OPTIMIZATION: Linear Programming
|
|
- Leslie Stokes
- 5 years ago
- Views:
Transcription
1 May 21, 2013 OPTIMIZATION: Linear Programming Linear programming (OPTIMIZATION) is the process of taking various linear inequalities (constraints) related to some situation, and finding the "best" value obtainable under those conditions lesson 1 A typical example: You produce various types of shoes. Your have limited materials and labor. Each type of shoe that you produce uses different amounts of material and labour. Each type of shoe sells for different amounts. You must determine the "best" production level to maximize your profits. Another example: You have limited amounts of flour and sugar and limited "oven" time (these are your "constraints") to make some cookies and some cakes. You sell the cookies and cakes for different prices...and you may make more money selling a cake but because of the constraints you are able to make/sell more cookies...you need to determine exactly how much you should make of each In "real life", linear programming is part of a very important area of mathematics called "optimization techniques"...in order to OPTIMIZE your profit. Linear programming is used every day in the organization and allocation of resources. The general process for solving linearprogramming problems is to first graph the inequalities (called the "constraints") to form a walled-off area on the x,y-plane These "real life" systems can have dozens or even hundreds of variables. In this class, though, you will only work with the simple (and graphable) twovariable linear case. This area is called the "feasibility region" or the "polygon of constraints".
2 The "feasibility region" (polygon of constraints) is the region that contains ALL the points that meet every single constraint set out at the start of the problem The second step is to determine the coordinates of the corners of this feasibility region (that is, find the intersection points of the various pairs of lines) A long time ago somebody proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region. Step three is to test these corner points in the "optimization equation" for which you're trying to find the highest or lowest value. consider the following example: Let's walk through an example: The three inequalities in the curly braces are the constraints. The polygon on the plane that they create will be the feasibility region. The formula "z = 3x + 4y" is the optimization equation. Find the (x, y) corner points of the feasibility region that return the largest and smallest values of z. This will be optimizing z.
3 The formula "z = 3x + 4y" is the optimization equation. This is an equation that right now is GIVEN to you, but in the context of a story you may have to deduce the optimization equation. My first step is to solve each inequality for the more-easily graphed equivalent forms: turn them into y=ax + b Graph it: this is from last year imagine they are straight lines--we did this last year--draw the solid or dashed line (according to the inequality) choose a "test point"...do a test...indicate if it is true or false shade appropriately according to your test find the area that is commonly covered by all three shadings we usually use (0,0) or (10.0) true: start at line and shade toward test point false: start at line and shade away from test point find the area that is covered by all three shadings--this is the feasibility region...or the "polygon of constraints" EVERY point that is in the shaded region (including on the lines) meets all the constraints of the situation. There are an infinite number of them.
4 To find the corner points -- which aren't always clear from the graph -- pair up the lines that meet (thus forming a system of linear equations) and solve: the corner points are (2, 6), (6, 4), and ( 1, 3). So, to find the solution to this problem (to find the optimal solution--either a maximum or a minimum) I only need to plug these three corner points into the optimization equation: "z = 3x + 4y" z = 3x + 4y (2, 6): z = 3(2) + 4(6) = = 30 (6, 4): z = 3(6) + 4(4) = = 34 (-1, -3): z = 3(-1) + 4(-3) = = -15 the maximum of z = 34 occurs at (6, 4) the minimum of z = 15 occurs at ( 1, 3). Can you outline the general steps solve this type of problem?
