be in the second column. That is okay. Then proceed in this manner until you get to the bottom. If along the way, any row turns into all zeros, then move it to the bottom. Take for instance, the following matrix and its reduced form. 1 2 1 2 3 5 1 3 9 using the set of rules 1 4 0 change into 0 4 2 0 1 0 0 5 1 2 0 3 16 Notice that we can go no further since all the numbers in the bottom row that we would work with are zeros. Let’s consider what this bottom means. Using the variables x, y, and z, the bottom row represents: 0x + 0y + 0z = 3 or 0 = 3. This statement is obviously false. Therefore, we conclude that the original system has no solution. 2-Math 130 Here’s a second type that you will encounter: 1 4 2 3 using the set of rules 1 4 0 7 change into 2 8 1 9 0 0 1 5 Once again we can go no further; the first row, first column is a one and under it is a zero. The first nonzero in the second row has a zero above it. Let’s consider what these two rows are telling us. The first row gives us x + 4y = -7 and the second row tells us that z = 5. We have no way to determine a unique solution for x and y, so this is an example where there are an infinite number of solutions. Let’s consider how we shall represent these solutions. Solving the first equation for x gives us x = -7 – 4y and since we know no information about y, we say that y can be any number. This is where the infinite number of solutions comes from. As a system, we now have: x = 7 – 4y y = any number z=5 Instead of writing out “any number” we use another letter called a parameter. The letter k is usually used if there is only one parameter. If there are two, then it is customary to use j and k. So, we would write the solution as: x = 7 – 4k y=k z=5 And in ordered triplet form, we would write: (-7 – 4k, k, 5). Note that we can now find as many particular solutions as we need (recall there are an infinite number) by choosing values for k. There’s no trick to this. I’m just picking whatever numbers I want for values of k. 1 4 If k = 0 then If k = 4 then If k = -1 then If k = then x = -7 – 4(0) = -7 x = -7 – 4(4) = -23 x = -7 – 4(-1) = -3 x = -7 – 4( y=0 y=4 y = -1 y= z=5 z=5 z=5 z=5 1 4 ) = -6 1 4 So we have identified four of the infinite number of solutions: (-7, 0, 5), (-23, 4, 5), (-3, -1, 5), and (-6, and we could find however many we need in this manner. 1 4 , 5) Section 2.4 Matrix Operations The textbook does an adequate job on this section. Please contact the instructor if you need additional help. Section 2.5 Multiplication of Matrices After Exam 1, in textbook section 2.6 and Lesson 4, you will learn a final method for solving a system of simultaneous equations. In this section you will learn how to combine matrices by multiplying them together; a skill that will be needed in the next section. The first type of multiplication is for multiplying a row matrix times a column matrix. Recall that the dimensions for a row matrix are of the form 1 x c and the dimensions for a column matrix are of the form r x 1. In order to multiply these matrices “c” must equal “r”. This type of multiplication of a row matrix times a column matrix is called a dot product and results in a number, not a matrix. 17 2-Math 130 Definition: The dot product of a row matrix times a column matrix is found as follows: c1 The number found by computing: c 2 rn r1 c1 r2 c2 r3 c3 rn cn r1 r2 cn 4 For example, 2 3 5 7 (2)(4) (3)(7) (5)(6) 8 21 30 17 6 To multiply two matrices together the number of columns of the first matrix must equal the number of rows of the second matrix, similar to the dot product, except the result is going to a matrix. (matrix A)(matrix B) = matrix C with sizes (r x n) (n x c) = (r x c) One helpful trick to keep track of your place is to use the index finger on your left hand to run under the rows in the first matrix while using the index finger of your right hand to follow down the columns in the second matrix. Try this on example 2, page 144 in the textbook. With enough practice you can do a lot of the arithmetic in your head. In this textbook example, using my index fingers as described above, I would think to myself (or talk out loud, if that helps), “7 (1 times 7) + -6 (3 times -2) + -6 (2 times -3) is 5,” which goes in the first row and first column. Using the first row again, but now with the second column, repeat the process. Then move on to the second row in the first matrix dotted with each column in the second matrix. Two final notes: 1. Notice, from example 5 (p. 146), that unlike multiplication with numbers, multiplication of matrices is order dependent, i.e. AB does not necessarily equal BA. 2. The identity matrix, discussed on pages 148 – 150 in the textbook, is an important concept that we will return to in section 2.6, Lesson 4. Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 2.3 (p. 120): Section 2.4 (p. 135): Section 2.5 (p. 151): # 31, 33, 37, 43, 47, 49, 57, 67, 69 #1, 5, 9, 11, 15, 19, 23 – 33 odd, 37, 43, 45, 47, 51, 55 #1 – 15 odd, 21 – 31 odd, 37, 41, 43, 47, 51, 53, 55, 57 Graded Assignment #3 Show all your work in a neat and organized manner or you will receive no credit. (12 problems) Section 2.3 P. 120): Section 2.5 (p. 151): #2, 4, 6, 8, 32, 34, 60 #14, 16, 28, 38, 54 18 2-Math 130 Exam 1 Information Make arrangements with your proctor to schedule Exam 1. Prior to taking this exam: • You must submit lessons 1, 2, and 3 to your instructor before taking this exam. • Please do not take this exam until you have received graded lessons 1, 2, and 3 back from your instructor. • Do not submit any subsequent lessons until you have taken this exam. Below, there is a test outline containing textbook exercises that are similar to what will be on the test. You should do all of these problems to prepare for the test. You may find that you need to do them more than once. Keep track of the problems you had to look up in the textbook, in this course guide, or on your past homework before you could solve the problems. Do those over again until you can do every problem without any references. Exam components: • This exam covers lessons 1, 2, and 3. • This is a closed-book, closed-notes exam. • Calculators are NOT allowed. The format and questions on this exam are similar to the textbook problems. • This exam is worth 100 points. • Time limit: 75 minutes. Items to take to the exam: • Photo identification • V number • Pencil and eraser Exam grades and comments: • See your Registration Confirmation Letter for how you will receive exam grades and comments from your instructor. • Graded exams will not be returned to you. However, arrangements can be made to view graded exams. Contact the ISI office for more information. Exam 1 Topic Outline (Refer to your lessons for page numbers) 1. Find the slope of a line, given 2 points on the line. [1.2 #15, 17] 2. Find the x- and y- intercepts of a line. [1.2 #63, 65] 3. Find an equation of a line. [1.2 #23, 31, 39, 51, 53, 55] 4. Linear Models [1.1 #1, 9, 13–23 odd; 1.2 #93, 95, 101, 103, 109] 5. Cost, revenue, and profit functions [1.3 #1–15 odd, 25, 27, 29, 37] 6. Elimination Method for 2 x 2 systems. [2.1 #17, 29, 31] 7. Gauss-Jordan Method for 3 x 3 systems. [2.2 #39, 41, 45; 2.3 #31, 35, 37] 8. Gauss-Jordan Method for general systems. [2.3 #33, 43, 47, 49, 57] 9. Systems’ Applications: Define variables and set up a system of equations. [2.1 #43, 45, 51, 59; 2.2 #53, 55, 57, 59, 63; 2.3 #67, 69] 10. Matrices [2.4 #11, 15, 19, 23, 27, 29, 31, 33, 37, 43, 45, 47, 55] 11. Matrix multiplication [2.5 #1, 5, 9–31 odd, 37, 41, 51, 53, 55] Extra Practice Problems are on the next page. 19 2-Math 130 Extra Practice The following problems do not necessarily cover all the topics. Make sure you can do the exam outline problems above. It is also a good idea to make sure you can do the examples that were presented in this course guide: pp. 61–63 #7, 13, 15c, e, f, 25, 27, 29, 31, 36, 39, 41, 43, 47, 51 pp. 202–204 #3, 7, 9, 11, 15–25 odd, 31, 33, 35, 37, 39, 41 20 2-Math 130 Lesson 4 Matrix Equations Reading Assignment Chapter 2, Linear Systems, Section 2.6 Overview In this section you will be learning another method for solving systems of equations. “What? Yet another method?” you say. Yes, but this method, which uses matrices, has many advantages over the methods you studied in the first three lessons. The most important of which is that this efficiently handles large numbers of variables. In real science and business applications, you will find that you may be dealing with dozens or even hundreds of variables. Using algebraic elimination or the Gauss-Jordan method can be very unwieldy. The method of matrix equations, however, is easily adaptable for the use of a computer. Consequently, this method is very important. Section 2.6 Inverse matrices When dealing multiplication of regular numbers the number 1 is called the identity because any number multiplied by 1 is itself, i.e. a 1 a and 1 a a . Also, for numbers with multiplication, the inverse of a 1 number is its reciprocal because any number times its reciprocal equals the identity, i.e. a 1 a 1 and a 1 . a In the last section, you saw that for matrices with multiplication, the square matrix with diagonal containing 1’s and 0’s in all the other entries is the identity matrix, i.e. A I A and I A A . Our immediate goal is to find the inverse of a square matrix, if one exists. The general notation for the inverse of a matrix A is to write A1 . Use caution here; this is not the algebraic notation meaning reciprocal. If A is a matrix, A1 1 and actually 1 is meaningless. A A So, given a matrix A, we are searching for a matrix A1 for which A A1 I and A1 A I . First, I will show a quick method that works only for 2 2 matrices. a b d b Let A . Find the determinant, D ad bc . Then A1 1 . Notice that a and d D c a c d swapped places, while b and c stayed in their positions but became opposite signs. 