...Name: ______________________ Class: _________________ Date: _________ ID: A Solving Real-World problems with System of Linear Equations ____ 1 Mr. Frankel bought 7 tickets to a puppet show and spent $43. He bought a combination of child tickets for $4 each and adult tickets for $9 each. Which system of equations below will determine the number of adult tickets, a, and the number of child tickets, c, he bought? A. a = c - 9 9a + 4c = 43 B. 9a + 4c = 43 a +c=7 C. a + c = 301 a +c=7 D. 4a + 4c = 50 a +c=7 2 Tyrone is packaging a mix of bluegrass seed and drought-resistant seed for people buying grass seed for their lawns. The bluegrass seed costs him $2 per pound while the drought-resistant grass seed costs him $3 per pound. a. Write an equation showing that Tyrone spent $68 altogether for the two types of grass seed. b. Write an equation showing that Tyrone bought a total of 25 lb of the two types of grass seed. c. Solve the system of equations to find out how many pounds of each type of grass seed Tyrone bought. Mr. Jarvis invested a total of $9,112 in two savings accounts. One account earns 7.5% simple interest per year and the other earns 8.5% simple interest per year. Last year, the two investments earned a total of $884.88 in interest. Write a system of equations that could be used to determine the amount Mr. Jarvis initially invested in each account. Let x represent the amount invested at 7.5% and let y represent the amount invested at 8.5%. A. x + y = 9, 112 0.075x +...
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...optional)** RESEARCH PAPER/THESIS/DISSERTATION APPROVAL TITLE (in all caps) By (Author) A Thesis/Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of (Degree) in the field of (Major) Approved by: (Name of thesis/dissertation chair), Chair (Name of committee member 1) (Name of committee member 2) (Name of committee member 3) (Name of committee member 4) Graduate School Southern Illinois University Carbondale (Date of Approval) AN ABSTRACT OF THE DISSERTATION OF NAME OF STUDENT, for the Doctor of Philosophy degree in MAJOR FIELD, presented on DATE OF DEFENSE, at Southern Illinois University Carbondale. (Do not use abbreviations.) TITLE: A SAMPLE RESEARCH PAPER ON ASPECTS OF ELEMENTARY LINEAR ALGEBRA MAJOR PROFESSOR: Dr. J. Jones (Begin the abstract here, typewritten and double-spaced. A thesis abstract should consist of 350 words or less including the heading. A page and one-half is approximately 350 words.) iii DEDICATION (NO REQUIRED FOR RESEARCH PAPER) (The dedication, as the name suggests is a personal dedication of one’s work. The section is OPTIONAL and should be double-spaced if included in the thesis/dissertation.) iv ACKNOWLEDGMENTS (NOT REQUIRED IN RESEARCH PAPER) I would like to thank Dr. Jones for his invaluable assistance and insights leading to the writing of this...
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...TUTOR CLASS Content: Covered all chapter 3, 4, 5. No: * Linear Approximation * Linear programming * Leontief model Chapter 3: INTRODUCTION TO LINEAR EQUATIONS ANDMATRICES (Gaussian elimination) 1. Solve the following system x1-4x2+x3=5-2x1-5x2+3x3=33x1+2x2-x3=1 Using: A, Gauss elimination method B, Cramer’s rule. (chapter 4) Chapter 4: EIGENVECTORS AND EIGENVALUES (Determinants, Cramer’s rule, Matrix Inverse, Eigenvectors, Eigenvalues) 2. Let A b the matrix defined by A=3 4 -1 04 -1 0 3-1 1 2 3 A, Applying the row operation 4 R4+R2→R2 to A, what is the resulting matrix B we get? B, Use row operations to transform A to the form C where C=0 a b c0 d e f0 g h i-1 1 2 3 And where the letters a, b, c, d, e, f, g, h, g and I are the numbers that you have to find out (depending on your row operations). Find a relation between the determinant of A and determinant of C. C, Evaluate the determinant of A, detA, by using the properties of determinants and cofactor expansions along rows and columns of your choice. 3. ( Matrix Inverse) Recall that if a matrix A is invertible, then the unique solution of the linear system of equations AX= b is given by X= A-1b Given the linear system of equations x+2y-9z =1-2x-4y+19z=0-y+2z =1 A, Find the inverse of the matrix A of coefficient of the linear system above, where A= 1 2 -9-2 -4 190...