5 here's another example: constraints: there are 6 of them this vertical line means nothing I put them in this form so I can graph them a little easier optimization equation z = - 0.4x + 3.2y Graph it. Shade it. From the graph, determine which lines cross to form the corners pair them up in order to find the coordinates. start at the "top" of the shaded area and work clockwise around the edges
6 Now I'll plug each corner point into the optimization equation, z = - 0.4x + 3.2y Let's look at a problem in context: (1, 6): z = 0.4(1) + 3.2(6) = = 18.8 (5, 2): z = 0.4(5) + 3.2(2) = = 4.4 (5, 0): z = 0.4(5) + 3.2(0) = = 2.0 (4, 0): z = 0.4(4) + 3.2(0) = = 1.6 (0, 2): z = 0.4(0) + 3.2(2) = = 6.4 (0, 5): z = 0.4(0) + 3.2(5) = = 16.0 Then the maximum is 18.8 at (1, 6) and the minimum is 2 at (5, 0). A calculator company produces a scientific calculator and a graphing calculator. Past sales indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and no more than 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits? The question asks for the optimal number of calculators, I need to determine HOW MANY of EACH KIND OF CALCULATOR to produce so my variables will be: x: # of scientific calculators produced y: # of graphing calculators produced Since they can't produce negative numbers of calculators, there are two non-negative constraints, x > 0 y > 0 What information does the story give us about the constraints? What are the limitations on "resources"?
7 Past sales indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and no more than 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped each day. In addition to the two non-negative constraints, x > 0 y > 0 Also, x > 100 y > 80 x < 200 y < 170 x + y > 200 or for graphing purposes y > x The revenue equation will be the optimization equation: maximize profit P = 2x + 5y loss for sci. calc. profit for the graph. calc {x > 100 y > 80 x < 200 y < 170 y > x Corner points are: (100, 170) either through observation or (200, 170) solving systems (200, 80) (120, 80) and (100, 100)
8 Test them all in R (100, 170) (200, 170) (200, 80) (120, 80) and (100, 100) R = 2x + 5y RESULT: the maximum value of R = at (x, y) = (, ) That is, scientific calculators and graphing calculators is the OPTIMAL production rate to maximize profit considering the constraints. Another example You need to buy some filing cabinets. You know that Cabinet X costs $10, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of each type of model should you buy, in order to maximize storage volume? The question ask for the number of cabinets needed to maximize storage,, but always taking constraints into consideration variables will be: x: number of model X cabinets y: number of model Y cabinets Naturally, x > 0 and y > 0. costs and floor space will be my constraints volume will be my optimization equation space: cost : cost
9 Naturally, x > 0 and y > 0. costs and floor space will be my constraints volume will be my optimization equation cost: 10x + 20y < 140 space: 6x + 8y < 72 Naturally, x > 0 and y > 0. costs and floor space will be my constraints volume will be my optimization equation cost: 10x + 20y < 140 y < - (1/2)x + 7 space: 6x + 8y < 72 y < - (3/4)x + 9 Test the corner points at (8, 3), (0, 7), and (12, 0) (8, 3) (0, 7) (12, 0) (8, 3) (0, 7) (12, 0) you should obtain a maximum volume of 100 cubic feet by buying eight of model X and three of model Y
Mathematics for Business and Economics - I. Chapter7 Linear Inequality Systems and Linear Programming (Lecture11)
Mathematics for Business and Economics - I Chapter7 Linear Inequality Systems and Linear Programming (Lecture11) A linear inequality in two variables is an inequality that can be written in the form Ax
More informationAlgebra 2 Notes Systems of Equations and Inequalities Unit 03b. Optimization with Linear Programming
Optimization with Linear Programming Big Idea Linear programming is one of the most practical uses of mathematics in the real world. The inequalities of the system represent the constraints in the problem
More informationLinear Programming. You can model sales with the following objective function. Sales 100x 50y. x 0 and y 0. x y 40
Lesson 9.7 Objectives Solve systems of linear inequalities. Solving Systems of Inequalities Suppose a car dealer nets $500 for each family car (F) sold and $750 for each sports car (S) sold. The dealer
More informationGraphing Linear Inequalities in Two Variables.
Many applications of mathematics involve systems of inequalities rather than systems of equations. We will discuss solving (graphing) a single linear inequality in two variables and a system of linear
More informationDid you ever think that a four hundred year-old spider may be why we study linear relationships today?
