LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH

Size: px
Start display at page:

Download "LINEAR PROGRAMMING PROBLEM: A GEOMETRIC APPROACH"

Transcription

1 59 LINEAR PRGRAMMING PRBLEM: A GEMETRIC APPRACH 59.1 INTRDUCTIN Let us consider a simple problem in two variables x and y. Find x and y which satisfy the following equations x + y = 4 3x + 4y = 14 Solving these equations, we get x=2 and y=2. What happens when the number of equations and variables are more? Can we find a unique solution for such system of equations? However, a unique solution for a set of simultaneous equations in n variables can be obtained if there are exactly n relations. What will happen when the number of relations is greater than or less then n? A unique solution will not exist, but a number of trial solutions can be found. Again, if the number of relations are greater than or less than the number of variables involved and the relation are in the form of inequalities. Can we find a solution for such a system? Whenever the analysis of a problem leads to minimizing or maximizing a linear expression in which the variable must obey a collection of linear inequalities, a solution may be obtained using linear programming techniques. ne way to solve linear programming problems that involve only two variables is geometric approach called graphical solution of the linear programming problem.

2 18 :: Mathematics 59.2 BJECTIVES After the study of this lesson you will be able to define feasible solution define optimal solution solve linear programming problems by graphical method SLUTIN F LINEAR PRGRAMMING PRBLEMS In the previous lesson we have seen the problems in which the number of relations are not equal to the number of variables and many of the relations are in the form of inequalities (i.e., or ) to maximize (or minimize) a linear function of the variables subject to such conditions. Now the question is how one can find a solution for such problems? To answer this questions, let us consider the system of equations and inequations (or inequalities). We know that x represents a region lying towards the right of y axis including the y axis. y x x x y Fig Similarly, the region represented by y, lies above the x axis including the x axis. y x y 0 x y Fig. 59.2

3 Linear Programming Problem: A Geometric Approach:: 19 The question arises what region will be represented by x and y simultaneously. bviously, the region given by x, y will consist of those points which are common to both x and y. It is the first quadrant of the plane. y x, y x x y Fig Next, we consider the graph of the equation x + 2y 8. For this, first we draw the line x + 2y = 8 and then find the region satisfying x + 2y 8. Usually we choose x = 0 and calculate the corresponding value of y and choose y = 0 and calculate the corresponding value of x to obtain two sets of values (This method fails, if the line is parallel to either of the axes or passes through the origin. In that case, we choose any arbitrary value for x and choose y so as to satisfy the equation). Plotting the points (0,4) and (8,0) and joining them by a straight line, we obtain the graph of the line as given in the figure 59.4 below. y B(0,4) x + 2y = 8 x A(8,0) x y Fig. 59.4

4 20 :: Mathematics We have already seen that x and y represents the first quadrant. The graph given by x + 2y 8 lies towards that side of the line x + 2y = 8 in which the origin is situated because any point in this region will satisfy the inequality. Hence the shaded region in the figure (59.5) represents x, y and x + 2y 8 simultaneously. y (0,4) x + 2y = 8 (8,0) x x y Similarly, if we have to consider the regions bounded by x, y and x + 2y 8, then it will lie in the first quadrant and on that side of the line x + 2y = 8 in which the origin is not located. The graph is shown by the shaded region, in figure (59.6) y Fig (0,4) x + 2y = 8 x (8,0) x y Fig The shaded region in which all the given constraints are satisfied is called the feasible region Feasible Solution A set of values of the variables of a linear programming problem