21 2-Math 130 3 2 For example: Find the inverse of A 5 4 (i) D (3)(4) (2)(5) 12 10 22 2 2 1 4 4 2 1 (ii) A1 22 22 11 11 22 5 3 5 3 5 3 22 22 22 22 (iii) You can easily check this by multiplying A A1 to get the identity matrix, I. 1 6 10 3 6 2 3 2 11 11 11 22 11 22 1 0 (iv) 5 4 5 3 20 20 10 12 0 1 22 22 22 22 22 22 Unfortunately, this short-cut method only works for 2 2 matrices. To find the inverse of any size square matrix, we use a process similar to the Gauss-Jordan method. Take matrix A and append the identity matrix to it: A I . Use the same rules as the Gauss-Jordan method to turn matrix A into the identity matrix. When you have accomplished that, matrix I will become the inverse matrix, A1 , i.e. A I G-J Rules I A1 3 2 For example, using the matrix A from the short-cut example above A 5 4 3 2 1 0 3 2 1 0 A I 5R1 3R2 R2 0 22 5 3 5 4 0 1 R2 11R1 R1 33 0 6 0 1 5 1 22 R2 R2 22 3 1 R1 3 33R1 22 1 0 0 1 6 33 5 22 3 33 = I A1 3 22 So, the inverse of matrix A, 3 2 1 6 33 33 11 11 A1 5 3 5 3 22 22 22 22 The same method can be used for any size square matrix. Your textbook section 2.6, example 4 (pp. 1634), shows you the procedure for a 3 3 matrix. One final note before we see how to solve equations using matrix inversion. Does every matrix have an inverse? Let’s start with a simpler question, “Does every number have an inverse?” Can you think of a number that doesn’t have a reciprocal? How about the number zero? So, not every number has an 22 2-Math 130 inverse when dealing with multiplication. Now, go back and look at the short-cut method for finding the inverse of a 2 2 matrix. What could go wrong? Do you see a possible problem? What if the determinant, D, is zero? In that case, 1 would not exist and the given matrix would not have an D inverse. For 4 6 , D (4)(3) (6)(2) 0 so this matrix has no inverse. For larger matrices using the 2 3 Gauss-Jordan method, if a matrix doesn’t have an inverse it will not be possible to turn the matrix into the identity, similar to the difficulties back in textbook section 2.3 (beginning on p. 107). Section 2.6 Matrix Equations Before proceeding, would you please look back in your self-study assignments for lesson 3, section 2.5 #53 and 55 and also graded problem #54. Notice how for each of these when a square matrix containing numbers is multiplied by a column matrix containing variables the result is a system of “equations” without the equals portion. So, our first step is to take a system of equations and rewrite it in Matrix Form. For instance, 3x 5 y 4 z 10 4 x 2 y 3z 12 Can be written as x z 2 3 5 4 x 10 4 2 3 y 12 1 0 1 z 2 This is the first step: to write a system of equations in matrix form A X B , where A is the square coefficient matrix, X is the column variable matrix, and B is the column constants matrix. Before discussing how to solve such a matrix, let’s look at how a mathematician solves a simple algebra equation if one had to show all the steps. This is the same process we will use to solve a matrix equation. Given: 3 x9 4 4 3 4 x 9 3 4 3 Multi Multiply both sides on the left by the reciprocal of 1x 6 A number times its inverse is the identity. x6 The identity times x is still x. 3 . 4 Given: A x B A1 A x A1 B Multiply both sides on the left by the inverse matrix, A1 I x A1 B A matrix times its inverse is the identity matrix, I. x A1 B The identity matrix times x equals x. 23 2-Math 130 At the beginning of this lesson we found, using two different methods, that 1 2 3 2 11 5 4 5 22 1 11 . 3 22 We can use this information to solve the following system: 3x 2 y 12 matrix 3 2 x 12 form 5 x 4 y 2 5 4 y 2 1 1 3 2 3 2 x 3 2 12 5 4 5 4 y 5 4 2 times by A1 2 x I 11 y 5 22 1 11 12 3 2 22 1 since A A I 24 2 22 x 11 11 11 2 y 60 6 66 3 22 22 22 Therefore x 2 and y 3 . Check by substituting into the original system: 3 2 2 3 6 6 12 5 2 4 3 10 12 2 The same process can be used on any n n system of equations. Your textbook section 2.6 example 7 shows you the procedure for solving a 3 3 system using a matrix equation. 24 2-Math 130 Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 2.6 (p. 170): #1, 3, 5, 9 – 33 odd, 37, 39, 41, 43, 47 For #3 and #5 you are to check that AB = I. You may use the short-cut, shown above, for finding the inverse of a 2 x 2 matrix. Graded Assignment #4 Show all your work in a neat and organized manner or you will receive no credit. (7 problems) Section 2.6 (p. 170): #14, 28c, 30c, 32, 34, 40, 46 You may use the short-cut, shown above, for finding the inverse of a 2 x 2 matrix. 25 2-Math 130 Lesson 5 Linear Programming: Geometric Approach Reading Assignment Chapter 3, Linear Programming, Sections 3.1, 3.2, 3.3, and 3.4 Overview Congratulations! You are now ready to delve into the application type we’ve been heading toward called linear programming (LP). Linear programming is a method or a process for finding optimal values and solutions. In business, one is concerned with maximizing profits and minimizing waste; what’s the optimal production schedule that will accomplish those goals? You will see many other scenarios in the applications where linear programming is applicable. We will be looking at two methods for solving LP problems: in Chapter 3 Linear Programming, a geometrical or graphical method, and in Chapter 4 Linear Programming: The Simplex Method, a numerical method which has similarities to matrices. The geometric method has the advantage that you can picture all the possible solutions and then choose the optimal one. Its major disadvantage is that it only works with two unknowns since we are restricted by our graphing ability. The method in Chapter 4 Linear Programming: The Simplex Method is advantageous because it works with many unknowns and is adaptable to the computer. 26 2-Math 130 Section 3.1 Linear Inequalities in Two variables This section is a review of material covered in algebra. Ask your grader if you need additional help on this material. Here is one additional example for you. Section 3.1 Exercise #22 p. 211 Let x = number of tables y = number of desks Then 2.5x 4 y 350 hours and x 0, y 0 (i) Sketch 2.5x 4 y 350 by finding the intercepts x-int : Let y 0 2.5x 0 350 x 140 (140,0) y-int: Let x 0 0 4 y 350 y 87.5 (0,87.5) Therefore, y = # of desks 87.5 10 x = # of tables 10 140 (ii) Determine whether the solution set is above or below this line by choosing any point, not on the line, to see if it satisfies the inequality. Choosing (10,10) we find: 2.5(10) 4(10) 25 40 65 350 , so (10, 10) is included in the solution and we shade the triangular region below the line segment. y 87.5 10 * x 10 140 27 2-Math 130 Section 3.2 Systems of Linear Inequalities; Feasible sets In order to solve LP problems, you are going to have to find the region where inequalities overlap and also the corners of such regions. The region of common points is called the feasible region. Here are two examples of feasible regions: bounded feasible region unbounded feasible region y y A A B C D x B 4 corners: A, B, C, and D 2 corners: A and B 28 x 2-Math 130 For example: Sketch the feasible region and find the corners for: x 2y 4 4 x 4 y 4 x 0, y 0 The basic procedure is to sketch each line using the intercepts. Then use a checkpoint to determine where to shade. y-int x-int (0,0) included? l: (0,2) (4,0) x 2y 4 0 0 4 , yes m: 4 x 4 y 4 (0,1) (-1,0) 0 0 4 , yes x 0, y 0 feasible set in the first quadrant only y l Corners: A (0, 0) the origin B (0, 1) the y-int of line m C l m see below D (4, 0) the x-int of line l m C B x A D l m : Point C is the intersection of lines l and m: x 2y 4 4 x 4 y 4 2R1 2x 4 y 8 4 x 4 y 4 6x 4 4 2 x 6 3 Then using line l: 2 2 y 4 3 10 5 2y y 3 3 Therefore, point C is ( 23 , 53 ) . As you will see in the next section, the importance of finding the corners of a feasible set is that mathematicians have proven that the optimal solution will be at a corner point. 29 2-Math 130 Section 3.3 Linear Programming We now are ready to combine all the steps into the process for solving a linear programming problem by graphing. 1. Define the variables. 