Show Me: Determine if a Function is Linear M8221 Did you ever think that a four hundred year-old spider may be why we study linear relationships today? Supposedly, while lying in bed Rene Descartes noticed
More informationCHAPTER 12: LINEAR PROGRAMMING
CHAPTER 12: LINEAR PROGRAMMING Previous Years Board Exam (Important Questions & Answers) MARKS WEIGHTAGE 06 marks 1. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the
More informationChapter 3 Linear Programming: A Geometric Approach
Chapter 3 Linear Programming: A Geometric Approach Section 3.1 Graphing Systems of Linear Inequalities in Two Variables y 4x + 3y = 12 4 3 4 x 3 y 12 x y 0 x y = 0 2 1 P(, ) 12 12 7 7 1 1 2 3 x We ve seen
More informationLinear Programming: A Geometric Approach
Chapter 3 Linear Programming: A Geometric Approach 3.1 Graphing Systems of Linear Inequalities in Two Variables The general form for a line is ax + by + c =0. The general form for a linear inequality is
More informationCHAPTER 4 Linear Programming with Two Variables
CHAPTER 4 Linear Programming with Two Variables In this chapter, we will study systems of linear inequalities. They are similar to linear systems of equations, but have inequalitites instead of equalities.
More informationWEEK 4 REVIEW. Graphing Systems of Linear Inequalities (3.1)
WEEK 4 REVIEW Graphing Systems of Linear Inequalities (3.1) Linear Programming Problems (3.2) Checklist for Exam 1 Review Sample Exam 1 Graphing Linear Inequalities Graph the following system of inequalities.
More informationChapter 4. Linear Programming
Chapter 4 Linear Programming For All Practical Purposes: Effective Teaching Occasionally during the semester remind students about your office hours. Some students can perceive that they are bothering
More informationExample Graph the inequality 2x-3y 12. Answer - start with the = part. Graph the line 2x - 3y = 12. Linear Programming: A Geometric Approach
Linear Programming: A Geometric Approach 3.1: Graphing Systems of Linear Inequalities in Two Variables Example Graph the inequality 2x-3y 12. Answer - start with the = part. Graph the line 2x - 3y = 12.
More informationSystems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses
5 5 Systems and Matrices Systems and Matrices 5.6 Systems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses Sections 5.6 5.8 2008 Pearson Addison-Wesley. All rights
More informationUNIT 6 MODELLING DECISION PROBLEMS (LP)
UNIT 6 MODELLING DECISION This unit: PROBLEMS (LP) Introduces the linear programming (LP) technique to solve decision problems 1 INTRODUCTION TO LINEAR PROGRAMMING A Linear Programming model seeks to maximize
More informationChapter 4 Linear Programming
Chapter Objectives Check off these skills when you feel that you have mastered them. From its associated chart, write the constraints of a linear programming problem as linear inequalities. List two implied
More informationLesson 1: Graphing Linear Inequalities in Two Variables Chapter 6.1
MATH 11 FOUNDATIONS Lesson 1: Graphing Linear Inequalities in Two Variables Chapter 6.1 Name: Date: A mathematical inequality must contain one of the following: Examples of linear inequalities in a single
More informationUnit 0: Extending Algebra 1 Concepts
1 What is a Function? Unit 0: Extending Algebra 1 Concepts Definition: ---Function Notation--- Example: f(x) = x 2 1 Mapping Diagram Use the Vertical Line Test Interval Notation A convenient and compact
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationG r a d e 1 1 A p p l i e d M a t h e m a t i c s ( 3 0 S ) Final Practice Examination
G r a d e 1 1 A p p l i e d M a t h e m a t i c s ( 3 0 S ) Final Practice Examination G r a d e 1 1 A p p l i e d M a t h e m a t i c s Final Practice Examination Name: Student Number: For Marker s Use
More informationLesson 6.1 Matrix Representations
Lesson. Matrix Representations. Supply the missing entries in each transition matrix..7 m r.9..7 a. [M] b. [R] c. [T] t. m. A survey of registered voters showed that of those people who voted in the presidential
More informationWrite all responses on separate paper. Show your work for credit. Write in complete sentences.