5 Linear Programming Problem: A Geometric Approach:: 21 which satisfies the set of constraints and the non negative restrictions is called a feasible solution of the problem ptimal Solution A feasible solution of a linear programming problem which optimizes its objective function is called the optimal solution of the problem. Note : If none of the feasible solutions maximize (or minimize) the objective function, or if there are no feasible solutions, then the linear programming problem has no solution. In order to find a graphical solution of the linear programming problem following steps be employe: Step 1: Step 2: Step 3: Steps 4: Step 5: Step 6: Formulate the linear programming problem. Graph the constraints inequalities (by the method discussed above) Identify the feasible region which satisfies all the constraints simultaneously. For less than or equal to constraints the region is generally below the lines and for greater than or equal to constraints, the region is above the lines. Locate the solution points on the feasible region. These points always occur at the vertex of the feasible region. Evaluate the objective function at each of the vertex (corner point) Identify the optimum value of the objective function. Example A : Minimize the quantity z = Solution : The objective function to be minimized is

6 22 :: Mathematics z = First of all we draw the graphs of these inequalities, which is as follows: B (0,1) x1 =1 As we have discussed earlier that the region satisfied by and is the first quadrant and the region satisfied by the line + along with will be on that side of the line + =1 in which the origin is not located. Hence, the shaded region is our feasible solution because every point in this region satisfies all the constraints. Now, we have to find optimal solution. The vertex of the feasible region are A(1,0) and B(0,1). The value of z at A=1 The value of z at B = 2 Take any other point in the feasible region say (1,1), (2,0), (0,2) etc. We see that the value of z is minimum at A(1,0). Example B : Fig Minimize the quantity z = A(1,0)

7 Linear Programming Problem: A Geometric Approach:: 23 Solution : The objective function to be minimized is z = First of all we draw the graphs of these inequalities (as discussed earlier) which is as follows: D E(0,1) ( 03, 4 ) C (½,½) 3, 0 2 x1 =1 A(1,0) 2 +4 =3 B ( 3, 0 2 ) Fig The shaded region is the feasible region. Every point in the region satisfies all the mathematical inequalities and hence the feasible solution. Now, we have to find the optimal solution. The value of z at B is 3 2 The value of z at C (½, ½) is The value of z at E (0,1) is 2 If we take any point on the line 2 +4 =3 between B and C 3 we will get and elsewhere in the feasible region greater than 2. f course, the reason any feasible point (between B and C) on 2 +4 =3 minimizes the objective function (equation) z = + 2 is that the two lines are parallel (both have slope

8 24 :: Mathematics 1 ). Thus this linear programming problem has infinitely many 2 solutions and two of them occur at the vertices. Example C : Maximize z = Solution : The objective function is to maximize z = First of all we draw the graphs of these inequalities, which is as follows D(0,240) C (0,150) B(90,105) A(160,0) 3 +2 =480 E (300,0) +2 =300 Fig The shaded region ABC is the feasible region. Every point in the region satisfies all the mathematical inequations and hence the feasible solutions.

9 Now, we have to find the optimal solution. The value of z at A(160, 0) is The value of z at B (90, 105) is The value of z at C (0, 150) is The value of z at (0, 0) is 0. Linear Programming Problem: A Geometric Approach:: 25 If we take any other value from the feasible region say (60, 120), (80,80) etc. we see that still the maximum value is obtained at the vertex B (90, 105) of the feasible region. Note: For any linear programming problem that has a solution, the following general rule is true. If a linear programming problem has a solution it is located at a vertex of the set of feasible solutions; if a linear programming problem has multiple solutions, at least one of them is located at a vertex of the set of feasible solutions. In either case, the value of the objective function is unique. Check point 1 : Example D : Maximize z = In a small scale industry a manufacturer produces two types of book cases. The first type of book case requires 3 hours on machine A and 2 hours on machines B for completion, whereas the second type of book case requires 3 hours on machine A and 3 hours on machine B. The machine A can run at the most for 18 hours while the machine B for at the most 14 hours per day. He earns a profit of Rs.30/ on each book case of the first type and R.40/ on each book case of the second type. How many book cases of each type should he make each day so as to have a maximum profit?