2. Write the objective function, sometimes called the target. This is the function that we are trying to optimize. 3. Write the constraint inequalities, sometimes called the restrictions. 4. Sketch the constraint inequalities to find the feasible region. 5. Find the corners of the feasible region. 6. Evaluate the objective function at each corner to find the optimal solution. Since there are many steps to solving one of these problems, please concentrate on doing your work in an organized manner. A NASA nutritionist wants to use two different tubes of food to meet the requirements of at least 42 units of protein, 20 units of carbohydrate, and 18 units of fat, while keeping the weight of the food at a minimum. Each tube of food A supplies 4 units of protein, 2 units of carbohydrate, 2 units of fat, and weighs 3 pounds. Each tube of food B contains 3 units of protein, 6 units of carbohydrate, and 1 unit of fat and weighs 2 pounds. How many tubes of each food should be given to the astronauts? Organize the data: A protein 4 carbs 2 fat 2 weight 3 B 3 6 1 2 total 42 30 18 min Define the variables: Let x number of tubes of food A Let y number of tubes of food B Write the objective function (target): Minimize W 3x 2 y Write the constraint inequalities (restrictions): y-int x-int (0,0)? l1 : 4 x 3 y 42 (0,14) (10.5,0) no 2 x 6 y 30 l2 : (0,5) (15,0) no l3 : 2x y 18 (0,18) (9,0) no x 0, y 0 Quadrant I 30 2-Math 130 Sketch the inequalities and the feasible set: y l3 18 A l1 14 B l2 5 C D x 9 10.5 15 Find the corners and evaluate the target: W 3x 2 y A: y-int of l3= (0,18) 0 36 36 B: l1 l3 (6, 6) 18 12 30 C: l1 l2 (9, 2) 27 4 31 D: x-int of l2 = (15,0) 45 0 45 [Use algebraic eliminations to find B and C] Solution To minimize the weight, she should supply 6 tubes of each food to each astronaut. 31 2-Math 130 Section 3.4 applications This section deals with setting up applications that have more than two variables. You will learn how to solve these in Chapter 4 Linear Programming: The Simplex Method. The three examples in the text illustrate the types of problems you are expected to be able to set up. Here is an alternative way to organize the data in example 3 (pp. 253-5): Calvert 80 x1 A 62 x2 x3 Hico 65 B 90 x4 If you find this diagram helpful, it will suffice for the “defining the variables” step, and you can set up the inequalities directly from the diagram. Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 3.1 (p. 211): Section 3.2 (p. 219): Section 3.3 (p. 240): Section 3.3 (p. 240): Section 3.4 (p. 254): #3, 7, 11, 13, 17, 19, 21, 27, 29, 31 #1, 3, 7, 9, 13, 15, 17, 25, 27 Set-up, but do not solve the L.P. problem for the following: #1, 39, 43, 45, 47, 49, 53, 59 Solve each of the following: #3, 5, 7, 9, 17, 19, 23, 27 #1, 3, 5, 7, 9 Graded Assignment #5 Show all your work in a neat and organized manner or you will receive no credit. (11 problems) Section 3.2 (p. 219): Section 3.3 (p. 240): Section 3.4 (p. 254): #14, 22, 26 #24, 26, 38, 40, 42 #4, 6, 10 32 2-Math 130 Lesson 6 Linear Programming: Simplex Method Reading Assignment Chapter 4, Linear Programming: The Simplex Method, Sections 4.1, 4.2, and 4.3 Overview The graphical method for solving LP problems that you learned in the previous lesson is nice because you have a picture and can see what is happening during the process. On the other hand, it is very limited in its usefulness since one can only solve problems with no more than two variables. In reality, LP problems usually have dozens of variables to consider. Those problems would be solved using the method in this lesson and a computer. Since you are just learning this method and doing the problems by hand, we will limit the number of initial variables to no more than four, with most problems having just three variables. The Simplex Method has variations depending upon the objective and constraints in the problem. We will only be looking at two of the many variations. The first is for a standard maximization problem and the second is a method for taking a standard minimization problem and turning it into a standard maximization problem. Some of the justifications and reasoning behind these methods are given in the textbook and in the overview below, but most of the mathematics behind these methods is beyond the scope of what we are studying. Section 4.1 and 4.2 The Simplex Method A Standard Maximization Problem has three components: 1. The objective function is to be maximized. 2. All (independent) variables must be non-negative ( 0 ) 3. Constraints must be of the form: expression non-negative constant, except for constraints which state that the variables themselves are non-negative (i.e. x 0 ). 33 2-Math 130 Consider the following example: Maximize the objective function: P 2 x1 3x2 x3 Subject to the constraints: x x x 7 1 2 3 3 x1 x2 2 x3 9 x1 0, x2 0, x3 0 This simplex method requires that we use equations, not inequalities, so we add slack variables to make up the difference. s1 takes up the slack ( s1 0) x1 x2 x3 7 becomes x1 x2 x3 s1 7 3x1 x2 2x3 9 becomes 3x1 x2 2 x3 s2 9 s2 takes up the slack (s2 0) This method also requires that the objective be in the equation form: expression + P = 0. So, P 2 x1 3x2 x3 becomes 2 x1 3x2 x3 P 0 by subtracting the variables over to the left. So, our standard maximization problem: Maximize: P 2 x1 3x2 x3 x1 x2 x3 7 Subject to: 3 x1 x2 2 x3 9 x1 0, x2 0, x3 0 Becomes: x1 x2 x3 s1 3 x1 x2 2 x3 2 x1 3 x2 x3 7 s2 9 P 0 x1 0, x2 0, x3 0, s1 0, s2 0 The next step is to set up a matrix from the system of equations we developed. This matrix is called the simplex tableau. x1 x2 x3 s1 s2 P x1 x2 x3 s1 7 Convert to matrix 3x1 x2 2 x3 s2 9 initial simplex tableau 2 x1 3x2 x3 P 0 1 1 1 1 1 2 0 3 2 3 1 0 0 1 0 0 7 0 9 1 0 In a simplex tableau, the variables with only 1’s and 0’s in their columns are called the basic variables; the other variables are called non-basic. In our set-up above, the variables s1 , s2 , and P are currently the basic variables. Some or all of the variables will transition between basic and non-basic as we manipulate the simplex tableau. A feasible solution (though not optimal) is at hand, if all non-basic variables are set to zero. If x1 0, x2 0, and x3 0, then x1 x2 x3 s1 7 becomes 0 0 0 s1 7. Therefore s1 7 and, similarly, s2 9 and P 0. The simplex method moves from a non-optimal solution to one at least as good, if not better. The method cycles through solutions until an optimal solution (maximum P value) is found. Section 4.1 (p. 262) deals with setting up the initial simplex tableau, and then section 4.2 (p. 273) deals with solving the tableau for the optimal solution. 34 2-Math 130 A solution is optimal if no negative numbers are in the bottom row, left of the vertical line. Negative numbers in the bottom row mean the maximum solution has not yet been found, so we must pivot. To pivot: 1. Choose the column that has the most negative number in the bottom row. This is called the pivot column. In the above example, it would be the x2 – column. 2. Form row ratios: For each row with a positive number in the pivot column, divide the right most entry in that row by the number in the pivot column, forming a row ratio. The pivot row has the smallest row ratio. 3. The pivot element is the intersection of the pivot column and the pivot row. 4. To pivot on the pivot element, use row operations to make a unit column. That is, make the pivot element a one (1) and all other entries in the pivot column zeros (0). 5. To determine if this is now the optimal, maximum solution, check for negative numbers in the bottom row, left of the line. If there are still negative numbers, repeat the process starting back at step one. 35 2-Math 130 Example for section 4.2: Maximize P 80 x1 70 x2 subject to 6 x1 3 x2 96 x1 x2 18 2 x1 6 x2 72 x1 0, x2 0 Step 1: Add the slack variables to make a system of equations and rewrite the objective equation. 6 x1 3 x2 s1 96 x1 x2 s2 18 2 x1 6 x2 s3 72 80 x1 70 x2 P 0 Note to remember: x1, x2 , s1, s2 , s3 0, and we want P to be as large as possible. Step 2: Set up the initial tableau. x1 x2 s1 s2 s3 P 3 6 1 1 2 6 80 70 1 0 0 0 0 1 0 0 0 0 1 0 0 96 0 18 0 72 1 0 In order to solve for the optimal solution recall: Pivot Column – The column with the most negative number in the bottom row. In the above example, that would be the x1 – column. Pivot Element – First, it must be positive. Second, choose the element with the smallest nonnegative ratio with the last column. In the above example, x1 ratio 96 / 6 16 smallest ratio, 96 6 1 ... 18 18 /1 18 so pivot on the 6 2 72 72 / 2 36 You have found the maximum value when the bottom row contains no negative numbers. Read through the textbook examples carefully, stopping to do each exercise as indicated. The simplex method is a long process, so take the time to master each step as you go along. 