Math 13 Liberal Arts Math HW4 Spring 13 Name Write all responses on separate paper. Show your work for credit. Write in complete sentences. 1. Graph the two lines described by the equations 2x + 7y = 61
More informationLINEAR PROGRAMMING INTRODUCTION 12.1 LINEAR PROGRAMMING. Three Classical Linear Programming Problems (L.P.P.)
LINEAR PROGRAMMING 12 INTRODUCTION ou are familiar with linear equations and linear inequations in one and two variables. They can be solved algebraically or graphically (by drawing a line diagram in case
More informationOPERATIONS RESEARCH. Linear Programming Problem
OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com 1.0 Introduction Linear programming
More informationUsing Linear Programming for Management Decisions
Using Linear Programming for Management Decisions By Tim Wright Linear programming creates mathematical models from real-world business problems to maximize profits, reduce costs and allocate resources.
More informationReview for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
More informationLesson Polygons
Lesson 4.1 - Polygons Obj.: classify polygons by their sides. classify quadrilaterals by their attributes. find the sum of the angle measures in a polygon. Decagon - A polygon with ten sides. Dodecagon
More informationI can solve simultaneous equations algebraically and graphically. I can solve inequalities algebraically and graphically.
B I can factorise and expand complex expressions. I can factorise Quadratics I can recognise the Difference of Two Squares (D.O.T.S) simultaneous equations algebraically and graphically. inequalities algebraically
More informationConcept: Solving Inequalities Name:
Concept: Solving Inequalities Name: You should have completed Equations Section 7 Part A: Solving Inequalities before beginning this handout. COMPUTER COMPONENT Instructions: In follow the Content Menu
More informationChapter 13-1 Notes Page 1
Chapter 13-1 Notes Page 1 Constrained Optimization Constraints We will now consider how to maximize Sales Revenue & Contribution Margin; or minimize costs when dealing with limited resources (constraints).
More informationQuestion 2: How do you solve a linear programming problem with a graph?
Question : How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.
More informationMath 20 Practice Exam #2 Problems and Their Solutions!
Math 20 Practice Exam #2 Problems and Their Solutions! #1) Solve the linear system by graphing: Isolate for in both equations. Graph the two lines using the slope-intercept method. The two lines intersect
More informationFoundations 110 EXAM Review
Foundations 110 EXAM Review Chapter 5 Systems of Linear Inequalities Graphing Linear Inequalities in Two Variables Things to note: Dashed lines will represent < or > signs Solid lines will represent or
More informationChapter 1 Polynomials and Modeling
Chapter 1 Polynomials and Modeling 1.1 Linear Functions Recall that a line is a function of the form y = mx+ b, where m is the slope of the line (how steep the line is) and b gives the y-intercept (where
More informationPrecalculus Notes: Unit 7 Systems of Equations and Matrices
Date: 7.1, 7. Solving Systems of Equations: Graphing, Substitution, Elimination Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities. Solution of a System
More informationd = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 Student Name: Date: Teacher Name: Sunil Dudeja Score:
Geometry EOC (GSE) Quiz Answer Key Equations and Measurement - (MGSE9 12.G.GPE.4) Use Coordinates For Theorems, (MGSE9 12.G.GPE.5 ) Prove Slope Criteria, (MGSE9 12.G.GPE.6) Find The Point, (MGSE9 12.G.GPE.7
More informationName: Date: Study Guide: Systems of Equations and Inequalities
Name: Date: Study Guide: Systems of Equations and Inequalities Systems of Equations Linear systems consist of two or more linear equations in the same variables. A solution to the linear system of equations
More informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More informationMath 2 Coordinate Geometry Part 3 Inequalities & Quadratics
Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics 1 DISTANCE BETWEEN TWO POINTS - REVIEW To find the distance between two points, use the Pythagorean theorem. The difference between x 1 and x
More informationA theme park charges $12 entry to visitors. Find the money taken if 1296 people visit the park.