10 26 :: Mathematics Solution : Let be the number of first type book cases and be the number of second type book cases that the manufacturer will produce each day. Since and are the number of book cases so (1) Since the 1 st type of book case requires 3 hours on machine A, therefore, book cases of first type will require 3 hours on machine A. 2 nd type of book case also requires 3 hours on machine A, therefore, book cases of 2 nd type will require 3 hours on machine A. But the working capacity of machine A is at most 18 hours per day, so we have or + 6 (2) Similarly, on the machine B, 1 st type of book case takes 2 hours and second type of book case takes 3 hours for completion and the machine has the working capacity of 14 hours per day, so we have (3) Profit per day is given by z = (4) Now, we have to determine and such that Maximize z = (objective function) subject to the conditions } constraints We use the graphical method to find the solution of the problem. First of all we draw the graphs of these inequalities, which is as follows:

11 Linear Programming Problem: A Geometric Approach:: 27 (0,6) C B(4,2) (0,0) A(6,0) Fig =14 =6 The shaded region ABC is the feasible region. Every point in the region satisfies all the mathematical inequations and hence known as feasible solution. We know that the optimal solution will be obtained at the vertices (0,0), A(6,0), B(4,2). Since the co ordinates of C are not integers so we don t consider this point. Co ordinates of B are calculated as the intersection of the two lines. Now the profit at is zero. Profit at A = = 180 Profit at B = = = 200 Thus the small scale manufacturer gains the maximum profit of Rs.200/ if he prepares 4 first type book cases and 2 second type book cases. Example E : Solve Example B of lesson 58 by graphical method. Solution : From Example B of lesson 58, we have (minimize) z = (objective function) subject to the conditions

12 28 :: Mathematics } 16 (Constraints) Fist we observe that any point (, ) lying in the first quadrant clearly satisfies the constraints, and, Now, we plot the bounding lines or 3 + = 24 x1 x2 + = B (0,24) Any point on or above the line 3 =24 satisfies the constraint 3 24 (Fig ) A(8,0) Fig Similarly any point on or above the line =16 satisfies the constraint 16 (Fig ) B (0,16) A(16,0) Fig Again any point on or above the line +3 =24 satisfies the constraints (Fig ) B(0,8) Fig A(24,0)

13 Thus combining all the above figures we get Linear Programming Problem: A Geometric Approach:: 29 D (0,24) (0,16) C (4,12) (0,8) B(12,4) (8,0) A(24,0) (16,0) 3 =24 =16 +3 =24 The region which is the common shaded area unbounded above. Now the minimum value of Fig z = is at one of the points A(24,0), B(12,4) C(4,12) and D(0,24) At A : z = = 144,000 At B : z = = 88,000 At C : z = = 72,000 At D : z = = 96,000 Thus, we see that z is minimum at C (4,12) where =4 and = 12. Hence, for minimum cost the firm should run plant P for 4 days and plant Q for 12 days. The minimum cost will be Rs.72, Check point 2 : Solve Example C of lesson 58 by graphical method. Example F : Maximize the quantity

14 30 :: Mathematics z = +2 1, Solution : First we graph the constraints 1, B (0,1) The shaded portion is the set of feasible solution. Now, we have to maximize the objective function. The value of z at A(1,0) is 1. The value of z at B(0,1) is 2. If we take the value of z at any other point from the feasible region, say (1,1) or (2,3) or (5,4) etc., then we notice that every time we can find another point which gives the larger value than the previous one. Hence, there is no feasible point that will make z largest. Since there is no feasible point that makes z largest, we conclude that this linear programming problem has no solution. Example G : A (1,0) Fig Solve the following problem graphically. =1 Minimize z =

15 Linear Programming Problem: A Geometric Approach:: 31 Solution : First we graph the constraints, 5 5, or 0, 5 5 (0,1) A +5 =5 feasible region Fig The shaded region is the feasible region. Here, we see that the feasible region is unbounded from one side. But it is clear from the figure (59.16) that the objective ( 5, 5 function attains its minimum value at the point A which is the 4 4 ) 4 point of intersection of the two lines =0 and +5 =5. Solving these we get = = Hence, z is minimum when =, =, and its minimum value is 2 10 = 10. Note: If we want to find max. z with these constraints then it is not possible in this case because the feasible region is unbounded from one side. INTEXT QUESTINS Minimize z = subject to the conditions