36 2-Math 130 Section 4.3 The Dual Method In this section, you will learn how to change a standard minimization problem into a standard maximization problem so that the simplex method can be used to solve it. A standard minimum problem still assumes that all variables are non-negative, but all the constraints are of the form: [expression non-negative constant]. First, you need a new concept called the transpose of a matrix. To transpose a matrix, you swap its rows and columns, so an n m matrix becomes an m n matrix. For example: 1 1 2 3 2 3 t 3 2 4 1 3 4 5 2 1 8 5 8 t We use this concept of transposition to change a standard minimum problem, called the primal system into a standard maximum problem, called its dual system. In general, using matrix notation: Primal system Dual System At Y C t A X B transposed X 0 Y 0 t min C X max B Y The dual system can be solved using the simplex method and the maximum value of the dual equals the minimum value of the primal. Example of transposing a primal system into a dual system: Primal System Dual System 2 x1 3x2 1 2 y1 3 y2 1y3 3 3x1 4 x2 2 3 ineq transposed 2 ineq 3 y1 4 y2 1y3 5 1x1 1x2 4 and and 2 var 3 var y1 0, y2 0, y3 0 x1 0, x2 0 max 1y1 2 y2 4 y3 min 3x1 5 x2 37 2-Math 130 Complete example 1: Step 1: Convert to the dual system. Primal System Dual System 3 y1 6 y2 18 3x1 5 x2 2 x3 4 5 y1 20 3i 2i 6 x1 8 x3 9 3v 2v 2 y1 8 y2 2 x1 0, x2 0, x3 0 y1 0, y2 0 min C 18 x1 20 x2 2 x3 max P 4 y1 9 y2 Step 2: Add slack variables x1 , x2 , and x3 . 3 y1 6 y2 x1 18 x2 5 y1 2 y1 8 y2 20 x3 4 y1 9 y2 2 P 0 Step 3: Set up the simplex tableau. y1 y2 x1 x2 x3 P 6 1 0 0 3 5 0 0 1 0 2 8 0 0 1 4 9 0 0 0 0 18 0 20 0 2 1 0 Step 4: Solve by pivoting as in section 4.2. y1 y2 x1 x2 x3 P 0 1 0 0 1 16 0 0 0 43 0 32 Primal Solution 1 10 1 5 2 5 1 10 0 0 1 0 0 5 0 4 0 34 1 29 Dual Solution max 29 when y1 4 y2 5 min 29 1 , x 0 when x1 32 , x2 10 3 Notice that the solution to the original minimum problem is found simply by reading the values in the bottom row under the x-columns. 38 2-Math 130 Complete example 2: Primal Dual y1 3 x1 x2 x3 20 y1 y2 5 2i 3i x2 2 x3 0 3v 2v y1 2 y2 1 x1 0, x2 0, x3 0 y1 0, y2 0 min C 3 x1 5 x2 x3 max P 20 y1 You may skip step 2 and go directly to the tableau, if you wish. y1 y2 x1 x2 x3 P 1 1 1 20 0 1 2 0 1 0 0 0 0 1 0 0 y1 3 5 pivoting 1 0 Solution: min 20 0 0 1 0 0 0 0 1 y2 x1 x2 x3 P 0 2 1 0 1 0 1 0 1 1 1 2 0 0 1 0 40 0 0 20 0 2 0 4 0 1 1 20 when x1 0, x2 0, x3 20 Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 4.1 (p. 269): #1 – 11 odd Section 4.2 (p. 288): #5 – 15 odd, 23, 29, 33, 35 Section 4.3 (p. 305): #1 – 15 odd; On #5 and #7, if you can do part (c) without first doing (a) & (b) then that is acceptable. Graded Assignment #6 Show all your work in a neat and organized manner or you will receive no credit. (12 problems) Section 4.1 (p. 269): Section 4.2 (p. 288): Section 4.3 (p. 305): #8, 10 #6, 22, 26, 28 #6c, 8c, 10, 12, 14, 16 39 2-Math 130 Exam 2 Information Make arrangements with your proctor to schedule Exam 2. Prior to taking this exam: • You must submit lessons 4, 5, and 6 to your instructor before taking this exam. • Please do not take this exam until you have received graded lessons 4, 5, and 6 back from your instructor. • Do not submit any subsequent lessons until you have taken this exam. Below, there is a test outline containing textbook exercises that are similar to what will be on the test. You should do all of these problems to prepare for the test. You may find that you need to do them more than once. Keep track of the problems you had to look up in the textbook, in this course guide, or on your past homework before you could solve the problems. Do those over again until you can do every problem without any references. Exam components: • This exam covers lessons 4, 5, and 6. • This is a closed-book, closed-notes exam. • Calculators are NOT allowed. The format and questions on this exam are similar to the textbook problems. • This exam is worth 100 points. • Time limit: 75 minutes. Items to take to the exam: • Photo identification • V number • Pencil and eraser Exam grades and comments: • See your Registration Confirmation Letter for how you will receive exam grades and comments from your instructor. • Graded exams will not be returned to you. However, arrangements can be made to view graded exams. Contact the ISI office for more information. Exam 2 Topic Outline (Refer to your lessons for page numbers) 1. The “short-cut” method for finding the inverse of a 2 x 2 matrix. [2.6 #9, 11] 2. The inverse of a square matrix. [2.6 #11, 13, 23, 25] 3. Matrix form [2.6 #27c, 29c, 31, 33] 4. Solving a system by matrix inversion. [2.6 #37, 41, 46] 5. Systems of inequalities; feasible set; corners [3.2 # 25, 27] 6. Geometric approach to linear programming [3.3 #21, 23, 25] 7. Converting a system of inequalities, with its objective, into a system of equations. [4.1 #5, 7] 8. Simplex tableau for a maximum problem [4.1 #9, 11] 9. Simplex method for linear programming [4.2 #7, 9, 11, 13, 15, 23, 25, 33] 10. The dual of a minimum problem [4.3#5c, 7c, 9, 11] 11. Solving a minimum linear programming problem using its dual. [4.3 #13, 15] 40 2-Math 130 Extra Practice Problems are on the next page. Extra Practice – The following problems do not necessarily cover all the topics. Make sure you can do the exam outline problems above. It’s also a good idea to make sure you can do the examples that were presented in this study guide. p. 203 #27, 29 pp. 258 – 259 #3, 5, 9, 11, 13, 17, 21, 23 pp. 358 – 359 #1 – 13 odd, 19, 27, 29, 37, 41 41 2-Math 130 Lesson 7 Sets and Counting, Part I Reading Assignment Chapter 6, Sets and Counting, Sections 6.1, 6.2, and 6.3, Overview Congratulations! You have now completed two-thirds of the course and will be studying completely different concepts. Chapters 6 and 7 deal with sets, counting, and probability. These chapters form a foundation for future study on statistics, which nearly all degrees now require. Before starting this new material, I invite you to re-read the paragraphs under the “self-study practice problems” on page 3 of this course study guide. Remember that false starts and practicing many problems are necessary to master new concepts in mathematics. Good Luck! Section 6.1 Sets A set is simply a group or a collection. This section covers the basic terms and notation when dealing with the concept of a set. Read it carefully; take notes to ensure you understand the key concepts as listed below: Set Element or member Empty or null set Note: We use either empty braces { } or the symbol to denote an empty set. Do not combine the two. is a set within a set and, therefore, it is not empty. (Strange, I know.) Equal sets Explicit or roster vs. implicit or set-builder notation e.g. a, e, i, o, u x x is a vowel of the English alphabet Note: To write simply x x is a vowel would not be a well-defined set since there are vowels in other languages besides a, e, i, o, u. Universe or universal set Subset Note: For our studies, the difference between the symbols and are not crucial, so we will use which is more general. Complement of a set Note 1: If you use any other texts or any material on the internet, you should be aware c that some authors use A and some use A for the complement of A. You may use whichever notation you wish. A Ac A c Note 2: Think of the complement as meaning “not.” Thus, E means “not E.” So, if E was the set of all even whole numbers in the Universe of all whole numbers, then E would be the set of all odd whole numbers. 42 2-Math 130 Union Note: Think “or,” but keep in mind that mathematically the word “or” means one or the other or both, whereas in many everyday usages of the word “or” we mean an exclusive “or” such as, “Would you like potatoes or rice with dinner?” Intersection Note: Think “AND.” It must be in both sets. Disjoint sets De Morgan’s Laws These are two rules having to do with complements, union, and intersection which are not covered in your text. You may find that they make some problems easier. The two laws are as follows: i) A B A B not (A or B) = (not A) and (not B) ii) A B A B not (A and B) = (not A) or (not B) Example for De Morgan’s Laws Let U a, b, c, d , e, f , g , A a, b, g, and B b, d , e, f Then A and B a, b, d , e, f , g c A B c, d , e, f a, c, g c Section 6.2 Counting Using a Venn Diagram Venn diagrams are a useful tool for counting elements in a set. We will use them again in the next chapter on probability, so this is an important technique. When given the number of elements in the overlapping region (intersection) of two or three sets, I always start with a Venn diagram as is shown in examples 2 (p. 433) and 6 (pp. 435-6). Start from the inside (intersection) of the sets and work your way out. Note that on example 4 (p. 434), I would start with a Venn diagram as illustrated on the bottom of page 434 in figure 6-9. After filling in the Venn diagram, I would use it to answer the questions (a) through (d). When the number of elements in the intersection of two sets is unknown, then you should use the formula as illustrated in example 3 (p. 433). You will not be given any problems involving three sets unless the number of elements in the intersection of all three sets is also given. You’re welcome. Section 6.3 Basic Counting Principles The most important concept in this section is the Multiplication Rule on page 443. You will use it repeatedly for the remainder of this course. While tree diagrams can be useful for visualizing the Multiplication Rule, their most important use won’t be seen until section 7.6 in lesson 10. 