Write an Equation An equation is a term used to describe a collection of numbers and variables related through mathematical operators. An algebraic equation will contain letters that relate to real quantities
More informationLINEAR PROGRAMMING: A GEOMETRIC APPROACH. Copyright Cengage Learning. All rights reserved.
3 LINEAR PROGRAMMING: A GEOMETRIC APPROACH Copyright Cengage Learning. All rights reserved. 3.1 Graphing Systems of Linear Inequalities in Two Variables Copyright Cengage Learning. All rights reserved.
More informationGraphical Methods in Linear Programming
Appendix 2 Graphical Methods in Linear Programming We can use graphical methods to solve linear optimization problems involving two variables. When there are two variables in the problem, we can refer
More informationI can solve simultaneous equations algebraically, where one is quadratic and one is linear.
A* I can manipulate algebraic fractions. I can use the equation of a circle. simultaneous equations algebraically, where one is quadratic and one is linear. I can transform graphs, including trig graphs.
More informationLinear Programming. ICM Unit 3 Day 1 Part 1
Linear Programming ICM Unit 3 Day 1 Part 1 Arrival: Tools of the Trade Pencils Grab a couple to SHARE with your partner. Graph Paper Two pieces per student for today. Ruler One per student. Get out a NEW
More informationConcept: Solving Inequalities Name:
Concept: Solving Inequalities Name: You should have completed Equations Section 7 Part A: Solving Inequalities before beginning this handout. COMPUTER COMPONENT Instructions: In follow the Content Menu
More informationFoundations of Mathematics 11
6.1 Graphing Linear Inequalities in Two Variables (Part 1) Review of Graphing Linear Equations: Most linear equations will be written in one of two different forms: 0 General form Slope y-intercept form
More informationSolving linear programming
Solving linear programming (From Last week s Introduction) Consider a manufacturer of tables and chairs. They want to maximize profits. They sell tables for a profit of $30 per table and a profit of $10
More informationEnd-of-Module Assessment Task
Name Date 1. The juice box pictured below is 4 inches high, 3 inches long, and 2 inches wide. a. In the grid above, the distance between grid lines represents one inch. Use the grid paper to sketch the
More information( )! 1! 3 = x + 1. ( ) =! x + 2
7.5 Graphing Parabolas 1. First complete the square: y = x 2 + 2x! 3 = x 2 + 2x + 1 ( )! 1! 3 = x + 1 ( ) 2! 4 The x-intercepts are 3,1 and the vertex is ( 1, 4). Graphing the parabola: 3. First complete
More informationELGIN ACADEMY Mathematics Department Evaluation Booklet (Main) Name Reg
ELGIN ACADEMY Mathematics Department Evaluation Booklet (Main) Name Reg CfEM You should be able to use this evaluation booklet to help chart your progress in the Maths department from August in S1 until
More information11.2 RECTANGULAR COORDINATES IN THREE DIMENSIONS
11.2 Rectangular Coordinates in Three Dimensions Contemporary Calculus 1 11.2 RECTANGULAR COORDINATES IN THREE DIMENSIONS In this section we move into 3 dimensional space. First we examine the 3 dimensional
More information5-8. Systems of Linear Inequalities. Vocabulary. Lesson. Mental Math
Lesson 5-8 Systems of Linear Inequalities Vocabulary feasible set, feasible region BIG IDEA The solution to a system of linear inequalities in two variables is either the empty set, the interior of a polygon,
More informationMath 1313 Prerequisites/Test 1 Review
Math 1313 Prerequisites/Test 1 Review Test 1 (Prerequisite Test) is the only exam that can be done from ANYWHERE online. Two attempts. See Online Assignments in your CASA account. Note the deadline too.