16 32 :: Mathematics 2. Maximize z = subject to the conditions , , 3. Solve 1(a) of the Intext Questions Solve 1(b) of the Intext Questions 58.1 WHAT YU HAVE LEARNT A set of values of the variables of a linear programming problem which satisfies the set of constraints and the non negative restrictions is called a feasible solution. A feasible solution of a linear programming problem which optimizes its objective function is called the optimal solution of the problem. If none of the feasible solutions maximizes (or minimizes) the objective function, or if there are no feasible solutions, then the linear programming problem has no solutions. If a linear programming problem has a solution, it is located at a vertex of the set of feasible solution. If a linear programming problem has multiple solutions, at least one of them is located at a vertex of the set of feasible solutions. But in all the cases the value of the objective function remains the same. TERMINAL QUESTINS 1. A dealer has Rs.1500/ only for the purchase of rice and wheat. A bag of rice costs Rs.180/ and a bag of wheat costs Rs.120/-. He has a storage capacity of ten bags only and the dealer gets a profit of Rs.11/ and Rs.8/ per bag of rice and wheat respectively. How he should spend his money in order to get maximum profit? 2. Solve the following problem : Max. z =

17 Linear Programming Problem: A Geometric Approach:: Solve the following problem : Maximize z = A machine producing either product A and B can produce A by using 2 units of chemicals and 1 unit of a compound and can produce B by using 1 unit of chemicals and 2 units of the compound. nly 800 units of chemicals and 1000 units of the compound are available. The profits available per unit of A and B are respectively Rs.30 and Rs.20. Find the optimum allocation of units between A and B to maximize the total profit. Find the maximum profit. 5. Maximize z = , ANSWERS T CHECK PINTS Check point 1: (0,40) C (0,30) B (20,20) feasible region Fig Max z = 140 at B(20,20) A (40,0) (60,0) + 2 =60 + =40

18 feasible region 34 :: Mathematics Check point 2 : (0,600) feasible region C (0,400) B (200,200) ANSWERS T INTEXT QUESTINS (400,0) A(300,0) =400 Fig =600 Max z = 1200 at C (0,400) D(0,24) (0,16) C(4,12) (0,8) B(12,4) (8,0) (16,0) A(24,0) 3 =24 =16 +3 =24 Fig Min. z =720 at C(4,12), =4, =12

19 Linear Programming Problem: A Geometric Approach:: (0,25) Feasible region D (0,10) C (3,9) B A(10,0) 5 +2 =50 Fig =12 Max. z = 330 at C(3,9), = 3, = 9 +3 = (, ) C(0,60) B(10,50) Feasible region A(20,0) 5 =100 =60 Fig Max. z = 1250 at B(10,50), = 10, = 50

20 36 :: Mathematics 4. B (0, ) Feasible region 2 +3 =8 Fig A(10,0) 4 +3 =40 Min. z = 40,000 at A(10,0), =10, =0 ANSWERS T TERMINAL QUESTINS 1. ( 0, 25 2) C (0,10) Feasible region B(5,5) ( ) (10,0) A25, 0 x 3 1 = =50 Fig Max z = 95 at B(5,5), = = 5

21 Linear Programming Problem: A Geometric Approach:: ( 0, 25 3) C(0,5) Feasible region B(4,3) (10,0) A 25, 0 4 Fig =50 +2 =10 Max z = 160 at B(4,3), = 4 = 3 3. (0,1100) (0,700) D (0,600) C B Feasible region =9600 A (900,0) =8400 Fig =9900 A (900,0) D(0,600) B (626, 335) & (0, 0) C(480, 420) Max z=8984 at B(626, 335) = 626, = 335