43 2-Math 130 You may find it helpful to think of the multiplication principle as “fill in the slots” problems. For instance, if I’m going to choose a hat, a scarf, and a coat to wear from my wardrobe which consists of 4 hats, 6 scarves, and 2 coats, then I have 3 “slots”: . I fill in the “slots” with the number hats scarves coats of available choices and then multiply those numbers to get the total number of possible outfits: 4 ∙ 2 ∙ 6 = 48 possible outfits. hats scarves coats Notice that we are choosing a hat AND a scarf AND a coat. It is the “AND” which tells us to use the Multiplication Rule. Note how this is different from the following problem which uses “OR”. Let’s suppose I’m choosing one dessert from five types of pies and three types of cakes, then I am choosing a piece of pie OR a piece of cake and the number of possibilities is 5 3 8. As always, there is no substitute for practice when studying the different ways that the multiplication principle can be applied. Here are some more examples to help you with this section. Example: How many ways can 5 CD’s be arranged on a shelf? 5 CD’s 5 slots 5 4 3 2 1 120 Example: There are 3 Jazz CD’s, 4 rock ‘n roll CD’s, and 3 country & western CD’s. a) How many ways can they all be arranged on a shelf? 10 CD’s 10 slots 10 9 8 7 6 5 4 3 2 1 3, 628,800 b) How many arrangements are there if the types of music must be grouped together? This type of problem involves using the multiplication principle more than once. For the order of the music types there are 3 2 1 6 orders. Then we have to count the order of the individual CD’s within each group: 3 2 1 4 3 2 1 3 2 1 6 5,184 Jazz R 'n R Country Example: How many three-letter “words” can be formed if a) repetition of letters is allowed? b) repetition is not allowed? 26 26 26 17,576 26 25 24 15,600 44 music type 2-Math 130 Textbook example 8 (pp. 444-5): Alternate solution: First arrange the three couples: 3 ∙ 2 ∙ 1 but then there are two orders for each couple to have sat down in (partner one, partner two OR partner two, partner one). So, the total possible seating orders are: 3 ∙ 2 ∙ 1 ∙ 2 ∙ 2 ∙ 2 = 48 This last example illustrates that it’s sometimes easier to count the number of ways to not do an arrangement (the complement) and then subtract that number from the total possible arrangements. We will revisit this method again in section 6.6 and in chapter 7. For now, it should help you do exercise #54 (p. 450). Example: How many ways can seven students be assigned seats in a row of seven desks if there are two particular students who are not allowed to sit next to each other? i) The total number of ways to arrange the seven students with no restrictions is 7 6 5 4 3 2 1 5040 . ii) To find the number of ways to arrange the seven students if the pair in question is sitting next to each other, first think of the pair as one student, leaving 6 students and six slots, so 6 5 4 3 2 1 , but then as before there are two orders for the pair. So, the total number of seating arrangements with the pair next to each other is 6 5 4 3 2 1 2 1440 . iii) Therefore, the number of arrangements for the particular pair of students to not sit beside each other is 5040 1440 3600 . Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 6.1 (p. 430): Section 6.2 (p. 436): Section 6.3 (p. 447): #1 – 19 odd, 23 – 53 odd, 57 – 71 odd #3 – 11 odd, 15 – 27 odd, 33, 39 #1 – 11 odd, 15 – 33 odd, 49 – 61 odd Graded Assignment #7 Show all your work in a neat and organized manner or you will receive no credit. (15 problems) Section 6.1 (p. 430): Section 6.2 (p. 436): Section 6.3 (p. 447): #42, 44, 52, 70, 72, 74 #12, 14, 20, 22, 24, 28 #50, 52, 54 45 2-Math 130 Lesson 8 Sets and Counting, Part II Reading Assignment Chapter 6, Sets and Counting, Sections 6.4, 6.5, and 6.6 Overview These next three sections from Chapter 6 Sets and Counting deal with two types of arrangements called permutations and combinations. These will be used extensively in the chapter on probability. *A note about the use of calculators: please use a calculator to perform the arithmetic in this section, but keep in mind that you will not have a calculator on the exam. The numbers on the exam will be of reasonable magnitude for you to compute by hand. You may need to do some cancelling such as: 10 9 8 7 10 2 9 3 8 27 2 3 2 7 3 2 7 42 5 4 3 2 1 5 4 3 2 1 1 2 1 Section 6.4 Permutations A permutation is a special case usage of the multiplication rule where each element is chosen from the same set, repetition of elements is not allowed, and the order of choice is important. Here are two more examples to accompany the ten examples given in the text. Example: How many ways are there for five people to get in line? “In line” order matters P(5, 5) or 5! or 5 4 3 2 1 120 Sometimes there’s more than one way to think about a problem. Example: How many ways are there to elect two officers, a president and vice president, from a group of five eligible people? “officers” order matters P(5, 2) or 5 4 20 46 2-Math 130 Section 6.5 Combinations A combination differs from a permutation only in the fact that the order of choice is not important. Keep the concept of a committee (a group of people whose order does not matter) in mind when dealing with combinations. You may skip the material on the Binomial Theorem, text pages 470 – 472. The first twelve examples, along with the four presented here sufficiently cover combinations. Example: How many ways are there to choose a three-person committee from a group of five people? “committee” order is irrelevant C (5,3) or P(5,3) 3! or 5 4 3 10 3! Example: How many different pizzas can be made with three toppings if there are six toppings available? “pizza toppings” mathematically it makes no difference as to the order we put the toppings on a pizza, so C (6,3) 654 20 3! Example: Twelve students apply for three assistantships. How many ways can these be awarded if a) no candidate is given preference? b) there are seven men and five women and at least one woman must be given a position? “assistantships” similar to a committee (a) C (12,3) 12 11 10 220 3! (b) three possibilities: (three women and no men) OR (two women and one man) OR (one woman and two men) C (5,3) C (7, 0) C (5, 2) C (7,1) C (5,1) C (7, 2) 5 4 3 5 4 7 6 1 7 5 3! 2! 2! 10 70 105 185 Notice and recall “AND” multiply “OR” add 47 2-Math 130 Example: From a group of ten students, how many ways are there to a) choose at least nine of them? b) choose at least two of them? c) choose no more than eight students? (a) Choosing: nine students OR all ten C (10,9) C (10,10) (b) Choosing: two students OR three students OR…all ten. There must be an easier way! (Review the last example from Lesson 7 Sets and Counting, Part I in this study guide.) at least 2 students C no student OR one student Now, think of each student as either being chosen or not being chosen. So, the number of ways to choose at least two students is possible of ways Number of ways Number Totalchoices to choose none to choose one 210 C (10, 0) C (10,1) (c) Similarly, it is easier to count the complement in this case. possible of ways Number of ways Number of ways Number Totalchoices to choose 8 to choose 9 to choose all 10 210 C (10,8) C(10,9) C(10,10) Section 6.6 A Mixture of Counting Problems This section will give you practice in distinguishing between permutations and combinations. Also, you will see some problems where a permutation and a combination are used jointly. Here is one more example to help you with the multiplication principle, permutations, combinations, “AND”, and “OR”. Example: There are seven math, eight science and five literature books in a pile. How many ways are there to: a) line them all up? b) line them all up, if the subjects must be grouped together? c) choose any three books? d) choose one book from each subject? e) choose two books from each subject? f) choose exactly two books from one subject? (a) 20 19 18 1 or 20! (b) 7! 8! 5! 