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...MATH 3330 INFORMATION SHEET FOR FINAL EXAM FALL 2011 FINAL EXAM will be in PKH 103 at 2:00-4:30 pm on Tues Dec 13 • See above for date, time and location of FINAL EXAM. Recall from the first-day handout that any student not obtaining a positive score on the FINAL EXAM will not pass this class. • The material covered will be the same as that covered on the homework from the start of the semester through Dec 6 (but not §6.3) inclusive. (Homework is listed at my website: www.uta.edu/math/vancliff/T/F11 .) • My remaining office hours are: 3:30-4:20 pm on Thurs Dec 8 and 3:30-5:30 pm on Mon Dec 12. • This test will be, in part, multiple choice, but you do NOT need to bring a scantron form. There will be several choices of answer per multiple-choice question and, for each, only one answer will be the correct one. You should do rough work on the test or on paper provided by me. No calculator is allowed. No notes or cards are allowed. BRING YOUR MYMAV ID CARD WITH YOU. • When I write a test, I look over the lecture notes and homework which have already been assigned, and use them to model about 85% of the test problems (and most of them are fair game). You should expect between 30 and 40 questions in total. • A good way to review is to go over the homework problems you have not already done & make sure you understand all the homework well by 48 hours prior to the test. You should also look over the past tests/midterms and understand those fully. In addition, this...
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...Course Syllabus MTH/208 – College Mathematics 1 Course: X Course Start Date: X Course End Date: X Campus/Learning Center : X |[pic] |Syllabus | | |College of Natural Sciences | | |MTH/208 Version 6 | | |College Mathematics I | Copyright © 2012, 2011, 2008, 2007, 2006, 2005 by University of Phoenix. All rights reserved. Course Description This course begins a demonstration and examination of various concepts of algebra. It assists in building skills for performing specific mathematical operations and problem solving. These concepts and skills serve as a foundation for subsequent quantitative business coursework. Applications to real-world problems are emphasized throughout the course. This course is the first half of the college mathematics sequence, which is completed in MTH/209: College Mathematics II. Policies Faculty and students will be held responsible for understanding and adhering to all policies contained within the following two documents: ...
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...**(This page is optional)** RESEARCH PAPER/THESIS/DISSERTATION APPROVAL TITLE (in all caps) By (Author) A Thesis/Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of (Degree) in the field of (Major) Approved by: (Name of thesis/dissertation chair), Chair (Name of committee member 1) (Name of committee member 2) (Name of committee member 3) (Name of committee member 4) Graduate School Southern Illinois University Carbondale (Date of Approval) AN ABSTRACT OF THE DISSERTATION OF NAME OF STUDENT, for the Doctor of Philosophy degree in MAJOR FIELD, presented on DATE OF DEFENSE, at Southern Illinois University Carbondale. (Do not use abbreviations.) TITLE: A SAMPLE RESEARCH PAPER ON ASPECTS OF ELEMENTARY LINEAR ALGEBRA MAJOR PROFESSOR: Dr. J. Jones (Begin the abstract here, typewritten and double-spaced. A thesis abstract should consist of 350 words or less including the heading. A page and one-half is approximately 350 words.) iii DEDICATION (NO REQUIRED FOR RESEARCH PAPER) (The dedication, as the name suggests is a personal dedication of one’s work. The section is OPTIONAL and should be double-spaced if included in the thesis/dissertation.) iv ACKNOWLEDGMENTS (NOT REQUIRED IN RESEARCH PAPER) I would like to thank Dr. Jones for his invaluable assistance and insights leading to the writing of this paper. My sincere thanks also goes to the...
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...O Solving Systems of Linear Equations using Inverse of a Matrix & Elementary Row Operations Consider: [pic] be a linear system of n equations in n unknowns and let [pic] be the coefficient matrix so that we can write the given system as AX = B where [pic]. If [pic] then the system has a unique solution. To solve for SLE of the form, [pic], [pic]. Recall: For SLE in 2 unknowns: For SLE in 3 unknowns: given [pic] , given [pic] [pic] [pic], where [pic] , [pic] is the transpose of the cofactors of A. Minor Any element [pic] of [pic] is associated with another determinant [pic]of [pic]order obtained by deleting the ith row and the jth column in [pic]. Example: [pic] , [pic], [pic] Cofactor The product of the minor [pic] is called the cofactor of the element of [pic]. That is, [pic]. Example: Given: [pic] [pic] , [pic], [pic] [pic] , [pic] , [pic] Examples: Solve each of the following systems of linear equations using Inverse: 1. [pic] 2. [pic] 3. [pic] 4. [pic] Solution: 1. [pic] The corresponding matrix for this is [pic] [pic], [pic], to solve: [pic] , ie. [pic] Therefore: [pic] and the s.s. = [pic] 2. [pic] The corresponding matrix for this is [pic] [pic], [pic], to solve: [pic] , ie. [pic] Therefore: [pic]...