22 38 :: Mathematics 4. Max. z = (0,800) C(0,500) B(200,400) Feasible region ABC A(400,0) (1000,0) 2 = =1000 Fig Max z = at B (200,400). =200, = (0,24) C(0,20) B(8,12) Feasible region ABC A(16,0) (20,0) =20 Fig =5760 Max z = 392 at B(8,12). = 8, = 12

The Graphical Method: An Example

The Graphical Method: An Example The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

More information

Chapter 3: Section 3-3 Solutions of Linear Programming Problems

Chapter 3: Section 3-3 Solutions of Linear Programming Problems Chapter 3: Section 3-3 Solutions of Linear Programming Problems D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming

More information

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

EdExcel Decision Mathematics 1

EdExcel Decision Mathematics 1 EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation

More information

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach

Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we

More information

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max

3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,

More information

Linear Programming. Solving LP Models Using MS Excel, 18

Linear Programming. Solving LP Models Using MS Excel, 18 SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

More information

All points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates?

All points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates? Classwork Example 1: Extending the Axes Beyond Zero The point below represents zero on the number line. Draw a number line to the right starting at zero. Then, follow directions as provided by the teacher.

More information

Chapter 8 Graphs and Functions:

Chapter 8 Graphs and Functions: Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes

More information

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved.

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved. 7.5 SYSTEMS OF INEQUALITIES Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities

More information

1. Briefly explain what an indifference curve is and how it can be graphically derived.

1. Briefly explain what an indifference curve is and how it can be graphically derived. Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles

More information

LINEAR EQUATIONS IN TWO VARIABLES

LINEAR EQUATIONS IN TWO VARIABLES 66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

More information

Lesson 22: Solution Sets to Simultaneous Equations

Lesson 22: Solution Sets to Simultaneous Equations Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise

More information

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

Module1. x 1000. y 800.

Module1. x 1000. y 800. Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,

More information

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

More information

2. THE x-y PLANE 7 C7

2. THE x-y PLANE 7 C7 2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

More information

Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1

Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1 Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear

More information

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6 Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

More information

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

More information

Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

More information

Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )

Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd ) (Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion

More information

TIME VALUE OF MONEY PROBLEM #8: NET PRESENT VALUE Professor Peter Harris Mathematics by Sharon Petrushka

TIME VALUE OF MONEY PROBLEM #8: NET PRESENT VALUE Professor Peter Harris Mathematics by Sharon Petrushka TIME VALUE OF MONEY PROBLEM #8: NET PRESENT VALUE Professor Peter Harris Mathematics by Sharon Petrushka Introduction Creativity Unlimited Corporation is contemplating buying a machine for $100,000, which

More information

Question 2: How do you solve a linear programming problem with a graph?

Question 2: How do you solve a linear programming problem with a graph? Question 2: 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 information

Largest Fixed-Aspect, Axis-Aligned Rectangle

Largest Fixed-Aspect, Axis-Aligned Rectangle Largest Fixed-Aspect, Axis-Aligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: February 21, 2004 Last Modified: February

More information

Graph Ordered Pairs on a Coordinate Plane

Graph Ordered Pairs on a Coordinate Plane Graph Ordered Pairs on a Coordinate Plane Student Probe Plot the ordered pair (2, 5) on a coordinate grid. Plot the point the ordered pair (-2, 5) on a coordinate grid. Note: If the student correctly plots

More information

Special Situations in the Simplex Algorithm

Special Situations in the Simplex Algorithm Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

More information

Class 9 Coordinate Geometry

Class 9 Coordinate Geometry ID : in-9-coordinate-geometry [1] Class 9 Coordinate Geometry For more such worksheets visit www.edugain.com Answer t he quest ions (1) Find the coordinates of the point shown in the picture. (2) Find

More information

Math Review Large Print (18 point) Edition Chapter 2: Algebra

Math Review Large Print (18 point) Edition Chapter 2: Algebra GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter : Algebra Copyright 010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

Question 2: How will changes in the objective function s coefficients change the optimal solution?