3! subject order (c) (d) (e) (f) C (20,3) 7 8 5 or C (7,1) C (8,1) C (5,1) C (7,2) C (8,2) C (5,2) two math OR two science OR two literature C (7,2) C (8,2) C (5,2) 48 2-Math 130 Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 6.4 (p. 459): Section 6.5 (p. 473): Section 6.6 (p. 482): #1 – 41 odd, 45, 57, 59 #1 – 11 odd, 19, 23, 25, 29, 31, 41, 43, 45, 47 #1, 3, 5, 9, 11, 13, 17, 19, 21, 23, 31 Graded Assignment #8 Show all your work in a neat and organized manner or you will receive no credit. (17 problems) Section 6.4 (p. 459): Section 6.5 (p. 473): Section 6.6 (p. 482): #8, 38, 60, 62 #34, 36, 38, 42 #10, 12, 14, 16, 18, 28, 30, 32, 34 49 2-Math 130 Lesson 9 Probability, Part I Reading Assignment Chapter 7, Probability, Sections 7.1, 7.2, and 7.3 Overview As you are about to start the last chapter for this course, if you haven’t given yourself a congratulatory treat in a while, do so now. It’s important to celebrate the small victories and prepare for the work ahead. Sections 7.1, 7.2, 7.3 Basic Probability The first three sections in this chapter contain a lot of new vocabulary and formulas. Instead of outlining each section separately, I’m going to list the most important terms and formulas so you’ll have them together as a reference and then conclude with a few examples to supplement those given in the text. 7.1 7.1 7.1 7.1 Experiment – An activity that has observable results. Sample space, S – The set of all possible outcomes of an experiment. Event, E – Any subset of a sample space. P( E ) – The relative frequency of the event. 7.1 0 P( E ) 1 7.1 P 0 7.1 PS 1 Ps1 Ps2 . . . Psn , where S s1 , s2 , . . . , sn number of ways E can occur , if the simple outcomes of E are equally likely to occur total number of possible outcomes 7.3 PE 1 PE 7.2 P( E ) 7.3 Mutually exclusive events – Two events, E and F, are mutually exclusive if, as sets, they are disjoint. i.e., E F and PE F 0 7.3 PE or F PE F PE PF PE F Note that if E and F are mutually exclusive then PE F PE PF . However, be very, very careful to not assume mutually exclusive events without the proper information. Peoples’ intuition is often wrong in these cases. Example: Experiment: Roll two distinguishable dice. (This is the same as rolling a single die twice.) a) Find the probability of rolling at least one four. b) Find the probability of rolling a sum of at least five. a) E 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 4, 1, 4, 2, 4, 3, 4, 5, 4, 6 11 (See the table on page 512 [Figure 7-5] of the textbook.) 36 b) E 1, 4, 2, 3, 3, 2, 4, 1 PE 50 2-Math 130 PE 4 1 36 9 Example: Experiment: Roll one 20-sided die. Find the probability of rolling a multiple of 3 or a multiple of 5. Let T = rolling a multiple of 3 and let F = rolling a multiple of 5. Then nT 6 and nF 4 . Caution here, the answer is not 10 ! Why not? Because T and F are not mutually 20 exclusive events. There is one number common to both events. The number 15 is a multiple of both 3 and 5. T 3, 6, 9, 12, 15, 18 and F 5, 10, 15, 20 So, Pmultiple of 3 OR multiple of 5 PT or F PT 6 4 1 9 PT PF PT F 20 20 20 20 F Example: Experiment: There are five red, four white, and one black marble in a sack. a) If one marble is chosen without looking, what is the probability that it is not white? b) If two marbles are chosen without looking, what is the probability that they are the same color? a) Pnot white Pred OR black 5 1 6 3 10 10 5 Or Pnot white Pred Pblack Pred and black 5 1 0 6 3 10 10 10 10 5 4 6 3 Or 1 Pwhite 1 10 10 5 C5, 2 C4, 2 5241 4213 10 6 16 b) Psame color Prr OR ww 109 C10, 2 45 45 2 1 Example: Experiment: A family is planning on having three children. What is the probability of having: a) Exactly two girls? b) At least two girls? We make the assumption in probability that a baby is equally likely to be born male or female. Because of this assumption, you can use a tree diagram, similar to the one in the textbook in example 5 of section 7.2 (pp. 513-4) to find all the possible gender outcomes. 3 a) Pexactly 2 girls Pggb or gbg or bgg 8 51 2-Math 130 4 1 8 2 Example: Experiment: Toss a coin ten times and observe the number of heads. Find the probability of getting: a) Exactly two head b) At least two heads C10, 2 C8, 8 10219 1 45 a) P2H P2H and 8T 10 0.0439 10 2 2 1024 b) P at least 2H 1 P 1H OR 0H 1 P 1H AND 9T OR 0H AND 10T b) Pat least 2 girls Pggb or gbg or bgg or ggg C10, 1 C9, 9 C10, 0 C10, 10 1 210 10 1 1024 11 1013 1 0.9893 10 2 1024 1024 1024 Example: A club is comprised of twelve females and eight males. Find the probability that a random committee of five contains: a) Two males b) At least one male c) At least two males. n2M and 3F C8, 2 C12, 3 6160 a) P2M and 3F 0.3973 ncommittees of 5 C20, 5 15,504 b) Pat least one male 1 Pno males 1 Pall females 1 c) C8, 0 C12, 5 C20, 5 1 792 0.9489 15,504 Pat least 2 males 1 P1 male or 0 males C8, 1 C12, 4 C8, 0 C12, 5 3960 792 1 1 0.6935 C20, 5 15,504 Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 7.1 (p. 506): Section 7.2 (p. 515): Section 7.3 (p. 528): #1, 5 – 19 odd, 23 #1, 3, 7, 9, 13, 17, 19, 21, 23, 25, 27, 37, 39, 41, 43, 55 #1 - 15 odd, 19, 21, 25, 27, 29, 33, 35, 37 Graded Assignment #9 Show all your work in a neat and organized manner or you will receive no credit. (15 problems) Section 7.1 (p. 506): #10, 14, 24 52 2-Math 130 Section 7.2 (p. 515): Section 7.3 (p. 528): #6, 10, 18, 20, 24, 26, 30, 58 #16, 18, 22, 34 53 2-Math 130 Lesson 10 Probability, Part II Reading Assignment Chapter 7, Probability, Sections 7.4, 7.5, and 7.6 Overview These next three sections will finish the topics that are considered basics of probability. Section 7.4 Conditional Probability I’m going to start right in with an example to reinforce the textbook’s introduction to the concept of conditional probability. Example: A sack contains two red and three black chips which are also numbered as shown here: 1 2 3 4 5 The experiment is to randomly choose one chip. Let E be the event that an even number is on the chip and let B be the event that the chip is black. Furthermore, let’s assume we have an accomplice who gives us some partial information about the chip. (Silly, I know, but there is a real concrete use for this concept, as you will see soon enough.) 1 Then, P chip is even numbered, given that we know it's black P E B 3 1 and P chip is black, given that we know it is even P B E 2 In general, P E F is said as, “the probability of E, given F”. That is, the conditional probability of E occurring given that F has already occurred. In essence, a conditional probability reduces the number of elements in the sample space. Also, notice that P E F does not necessarily equal P E F . Example: Textbook exercise #18 (p. 543): a) P 4th grader 678 1205 b) P 4th grader AND above grade level 216 1205 c) P 5th grader AND at grade level OR 5th grader AND above grade level 324 98 422 1205 1205 120 225 120 P scoring below grade level, given a 4th grader 678 d) P 4th grader, given scoring below grade level e) 54 2-Math 130 When the information doesn’t fit in a nice chart, as in the above example, the formula presented in the textbook can be used: PE F PE F PF Pay close attention to the textbook examples #4 (p. 536), 6 (pp. 538-9), and 8 (p. 540). Section7.5 Independent Events Two events E and F are called independent events if any one of the following is true: i) P E F P E ii) P F E P F iii) P E F P E P F You must be very careful to NOT use your everyday understanding of “independent”; intuition is often wrong here. You must verify, mathematically, that one of the above three conditions is true. Also, be careful to not confuse mutually exclusive events and independent events. Exercise #1 of section 7.5 (p. 559) presents you with examples to show that two sets may be: mutually exclusive and independent mutually exclusive and dependent not mutually exclusive and independent not mutually exclusive and dependent. Section 7.6 Bayes’ Rule For the problems in this section, I require you to draw tree diagrams. I have redone the textbook’s example #1 below to show you how to use the tree diagram as the first step in solving the problem. Example: Textbook example #1 (pp. 567-8) D P D 0.65 0.05 0.0325 0.05 I 0.65 0.95 0.35 D′ P D 0.35 0.15 0.0525 D 0.15 II 0.85 P I D D′ PI D P D 0.0325 0.0325 325 13 0.0325 0.0525 0.085 850 34 55 2-Math 130 Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 7.4 (p. 541): Section 7.5 (p. 559): Section 7.6 (p. 574): Section 7.6 (p. 574): #3, 5, 9 – 17 odd, 23 – 31 odd, 35, 37, 39 #1 – 15 odd, 19 – 27 odd #1, 9, 11, 13, 15, 23, 31 [First draw a tree diagram to solve each of these.] #25 Graded Assignment #10 Show all your work in a neat and organized manner or you will receive no credit. (19 problems) Section 7.4 (p. 541): Section 7.5 (p. 559): Section 7.6 (p. 574): Section 7.6 (p.574): #4, 6, 10, 12, 14, 16, 28, 36 #6, 10, 24, 26 #2, 6, 18, 24, 38 [First draw a tree diagram to solve each of these.] #26 56 2-Math 130 Exam 3 Information Make arrangements with your proctor to schedule Exam 3. Prior to taking this exam: • You must submit lessons 7, 8, 9, and 10 to your instructor before taking this exam. • Please do not take this exam until you have received graded lessons 7, 8, 9, and 10 back from your instructor. • Do not submit any subsequent lessons until you have taken this exam. Below, there is a test outline containing textbook exercises that are similar to what will be on the test. You should do all of these problems to prepare for the test. You may find that you need to do them more than once. Keep track of the problems you had to look up in the textbook, in this