More informationFair Game Review. Chapter 15. Name Date. Find the area of the figure ft
Name Date Chapter 15 Fair Game Review Find the area of the figure. 1. 3 m 3 m 2. 5 m 7 m 14 m 9 m 3 m 3. 4 in. 1 in. 4. 12 in. 5 in. 9 in. 12 in. 7 in. 12 in. 5. 6. 5 ft 3 ft 15 ft 1 ft 4 in. 10 in. 8
More informationLINEAR PROGRAMMING. Chapter Overview
Chapter 12 LINEAR PROGRAMMING 12.1 Overview 12.1.1 An Optimisation Problem A problem which seeks to maximise or minimise a function is called an optimisation problem. An optimisation problem may involve
More informationVolume and Surface Area of Rectangular Prisms All Boxed Up
Volume and Surface Area of Rectangular Prisms SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Use Manipulatives, Visualization, Group Presentation, Think/Pair/Share The shape in Figure is a
More informationMatrices and Systems of Equations
1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 6: Systems of Equations and Matrices Section 6.3 Matrices and Systems of Equations Matrices
More informationUNIT 2 LINEAR PROGRAMMING PROBLEMS
UNIT 2 LINEAR PROGRAMMING PROBLEMS Structure 2.1 Introduction Objectives 2.2 Linear Programming Problem (LPP) 2.3 Mathematical Formulation of LPP 2.4 Graphical Solution of Linear Programming Problems 2.5
More informationUsing the Graphical Method to Solve Linear Programs J. Reeb and S. Leavengood
PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8719-E October 1998 $2.50 Using the Graphical Method to Solve Linear Programs J. Reeb and S. Leavengood A key problem faced by managers is how to
More informationSetup and graphical solution of Linear Programming Problems [2-variables] Mathematical Programming Characteristics
Setup and graphical solution of Linear Programming Problems [2-variables] Mathematical Programming Characteristics Decisions must be made on the levels of a two or more activities. The levels are represented
More informationLINEAR PROGRAMMING (LP), GRAPHICAL PRESENTATION GASPAR ASAMPANA
LINEAR PROGRAMMING (LP), GRAPHICAL PRESENTATION GASPAR ASAMPANA INTRODUCTION Linear Programming a is combination of a linear objective function and set of linear constraints. The linear constraints express
More informationObjectives: Find a function that models a problem and apply the techniques from 4.1, 4.2, and 4.3 the find the optimal or best value.
Objectives: Find a function that models a problem and apply the techniques from 4.1, 4., and 4.3 the find the optimal or best value. Suggested procedure: Step 1. Draw a picture! Label variables and known
More informationStudent Name: Date: Teacher Name: Sunil Dudeja. Score:
Geometry EOC (GSE) Quiz Equations and Measurement - (MGSE9 12.G.GPE.4) Use Coordinates For Theorems, (MGSE9 12.G.GPE.5 ) Prove Slope Criteria, (MGSE9 12.G.GPE.6) Find The Point, (MGSE9 12.G.GPE.7 ) Use
More informationGrade 7 Math (Master) Essential Questions Content Skills
Wilmette Public Schools, District 39 Created 2006-2007 Fall Grade 7 Math (Master) Why is it important to differentiate between various multiplication methods? How can a procedure lead you to an accurate
More informationVertical Line Test a relationship is a function, if NO vertical line intersects the graph more than once
Algebra 2 Chapter 2 Domain input values, X (x, y) Range output values, Y (x, y) Function For each input, there is exactly one output Example: Vertical Line Test a relationship is a function, if NO vertical
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I. 2 nd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for
More informationLesson 08 Linear Programming
Lesson 08 Linear Programming A mathematical approach to determine optimal (maximum or minimum) solutions to problems which involve restrictions on the variables involved. 08 - Linear Programming Applications
More informationMathematics. Linear Programming
Mathematics Linear Programming Table of Content 1. Linear inequations. 2. Terms of Linear Programming. 3. Mathematical formulation of a linear programming problem. 4. Graphical solution of two variable
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2015 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationSection Graphing Systems of Linear Inequalities
Section 3.1 - Graphing Systems of Linear Inequalities Example 1: Find the graphical solution of the inequality y x 0. Example 2: Find the graphical solution of the inequality 5x 3y < 15. 1 Example 3: Determine
More informationChapter 1: Variables, Expressions, and Integers
Name: Pre-Algebra Period: 8 Chapter 1: Variables, Expressions, and Integers Outline 1.1: p. 7 #12-15, 20-27, 32, 33, 34, 36 Date 1.2: p. 12 #16-20, 25-28, 30, 31, 36 1.3: p. 19 #10-18, 21-25, 31 1.4: p.