Question 2: How will changes in the objective function s coefficients change the optimal solution? Question 2: How will changes in the objective function s coefficients change the optimal solution? In the previous question, we examined how changing the constants in the constraints changed the optimal

More information

What is Linear Programming?

What is Linear Programming? Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

More information

MathQuest: Linear Algebra. 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7

MathQuest: Linear Algebra. 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7 MathQuest: Linear Algebra Systems of Equations 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7 (a) x = 4 and y = 5 (b) x = 4 and y = 5 (c) x = 2 and y = 1 (d) x = 2 and

More information

Linear functions Increasing Linear Functions. Decreasing Linear Functions

Linear functions Increasing Linear Functions. Decreasing Linear Functions 3.5 Increasing, Decreasing, Max, and Min So far we have been describing graphs using quantitative information. That s just a fancy way to say that we ve been using numbers. Specifically, we have described

More information

Chapter 3 Consumer Behavior

Chapter 3 Consumer Behavior Chapter 3 Consumer Behavior Read Pindyck and Rubinfeld (2013), Chapter 3 Microeconomics, 8 h Edition by R.S. Pindyck and D.L. Rubinfeld Adapted by Chairat Aemkulwat for Econ I: 2900111 1/29/2015 CHAPTER

More information

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

Week 1: Functions and Equations

Week 1: Functions and Equations Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

More information

MATHEMATICS CLASS - XII BLUE PRINT - II. (1 Mark) (4 Marks) (6 Marks)

MATHEMATICS CLASS - XII BLUE PRINT - II. (1 Mark) (4 Marks) (6 Marks) BLUE PRINT - II MATHEMATICS CLASS - XII S.No. Topic VSA SA LA TOTAL ( Mark) (4 Marks) (6 Marks). (a) Relations and Functions 4 () 6 () 0 () (b) Inverse trigonometric Functions. (a) Matrices Determinants

More information

The Cartesian Plane The Cartesian Plane. Performance Criteria 3. Pre-Test 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13

The Cartesian Plane The Cartesian Plane. Performance Criteria 3. Pre-Test 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13 6 The Cartesian Plane The Cartesian Plane Performance Criteria 3 Pre-Test 5 Coordinates 7 Graphs of linear functions 9 The gradient of a line 13 Linear equations 19 Empirical Data 24 Lines of best fit

More information

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts: Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b. PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

More information

SPIRIT 2.0 Lesson: A Point Of Intersection

SPIRIT 2.0 Lesson: A Point Of Intersection SPIRIT 2.0 Lesson: A Point Of Intersection ================================Lesson Header============================= Lesson Title: A Point of Intersection Draft Date: 6/17/08 1st Author (Writer): Jenn

More information

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

More information

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Unit 1 Equations, Inequalities, Functions

Unit 1 Equations, Inequalities, Functions Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious

More information

Quadratic Modeling Business 10 Profits

Quadratic Modeling Business 10 Profits Quadratic Modeling Business 10 Profits In this activity, we are going to look at modeling business profits. We will allow q to represent the number of items manufactured and assume that all items that

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Chapter 2: Systems of Linear Equations and Matrices:

Chapter 2: Systems of Linear Equations and Matrices: At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

More information

SECTION 8-1 Systems of Linear Equations and Augmented Matrices

SECTION 8-1 Systems of Linear Equations and Augmented Matrices 86 8 Systems of Equations and Inequalities In this chapter we move from the standard methods of solving two linear equations with two variables to a method that can be used to solve linear systems with

More information

Lecture 2: August 29. Linear Programming (part I)

Lecture 2: August 29. Linear Programming (part I) 10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

More information

CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

More information

Sect The Slope-Intercept Form

Sect The Slope-Intercept Form Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

More information

Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

More information

UNIT 1 LINEAR PROGRAMMING

UNIT 1 LINEAR PROGRAMMING OUTLINE Session : Session 2: Session 3: Session 4: Session 5: Session 6: Session 7: Session 8: Session 9: Session 0: Session : Session 2: UNIT LINEAR PROGRAMMING Introduction What is Linear Programming