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...Linear Algebra Applied to Economics Input-Output Analysis Introduction: There are several non-mathematical subjects that linear algebra can be applicable too. Economics is a topic that linear algebra can be used to make a formal application, for example in Input-Output Analysis, econometrics, Game theory, and break-even point analysis. As a group we are going to be focusing on the Input-Output analysis, a type of analysis created for the purpose of describing and making predictions of complicated mathematical models using systems of linear equations. It was established by “W. Leontief, who won the 1973 Nobel Prize in Economics” (Hefferon, p.60). In this paper, mathematical and linear algebra formulas, calculations, graphs, diagrams, pictures, etc., will be clearly shown as to further understand the applicability of linear algebra in economics. Calculations and mathematical examples used in economics will be provided in the context of this paper for better understanding. Also, terms and notations used will be explained, derivation and origin of mathematical results will be shown. Definitions: Economics is a branch of knowledge concerned with the production, distributions, and consumption of goods and services. Linear algebra is a branch of mathematics with the properties of finite dimensional vector spaces and linear mapping between the spaces. The equations are represented using matrices and vectors and consist of several unknowns. Econometrics is branch of economics...
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...inear systems of equations arise in many computational tasks in many different disciplines. They arise naturally when a continuous mathematical model is converted to a discrete numerical algorithm. However, there are also a huge number of initially discrete models where they also arise. As Wassily Leontief wrote in Scientific American in 1951 [153]: This article is concerned with a new effort to combine economic facts and theory known as “interindustry” or “input-output” analysis. Essentially it is a method of analysis that takes advantage of the relatively stable pattern of the flow of goods and services among the elements of our economy to bring a much more detailed statistical picture of the system into the range of manipulation by economic theory. As such, the method has had to await the modern high-speed computing machine as well as the present propensity of government and private agencies to accumulate mountains of data. It is now advancing from the phase of academic investigation and experimental trial to a broadining sphere of application in grand-scale problems of national economic policy.1 Gaussian elimination is the oldest and the simplest — but not always the fastest — algorithm for solving matrix equations. The title of this chapter is quite long because a matrix equation can be solved by many different algorithms. The only ones we discuss are Gaussian elimination and a variant which is faster in certain circumstances. Frequently in physical systems...
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...Math 116 Algebra 1a wk 7 ch. 8 Topic: Systems of Equations Refer to these examples: Ex. 2: EXAMPLE 2 Blending Flower Seeds. Tara’s website, Garden Edibles, specializes in the sale of herbs and flowers for colorful meals and garnishes. Tara sells packets of nasturtium seeds for $0.95 each and packets of Johnny-jumpup seeds for $1.43 each. She decides to offer a 16-packet spring-garden combination, combining packets of both types of seeds at $1.10 per packet. How many packets of each type of seed should be put in her garden mix? |Let the garden mix have x packets of Nasturtium seeds and y packets of Johnny-jumpup seeds. | |Then, by data we have | |x + y = 16 ( y = 16 – x | |0.95x + 1.43y = 16(1.10) | |0.95x + 1.43(16 – x) = 17.6 | |0.95x – 1.43x = 17.6 – 1.43(16) | |-0...
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...How will computers be programmed to control traffic flow and avoid congestion? They will be required to solve systems continually based on the following premise: If traffic is to keep moving, during any period of time the number of cars entering an intersection must equal the number of cars leaving that intersection. Let’s see what this means by looking at the intersections of four one-way city streets. Traffic Control Figure 6.3shows the intersections of four one-way streets.As you study the figure,notice that 300 cars per hour want to enter intersection from the north on 27th Avenue.Also, 200 cars per hour want to head east from intersection on Palm Drive. The letters and stand for the number of cars passing between the intersections. a. If the traffic is to keep moving, at each intersection the number of cars entering per hour must equal the number of cars leaving per hour. Use this idea to set up a linear system of equations involving and b. Use Gaussian elimination to solve the system. c. If construction on 27th Avenue limits to 50 cars per hour, how many cars per hour must pass between the other intersections to keep traffic flowing? Solution a. Set up the system by considering one intersection at a time, referring to Figure 6.3. For Intersection Because cars enter and cars leave the intersection, then For Intersection Because cars enter the intersection and cars leave then For Intersection Figure 6.3 indicates that cars enter and leave, so For...