course guide, or on your past homework before you could solve the problems. Do those over again until you can do every problem without any references. Exam components: • This exam covers lessons 7, 8, 9, and 10. • This is a closed-book, closed-notes exam. • Calculators are NOT allowed. The format and questions on this exam are similar to the textbook problems. • This exam is worth 100 points. • Time limit: 75 minutes. Items to take to the exam: • Photo identification • V number • Pencil and eraser Exam grades and comments: • See your Registration Confirmation Letter for how you will receive exam grades and comments from your instructor. • Graded exams will not be returned to you. However, arrangements can be made to view graded exams. Contact the ISI office for more information. Exam 3 Topic Outline (Refer to your lessons for page numbers) 1. Set Operations [6.1 #43 – 53 odd] 2. Counting with Venn diagrams [6.2 #3 – 11 odd, 23, 28] 3. Multiplication Rule [6.3 #5, 7, 9, 11, 17, 29, 49, 53] 4. Permutations [6.4 #7, 11, 17, 23b, 29, 37, 43, 57] 5. Combinations [6.5 #19, 23, 27, 41, 43] 6. Mixed counting practice [6.6 #3, 5, 11, 13, 17, 21] 7. Basic Probability [7.1 #1; 7.2 #1, 3, 7, 9, 13, 19, 21, 23, 27, 39, 41] 8. Compound Events [7.3 #5, 7, 9, 21, 25, 29, 33, 35] 9. Conditional Probability [7.4 #5, 9, 17] 10. Independence [7.5 #1, 3, 5, 7, 9, 11] 11. Bayes’ Rule [7.6 #9, 11, 13, 25] 57 2-Math 130 Extra Practice Problems are on the next page. Extra Practice – These problems do not necessarily cover all the topics. Make sure you can do the exam outline problems above. It’s also a good idea to make sure you can do the examples that were presented in this study guide. pp. 491 – 494 #5, 9, 11, 15, 17, 21, 23, 25, 27, 29, 35, 43, 51 pp. 593 – 596 #3, 5, 7, 15, 21, 27, 29, 33, 35, 37 58 2-Math 130 Lesson 11 [Self-study] Markov Chains Reading Assignment Section 7.7 Overview Section 7.7 Markov Chains This last section is going to show you an application that uses both matrices and probabilities. Are you excited? Go ahead, you can admit it, no one will know. A Markov chain, also called a Markov sequence or process, is a mathematical model used to describe an experiment or measurement that is repeated at regular time intervals where the possible outcomes for each trial are the same and the probabilities for one trial depends solely on the immediately preceding trial. Here is one more example to accompany the ones presented in the textbook: Suppose the data shows that 80% of the daughters of working women go on to be working mothers while only 30% of the daughters of non-working women go on to become working mothers. (Note that “working” here implies working outside of the home.) This data can be represented by the following matrix: not working working 0.8 T 0.3 0.2 0.7 working not working Suppose that at generation 0, 40% of women work. This can be represented by the row matrix: d0 0.4 0.6 We wish to find the percent of women working in the next two generations. 0.8 0.2 d1 d0 T 0.4 0.6 0.5 0.5 0.3 0.7 50% of women will be working. 0.8 0.2 d 2 d1 T 0.5 0.5 0.55 0.45 0.3 0.7 55% of women will be working. 59 2-Math 130 The following table shows the data after many generations: Generation % working 0 40.00 1 50.00 2 55.00 3 57.50 4 58.75 5. 59.38 . . . . . 10 11 12 13 14 59.98 59.99 60.00 60.00 60.00 . . . . . . The long term trend is for 60% of women to work. Of course, we would like to find the long term trend without doing the repeated matrix multiplications. We can do that as follows: Let x y be the stable distribution. 0.8 0.2 y x y 0.3 0.7 0.8 x 0.3 y x 0.2 x 0.3 y 0 So we have 0.2 x 0.7 y y 0.2 x 0.3 y 0 Then x y T x y x Notice that both of these equations are equivalent to 2x 3 y 0 . We need another equation in order to get a unique solution. Notice that for the stable distribution x y 1. So, x y 1 1 1 R2 R2 2 R1 1 1 1 R1 5 R1 R2 5 1 0 5 2 0 2 x 3 y 0 2 3 0 0 5 3 1 0 R1 15 R1 1 2 R2 5 R2 0 1 3 5 2 5 So, x 3 60% 5 Now, you are due a huge congratulations! And, good luck on the final exam. Self-Study Practice Problems These are not to be handed in. They are for you to check your understanding of the material; to exercise your math brain. Make sure you attempt each problem without looking in the solutions manual. It is crucial for you to practice, to do math, and not just read about it. Section 7.7 (p. 589): #1, 3, 5, 7, 9, 13, 17, 21, 39 The material will be covered on the final exam so be sure and do the self-study problems. There is no graded assignment for this section. Do not send anything in for this section. 60 2-Math 130 Final Exam Information Make arrangements with your proctor to schedule the Final Exam. Prior to taking this exam: • You must submit lessons 1-10 to your instructor before taking this exam. • Please do not take this exam until you have received graded lessons 1-10 back from your instructor. On the next pages there are two sample final exams, followed by their answers, which are similar to the final exam that you will take. Before you even try these, go back to lesson 1 and review all of the examples in this study guide. Then work whichever problems you can without looking at the answers. This is hard to do, I know, but if you resist you will get a better indicator of which topics you need to review the most. Keep track of the problems you had to look up in the textbook, in this guide, or on your past homework before you could solve them. You will know those are the topics that you need to go back and review the most. Exam components: • This exam covers lessons 1-11. • This is a closed-book, closed-notes exam. • Calculators are NOT allowed. The format and questions on this exam are similar to the textbook problems. • This exam is worth 100 points. • Time limit: 120 minutes (2 hours). Items to take to the exam: • Photo identification • V number • Pencil and eraser Exam grades and comments: • See your Registration Confirmation Letter for how you will receive exam grades and comments from your instructor. • Graded exams will not be returned to you. However, arrangements can be made to view graded exams. Contact the ISI office for more information. Final Exam reviews, and answers, continue on the following pages 61 2-Math 130 Final Exam Review #1 1. Find the equation of the line that passes through (–3, 4) and (2, 24). x y 2z 1 2. Use the Gauss-Jordan Method to solve: 3 x 2 y z 6 x y 4 z 11 v 4w 3. Find all the solutions to the following system of linear equations: v 4w 2 4. Compute the inverse of A 3y 1 x 2y 2 3y z 5 6 in two ways. 3 10 1 0 0 5. (a) Use the Gauss-Jordan Method to compute the inverse of the matrix: A 0 1 2 2 3 4 x 1 y 2z (b) Use the answer in Part (a) to solve: 6 2 x 3 y 4 z 11 6. Use the Geometric (graphical) method to solve the linear programming problem: Minimize C 3x 5 y Subject to: 2 x y 8 x 2 y 10 x, y 0 7. A mining company owns two mines which produce three grades of ore. Each day, mine A produces 1 ton of high-grade ore, 2 tons of medium-grade ore, and 3 tons of low-grade ore. Mine B produces 1 ton of high-grade ore, 1 ton of medium-grade ore, and 2 tons of low-grade ore daily. An order is placed for 70 tons of high-grade ore, 40 tons of medium-grade ore, and 60 tons of low-grade ore. If the operating costs for production are $500 for mine A and $350 for mine B, find the number of days each mine should be used to fill this order at the lowest cost. (a) Define the variables you will use. (b) Set up the LP problem (do not solve). 62 2-Math 130 8. Use the simplex method to solve the linear programming problem: Maximize: P 2 x1 3x2 6 x3 Subject to: 2 x1 3 x2 x3 10 2 x2 3x3 6 x1 0, x2 0, x3 0 9. For the given linear programming problem, set up the initial simplex tableau for the dual problem, and label each column with the appropriate variable. Do not solve. Minimize: C 30 x1 60 x2 50 x3 5 x1 3 x2 x3 2 Subject to: x1 2 x2 x3 3 x1 0, x2 0, x3 0 10. Find the solution to an original minimum problem if the final tableau is: y1 y2 y3 x1 x2 P 1 35 0 13 17 8 35 0 8 20 16 1 92 39 35 1 0 1 7 32 35 0 1 96 0 0 11. A committee of 4 people is to be selected from a group of 7 men and 3 women. In how many ways can the committee be selected if (a) 2 men and 2 women are selected? (b) at least 2 women are selected? 12. A campus radio station surveyed 190 students. The survey determined that 114 liked rock music, 50 liked country music, and 41 liked classical music. Moreover, 14 liked rock and country, 15 liked rock and classical, 11 liked country and classical, and 5 liked all three types. How many students (a) do not like any of the three types of music? (b) like only country music? 13. A survey of drivers determined that 23% had received a speeding ticket, 37% had received a parking ticket and 9% had received both. Given that a randomly chosen driver had received a parking ticket, what is the probability that the driver had also received a speeding ticket? 