More information

Lines That Pass Through Regions

Lines That Pass Through Regions : Student Outcomes Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe

More information

The application of linear programming to management accounting

The application of linear programming to management accounting The application of linear programming to management accounting Solutions to Chapter 26 questions Question 26.16 (a) M F Contribution per unit 96 110 Litres of material P required 8 10 Contribution per

More information

Elements of a graph. Click on the links below to jump directly to the relevant section

Elements of a graph. Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

More information

1. Graphing Linear Inequalities

1. Graphing Linear Inequalities Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means

More information

Linear Programming Supplement E

Linear Programming Supplement E Linear Programming Supplement E Linear Programming Linear programming: A technique that is useful for allocating scarce resources among competing demands. Objective function: An expression in linear programming

More information

2.4 Motion and Integrals

2.4 Motion and Integrals 2 KINEMATICS 2.4 Motion and Integrals Name: 2.4 Motion and Integrals In the previous activity, you have seen that you can find instantaneous velocity by taking the time derivative of the position, and

More information

n 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1

n 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1 . Calculate the sum of the series Answer: 3 4. n 2 + 4n + 3. The answer in decimal form (for the Blitz):, 75. Solution. n 2 + 4n + 3 = (n + )(n + 3) = (n + 3) (n + ) = 2 (n + )(n + 3) ( 2 n + ) = m ( n

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood

Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 $3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

More information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

Algebra 1 Topic 8: Solving linear equations and inequalities Student Activity Sheet 1; use with Overview

Algebra 1 Topic 8: Solving linear equations and inequalities Student Activity Sheet 1; use with Overview Algebra 1 Topic 8: Student Activity Sheet 1; use with Overview 1. A car rental company charges $29.95 plus 16 cents per mile for each mile driven. The cost in dollars of renting a car, r, is a function

More information

CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

More information

Elasticity. I. What is Elasticity?

Elasticity. I. What is Elasticity? Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in

More information

Using Linear Programming in Real-Life Problems

Using Linear Programming in Real-Life Problems Name Date A C T I V I T Y 4 Instructions Using Linear Programming in Real-Life Problems Mr. Edwards is going to bake some cookies for his algebra class. He will make two different kinds, oatmeal-raisin

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

More information

Linear Programming. March 14, 2014

Linear Programming. March 14, 2014 Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

More information

CHAPTER 1 Linear Equations

CHAPTER 1 Linear Equations CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or

More information

Exponentials. 1. Motivating Example

Exponentials. 1. Motivating Example 1 Exponentials 1. Motivating Example Suppose the population of monkeys on an island increases by 6% annually. Beginning with 455 animals in 2008, estimate the population for 2011 and for 2025. Each year

More information

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

More information

with functions, expressions and equations which follow in units 3 and 4.

with functions, expressions and equations which follow in units 3 and 4. Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model

More information

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

More information

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved. 1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

More information

Basic Components of an LP:

Basic Components of an LP: 1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic

More information

Grade 8 Mathematics Item Specification C1 TD Task Model 3

Grade 8 Mathematics Item Specification C1 TD Task Model 3 Task Model 3 Equation/Numeric DOK Level 1 algebraically, example, have no solution because 6. 3. The student estimates solutions by graphing systems of two linear equations in two variables. Prompt Features:

More information

c sigma & CEMTL

c sigma & CEMTL c sigma & CEMTL Foreword The Regional Centre for Excellence in Mathematics Teaching and Learning (CEMTL) is collaboration between the Shannon Consortium Partners: University of Limerick, Institute of Technology,

More information

Norwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction

Norwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction 1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate

More information

Let s explore the content and skills assessed by Heart of Algebra questions.

Let s explore the content and skills assessed by Heart of Algebra questions. Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

More information

Solving Linear Programs

Solving Linear Programs Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,

More information