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...*Week of November 18 -24 Thanksgiving Recess th th Required Text(s)/Software/Tools: Finite Mathematics & its Applications and MyMathLab Student Access Kit (packaged with the th textbook), 10 Edition, Goldstein, Schneider, and Siegel, M.J., Prentice Hall, ISBN 9780321645098 (with solutions manual), or ISBN 978-0321744586 (without solutions manual) Course Prerequisites: MTH 2002 College Algebra 2 Course Description This course offers students an opportunity to develop skills in linear mathematics and probability. Topics include matrices, inverses, input-output analysis, linear programming, sets, counting, probability, and the mathematics of finance. Applications will be developed in business, economics, and the sciences. Course Outcomes Students will have the opportunity to 1. Develop competency in solving systems of equations using matrices 2. Understand how to set up and solve linear programming problems 3. Develop competency in using counting techniques, including the inclusion-exclusion principle, Venn Diagrams, and the Multiplication Principle 4. Differentiate between and to use Permutations and Combinations in counting 5. Become competent in calculating probabilities using various methods 6. Recognize and apply Markov Processes 7. Learn how to set up and solve Interest, Annuities, and Amortization problems Course Methodology Each week, you will be expected to: 1. Review the week's learning objectives 2. Complete all assigned readings 3. Complete all lecture...
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...1.0 Introduction In this era of information technology, civil engineers rely heavily on software to perform their design tasks. Unfortunately, most commercial structural analysis packages are closed-source, which means that the operations that the program performs cannot be inspected by the user. Moreover, such software packages are invariably very pricey, and, hence, are generally not affordable for students and smaller engineering firms. The objective of this design project was to design a structural analysis program that would be free of charge and available to all. This computer program was to be open source and well commented, so that its users could comprehend the operations performed in the analysis of a given structure. To accomplish these objectives, the generalized stiffness method of structural analysis was implemented into a computer algorithm. This algorithm, called “TrussT Structural Analysis”, is a collection of visual basic modules embedded in a Microsoft Excel document using Visual Basic for Applications (VBA). This design report outlines the theory behind TrussT Structural Analysis, as well as the methods by which that theory was implemented into computer algorithms. The first two sections of this report present the theory of the generalized stiffness method of structural analysis and its implementation into a computer algorithm. The following sections present the procedures by which the stiffness method was modified to incorporate the analysis...
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...Question #3 (a)Clearly T has a linear transformation Kernel of T is the set of all vectors | x1 | x2 | | | in R2 such that | (*) | T ( | x1 | x2 | | | ) = 0R2 | This yields the equation 1 x1 +1 x2 | 1 x1 +0x2 | | | = | 0 | 0 | | | The matrix equation above is equivalent to the following homogeneous system of equations 1 x1 | +1 x2 | = | 0 | 1 x1 | +0 x2 | = | 0 | | | We now transform the coefficient matrix of the homogeneous system above to the reduced row echelon form to determine the solution space. 1 | 1 | 1 | 0 | | can be transformed by a sequence of elementary row operations to the matrix 1 | 0 | 0 | 1 | | The reduced row echelon form of the augmented matrix is 1 | 0 | 0 | 1 | | which corresponds to the system 1 x1 | | = | 0 | | 1 x2 | = | 0 | The leading entries in the matrix have been highlighted in yellow. x1 | = | 0 | x2 | = | 0 | This means the kernel consists only of the zero vector, and consequently has no basis. Comments | * The nullity of T is 0. This is the dimension of the kernel of L. * T is a one-to-one transformation since ker T = {0R2}. * T is not a one-to-one onto ransformation.Question #1 T=R3 _ R3T(x)=A(x) A X 1 2 1 | 1 -1 12 1 1 | | | X | 2 | 1-4 | | | 0 | -31 | | | | | | Question # 2References http://www.calcul.com/http://www...
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...ESSEX COUNTY COLLEGE Engineering Technologies and Computer Sciences Division ELC 230 – Circuits and Systems for Engineering Course Outline Course Number & Name: ELC 230 Circuits and Systems for Engineering Credit Hours: 3.0 Contact Hours: 3.0 Lecture: 3.0 Lab: N/A Other: N/A Prerequisites: Grades of “C” or better in MTH 122, PHY 104, and CSC 112 or CSC 121 Co-requisites: None Concurrent Courses: None Course Outline Revision Date: Fall 2010 Course Description: This is a calculus-based course in electric circuit theory and analysis for Engineering AS degree program students interested in pursuing computer or electrical engineering. It includes DC and AC principles with an emphasis on Kirchhoff's Laws, network theorems for resistive, capacitive, and inductive networks, mesh and nodal analysis, and sinusoidal steady-state analysis. Also, power, resonance, and ideal transformers are studied. The theory is reinforced with instructor-run demos. Assignments include the use of circuit analysis computer software. Course Goals: Upon successful completion of this course, students should be able to do the following: 1. analyze passive electric circuits to predict their behavior; 2. identify, analyze, and solve technical problems in linear systems; and 3. use state-of-the-art technology to solve problems in linear systems. Measurable Course Performance Objectives (MPOs): Upon successful completion of this course, students should...
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