More information, 6.7,, Order the numbers from least to greatest. 1. 1, 0, 2, 5, 4. Simplify the expression. 10.
Getting Ready for Pre-Algebra or Algebra Summer Math Practice The following are practice questions to evaluate the students understanding of concepts and skills taught in seventh grade as a readiness for
More informationLesson 10: Interpreting Graphs of Proportional Relationships
Classwork Example 1 Grandma s Special Chocolate Chip Cookie recipe, which yields 4 dozen cookies, calls for 3 cups of flour. Using this information, complete the chart: Create a table comparing the amount
More informationSection 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
More informationBCN Decision and Risk Analysis. Syed M. Ahmed, Ph.D.
Linear Programming Module Outline Introduction The Linear Programming Model Examples of Linear Programming Problems Developing Linear Programming Models Graphical Solution to LP Problems The Simplex Method
More informationCHAPTER 2 LINEAR PROGRAMMING: BASIC CONCEPTS
CHAPER 2 LINEAR PROGRAMMING: BASIC CONCEPS rue-alse Questions 2-1 Linear programming problems may have multiple goals or objectives specified. 2-2 Linear programming allows a manager to find the best mix
More informationStudents interpret the meaning of the point of intersection of two graphs and use analytic tools to find its coordinates.
Student Outcomes Students interpret the meaning of the point of intersection of two graphs and use analytic tools to find its coordinates. Classwork Example 1 (7 minutes) Have students read the situation
More informationGraphical Analysis. Figure 1. Copyright c 1997 by Awi Federgruen. All rights reserved.
Graphical Analysis For problems with 2 variables, we can represent each solution as a point in the plane. The Shelby Shelving model (see the readings book or pp.68-69 of the text) is repeated below for
More informationFinite Mathematics MAT 141: Chapter 3 Notes
Finite Mathematics MAT 141: Chapter 3 Notes Linear Programming David J. Gisch Graphing Linear Inequalities Linear Inequalities Graphing with Intercepts Find the -intercept. Substitute 0and solve for. Find
More informationThe graph of the region that shows the number of packages of each item Kala can purchase is
2. Solve each system of inequalities by graphing. The graph of the system of inequalities is 4. CCSS REASONING The most Kala can spend on hot dogs and buns for her cookout is $35. A package of 10 hot dogs
More informationYear 10 Term 2 Homework
Yimin Math Centre Year 10 Term 2 Homework Student Name: Grade: Date: Score: Table of contents 5 Year 10 Term 2 Week 5 Homework 1 5.1 Graphs in the number plane................................ 1 5.1.1 The
More information9 ft. 10 cm. 8 ft. 9 cm. 17 cm. 10 cm. 8 cm. 15 cm. 18 m 16 m. 14 m. Geometry: Homework Problems. Rectangles & Parallelograms
Geometry: Homework Problems Rectangles & Parallelograms Homework Find the area of each figure. 11. 9 ft 15. 8 ft 10 cm 9 cm 17 cm 12. 14 mm 16. The length of a rectangle is 17 m and the width is 5.25 m.