14. E and F are events from an experiment with P( E ) 0.6, P F 0.3, and P( E F ) 0.4. Compute P( F ), P( F E ), and P( E F ). Are E and F independent events? Justify mathematically. 15. A company manufactures calculators at three different production lines. 50% are produced in line 1, and of these 5% are defective. 20% are produced in line 2 of which 3% are defective, and 30% are produced in line 3 of which 1% are defective. First draw a tree diagram and then answer the following. What is the probability that (a) a calculator is defective? (b) given a defective calculator, it was produced in line 2? 63 2-Math 130 16. The matrix given below represents the transition of college students in a big city university between living on campus (C) and living at home (H) at the end of a semester. Find the percentage of those students that live at home when the stable distribution is achieved. From C C 0.9 To H 0.1 H 0.4 0.6 64 2-Math 130 Final Exam Review #2 1. Find the point of intersection of the lines: 3x 4 y 10 2x y 8 x 2y 1 2. Use the Gauss-Jordan Method to solve: 2 x 5 y z 3 3x 7z 2 w x 3. Use the Gauss-Jordan Method to find all the solutions for: 5w x 2 y w 2 4 z 1 1 4 4. Compute the inverse of the matrix A . 2 6 1 3 2 1 6 5. Let A , and B 3 5 2 2 4 (a) Find AB if possible. If not possible, state why. (b) Find BA if possible. If not possible, state why. 1 23 8 3 6. (a) Show that the matrices and are inverses of each other. 6 2 3 4 (b) Use part (a) to solve the system of equations: x 3 y6 2 3 x 4 y 8 1 0 5 6 10 5 7. Use the fact that the inverse of the matrix A 1 1 0 is the matrix A1 6 9 5 3 2 6 1 2 1 x to solve x y 5z 2 1 3x 2 y 6 z 1 65 2-Math 130 8. Use the Geometric (graphical) method to solve the linear programming problem: Maximize: P x y Subject to: 2 x y 12 2 x 3 y 24 x 0, y 0 9. Use the simplex method to solve the linear programming problem: Maximize: M x1 2 x2 3x3 Subject to: x1 2 x2 x3 40 x1 x2 x3 30 x1 0, x2 0, x3 0 10. For the given linear programming problem, set up the initial simplex tableau for the dual problem, and label each column with the appropriate variable. Do not solve. Minimize: C 20 x1 16 x2 Subject to: 7 x1 2 x2 12 3 x1 5 x2 8 6 x1 10 x2 9 x1 0, x2 0 11. At the left below is the primal problem and on the right is the final simplex tableau for the dual problem. Give the solution to the original primal problem. Minimize: C 8 x1 12 x2 y1 y2 Subject to: x1 3 x2 2 0 1 2 x1 2 x2 3 1 x1 0, x2 0 0 0 0 x1 x2 P 3 4 14 0 3 1 2 1 2 0 2 5 4 1 4 1 13 12. Set up the following problem. Do not solve. A farmer is going to plant at least 100 acres with 2 crops, wheat and lentils. Each acre of wheat costs $40 and takes 2 hours to plant. Each acre of lentils costs $30 and takes 3 hours to plant. The farmer has at most $12,000 and 240 hours to spend on planting. If she expects to make $50 for each acre of wheat and $75 for each acre of lentils, how many acres of each crop should she plant in order to maximize her profits? (a) Identify the unknowns. (b) Continue by forming the objective function and a system of constraints to model this problem. 13. To help plan the number of meals to be prepared in a college cafeteria, a survey was conducted and the following data were obtained: 130 students ate breakfast, 180 students ate lunch, 275 students ate dinner, 68 students ate breakfast and lunch, 112 students ate breakfast and dinner, 90 students ate lunch and dinner, and 58 students ate all three meals. How many students ate (a) only dinner in the cafeteria? (b) exactly two meals in the cafeteria? 66 2-Math 130 14. In how many ways can four married couples attending a concert be seated to a row of eight chairs if: (a) there are no restrictions? Do not simplify your answer. (b) each married couple is seated together? Do not simplify your answer. 15. An organization is comprised of 12 men and 10 women. They need to form a committee of 2 men and 3 women to plan a party. In how many ways can the committee be formed? Do not simplify your answer. 16. Let E and F be events with P( E ) 0.4, P( F ) 0.3, and P E F 0.9 . (a) Find P( F ), P( E F ), P( E F ), and P( E F ). (b) Are E and F independent events? Justify mathematically. (c) Are E and F mutually exclusive? Justify mathematically. 17. The attendant at a toll booth of an airport parking lot has been noticing the types of vehicles and drivers that pass through his station. He has determined that the probability that the vehicle is a pickup is 0.4 and the probability that the driver is a male is 0.6. The probability that both occur is 0.3. (a) Find the probability that either the vehicle is a pickup or that the driver is a male. (b) If he sees that the next vehicle is a pickup, what is the conditional probability that the driver is a male? 18. A professor has two assistants who grade for him. Mr. Goofy grades 70% of the papers, while Mr. Sloppy grades the remaining 30% of the papers. Of the papers Mr. Goofy grades, 10% are graded incorrectly. Of the papers Mr. Sloppy grades, 20% are graded incorrectly. (a) Draw a tree diagram for this situation. (b) A grading error is found in a paper. What is the probability that it was graded by Mr. Sloppy? 19. Birds migrate between habits I and II each year as follows: Of the birds in habitat I, 70% will migrate to habitat II the next year and 30% will remain in habitat I. Of the birds in habitat II, 50% will migrate to habitat I and 50% will remain in habitat II. (a) Find the transition matrix for this Markov process. (b) If in one year 80% of the birds are in habitat I and 20% are in habitat II, give the percentages of birds in each habitat the following year. 67 2-Math 130 Final Exam Review #1 Answers 1. y 4 28 ( x 3) 5 2. x 2, y 1, z 2 3. v 1 4w 3 y w any number or y any number z5 x 2 2y (1 4 j 3k , j, 2 2k , k , 5) 4. The two methods are the Gauss-Jordan method and the short-cut for 2x2’s. In either case, you 5 should get A1 3 2 3 1 0 which you can check by showing A A1 . 1 0 1 1 0 0 5. (a) A 2 2 1 3 1 1 2 2 1 (b) Recall A X B X A1 B x 1, y 21, z 27 2 6. Your feasible set should have 3 corners. The minimum value is C 26 when x 2 and y 4 . 7. (a) Let x1 number of days mine A should be used and x2 number of days mine B should be used (b) Minimize: C 500 x1 350 x2 Subject to: x1 x2 70 2 x1 x2 40 3 x1 2 x2 60 x1 0, x2 0 68 2-Math 130 8. The maximum is P 20 when x1 4, x2 0, and x3 2. 9. y1 y2 x1 5 1 1 3 2 1 2 x2 x3 P 0 0 0 30 0 1 0 0 60 1 0 0 1 0 50 3 0 0 0 1 0 10. The minimum is C = 92 when x1 20 and x2 16. 11. (a) C 7,2 C 3,2 7 6 3 2 63 2! 2! (b) C 3,2 C 7,2 C 3,3 C 7,1 3 2 7 6 1 7 70 2! 2! 12. From a Venn diagram: (a) 20 (b) 30 13. P S P 14. (a) 0.7, 2 3 P S P 9 P P 37 , 4 7 (b) No, since P(F) P(F E) 15. (a) 0.034 16. 1 5 (b) 6 34 3 17 20% 69 2-Math 130 Final Exam Review #2 Answers 1. (2, 4) 2. x 3, y 2, z 1 3. x 2 w y 3 2w z 1 w w any number 4. 3 2 1 1 2 5. (a) 11 14 8 22 12 4 (b) Not possible: Number of columns of B number of rows of A. 1 0 . 0 1 6. (a) Show that the product of the two matrices is 8 3 6 (b) Multiply to get x 24 and y 20 . 6 2 8 2 7. Use A 1 to get x 3, y 2, and z 1 . 1 1 8. The feasible set should have 3 corners. Maximum is P 12 when x 12 and y 0 . 9. Maximum is M 90 when x1 0, x2 0, and x3 30 . 10. y1 y2 y3 x1 x2 P 7 3 6 1 0 0 20 2 5 10 0 1 0 16 12 8 9 0 0 1 0 11. Minimum is C 13 when x1 5 4 and x2 1 4 70 2-Math 130 12. (a) x number of acres of wheat y number of acres of lentils (b) Maximize P 50 x 75 y Subject to: x y 100 40 x 30 y 12000 2 x 3 y 240 x 0, y 0 13. Use a Venn diagram to find (a) 131 and (b) 96. 14. (a) 8! (b) 2∙2∙2∙2∙4! 15. C(12, 2) C(10,3) 16. (a) 0.7, 0.2, 2 7 , 0.2 (b) No, P( E F ) P( E ) P( F )or P( E F ) P( E ) (c) No, P( E F ) 0 17. (a) 0.7 (b) 3 4 I 18. (a) 0.1 (b) P ( S I ) 6 13 G 0.7 0.9 C I 0.2 0.3 S 0.8 0.7 0.3 0.5 0.5 19. (a) T C (b) 66% in habitat I and 34% in habitat II 71 2-Math 130 72 2-Math 130 University of Idaho 875 Perimeter Drive MS 3081 Moscow ID 83844-3081 Local: 208-885-6641 Toll-free: 877-464-3246 Fax: 208-885-5738 [email protected] www.uidaho.edu/isi LESSON SUBMISSION FORM STUDENT: complete this section. Please print. To print a Lesson Submission Form visit the ISI website, select Forms. Name__________________________________________________ V number _____________________________ Daytime phone ____________________________________________ Email ________________________________ Course subject and name __________________________________________________________________________ Instructor name _________________________________________________________________________________ □ I have retained a copy of each lesson. _________________________ Lesson number _________________________ □ I have used a separate form for each course. ____________________ Lesson number _________________________ □ I have checked for correct postage before mailing. _______________ Lesson number _________________________ □ I have enclosed a self-addressed stamped envelope with enough postage to return the graded lesson(s). INSTRUCTOR: complete this section. Please print. 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