More informationSection 4.5 Linear Inequalities in Two Variables
Section 4.5 Linear Inequalities in Two Variables Department of Mathematics Grossmont College February 25, 203 4.5 Linear Inequalities in Two Variables Learning Objectives: Graph linear inequalities in
More informationUnit 1, Lesson 1: Tiling the Plane
Unit 1, Lesson 1: Tiling the Plane Let s look at tiling patterns and think about area. 1.1: Which One Doesn t Belong: Tilings Which pattern doesn t belong? 1 1.2: More Red, Green, or Blue? m.openup.org//6-1-1-2
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Pre-Calculus Mid Term Review. January 2014 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the graph of the function f, plotted with a solid
More informationChapter 7: Linear Functions and Inequalities
Chapter 7: Linear Functions and Inequalities Index: A: Absolute Value U4L9 B: Step Functions U4L9 C: The Truth About Graphs U4L10 D: Graphs of Linear Inequalities U4L11 E: More Graphs of Linear Inequalities
More informationSomeone else might choose to describe the closet by determining how many square tiles it would take to cover the floor. 6 ft.
Areas Rectangles One way to describe the size of a room is by naming its dimensions. So a room that measures 12 ft. by 10 ft. could be described by saying its a 12 by 10 foot room. In fact, that is how
More information422 UNIT 12 SOLID FIGURES. The volume of an engine s cylinders affects its power.
UNIT 12 Solid Figures The volume of an engine s cylinders affects its power. 422 UNIT 12 SOLID FIGURES Gas-powered engines are driven by little explosions that move pistons up and down in cylinders. When
More informationMathematics RIT Score:
Mathematics RIT Score: 201-210 Number Sense and Operations Whole Numbers Understand the concept of division using pictorial representation Use front-end estimation strategy for multiplication and division
More informationUnit 2 Day 9. FRED Functions
Unit 2 Day 9 FRED Functions 1 1. Graph 2. Test a point (0,0) 3. Shade Warm Up You may want to try the problems on this slide by hand! Practice for the non-calculator part of the test! 2 2 1. 2. y x 2x
More informationUnit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form
Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Imagine the path of a basketball as it leaves a player s hand and swooshes through the net. Or, imagine the path of an Olympic diver
More informationSection Graphing Systems of Linear Inequalities
Section 3.1 - Graphing Systems of Linear Inequalities Example 1: Find the graphical solution of the inequality y x 0. Example 2: Find the graphical solution of the inequality 5x 3y < 15. 1 How to find
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More information1. List the Intercepts of the Inequality and then sketch the graph using the Math 125 method of shading
Math 125 Final Exam Practice 1. List the Intercepts of the Inequality and then sketch the graph using the Math 125 method of shading 18x10y 90 2. USING ALGEBA(meaning no calculators), find the solution
More informationQuadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0
Quadratic Equations Learning Objectives 1. Graph a quadratic function using transformations. Identify the vertex and axis of symmetry of a quadratic function 3. Graph a quadratic function using its vertex,
More informationIntroduction to Linear Programming
Introduction to Linear Programming Linear Programming Applied mathematics is all about applying mathematical techniques to understand or do something practical. Optimization is all about making things
More informationThese notes are in two parts: this part has topics 1-3 above.
IEEM 0: Linear Programming and Its Applications Outline of this series of lectures:. How can we model a problem so that it can be solved to give the required solution 2. Motivation: eamples of typical
More informationReady To Go On? Skills Intervention 3-1 Using Graphs and Tables to Solve Linear Systems
Read To Go On? Skills Intervention 3-1 Using Graphs and Tables to Solve Linear Sstems Find these vocabular words in Lesson 3-1 and the Multilingual Glossar. Vocabular sstem of equations linear sstem consistent
More informationSECTION 1.2 (e-book 2.3) Functions: Graphs & Properties
SECTION 1.2 (e-book 2.3) Functions: Graphs & Properties Definition (Graph Form): A function f can be defined by a graph in the xy-plane. In this case the output can be obtained by drawing vertical line
More information6th Grade Math. Parent Handbook
6th Grade Math Benchmark 3 Parent Handbook This handbook will help your child review material learned this quarter, and will help them prepare for their third Benchmark Test. Please allow your child to
More informationSection 2.0: Getting Started
Solving Linear Equations: Graphically Tabular/Numerical Solution Algebraically Section 2.0: Getting Started Example #1 on page 128. Solve the equation 3x 9 = 3 graphically. Intersection X=4 Y=3 We are
More information