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Unit 3Linear Programming

Solving Systems of Equations with 3 VariablesInverses & Determinants of Matrices

Cramer’s Rule

Linear ProgrammingWhat is it?

Technique that identifies the minimum or maximum value of a quantity

Objective functionLike the “parent function”

Constrains (restrictions)Limits on the variablesWritten as inequalities

What is the name of the region where our possible solutions lie?Feasible region

Contains all of the points which satisfy the constraints

Vertex Principle of Linear ProgrammingIf there is a max or a min value of the linear

objective function, it occurs at one or more vertices of the feasible region

Testing VerticesFind the values of x and y that maximize and

minimize P?

What is the value of P at each vertex?

yxP 23

0,0

7

32

3

yx

xy

xy

1. Graph the constraints

2. Find coordinates of each vertex3. Evaluate P at each vertex

when x=4 and y=3 P has a max value of 18

0,0

7

32

3

yx

xy

xy

0,0 0,2

7,0

3,4

yxP 23 0)0(2)0(3 P6)0(2)2(3 P14)7(2)0(3 P18)3(2)4(3 P

Furniture ManufacturingA furniture manufacturer can make from 30

to 60 tables a day and from 40 to 100 chairs a day. It can make at most 120 units in one day. The profit on a table is $150, and the profit on a chair is $65. How many tables and chairs should they make per day to maximize profit? How much is the maximum profit?

Define our variables:X: number of tables Y: number of chairs

10040

6030

120

y

x

yx

40,30 40,60

60,60

90,30

yxP 65150

7100406530150 P

10350906530150 P

12900606560150 P

11600406560150 P

Practice Problem Teams chosen from 30 forest rangers

and 16 trainees are planting trees. An experienced team consisting of two rangers can plant 500 trees per week. A training team consisting of one ranger and two trainees can plant 200 trees per week.

1. Write an objective function and constraints for a linear program that models the problem.

2. How many of each type of team should be formed to maximize the number of trees planted? How many trainees are used in this solution? How many trees are planted?

3. Find a solution that uses all the trainees. How many trees will be planted in this case?

Experienced Teams

Training

TeamsTotal

# of Teams x y x+y

# of Ranger

s2x y 30

# of Trainee

s0 2y 16

# of trees

planted500x 200y 500x+20

0y

Ranger Problem1. Write an objective function and constraints for a linear

program that models the problem.

2. How many of each type of team should be formed to maximize the number of trees planted? How many trainees are used in this solution? How many trees are planted?

3. Find a solution that uses all the trainees. How many trees will be planted in this case?

0

0

162

302

y

x

y

yx

yxP 200500

15 experienced teams, 0 training teams

none 7500 trees

11 experienced teams, 8 training teams

7100 trees

AnnouncementsHomework due Wednesday

Unit 3 Test on Tuesday 10/8

Solving Systems of Equations with 3 VariablesWe are going to focus on solving in two ways

Solving by EliminationSolving by Substitution

EliminationEnsure all variables in all equations are

written in the same orderSteps:1. Pair the equations to eliminate a variable

(ex: y)2. Write the two new equations as a system

and solve for final two variables (ex: x and z)3. Substitute values for x and z into an original

equation and solve for yAlways write solutions as: (x,y,z)

Example

1934

1532

433

zyx

zyx

zyx

1532

433

zyx

zyx

1532

1934

zyx

zyx

2,1,5

Practice

102

732

32

zyx

zyx

zyx

3,4,1

Substitution1. Choose one equation and solve for the

variable2. Substitute the expression for x into each of

the other two equations3. Write the two new equations as a system.

Solve for y and x4. Substitute the values for y and z into one of

the original equations. Solve for x

Example

1022

124

42

zyx

zyx

zyx 4,1,2

Practice

12

752

64

zyx

zyx

zyx

6,1,4

Unit 4Working with Matrices

Inverses and Determinates (2x2)Square matrix

Same number of rows and columnsIdentity Matrix (I)

Square matrix with 1’s along the main diagonal and 0’s everywhere else

Inverse MatrixAA-1=I

If B is the multiplicative inverse of A then A is the inverse of B

To show they are inverses AB=I

100

010

001

Verifying Inverses for 2x2A= B=

AB= =

21

32

21

32

21

32

21

32

10

01

4322

6634

Determinates for 2x2Determinate of a 2x2 matrix is ad-bc

Symbols: detA

Ex: Find the determinate of = -3*-5-(4*2) =15-8 =7

dc

ba

dc

ba

52

43

Inverse of a 2x2 Matrix

Let If det A≠0, then A has an inverse.

A-1=

dc

baA

ac

bd

bcadac

bd

A

1

det

1

If det A=0 then there is NOT a unique solution

Ex: Determine if the matrix has an inverse. Find the inverse if it exists.

45

22M

21085242det bcadM

Since det M does not equal 0 an inverse exists!

25

24

det

1

45

22

1

1

1

MM

M

152

12

25

24

2

11M

Systems with MatricesSystem of Equations Matrix equation

1453

52

yx

yx

14

5

53

21

y

x

Coefficient matrix A

Variable matrix X

Constant matrix B

Solving a System of Equations with Matrices1. Write the system as a matrix equation

2. Find A-1

3. Solve for the variable matrix

14

5

53

21

y

x

13

25

13

251

13

25

65

1

BAy

x 1

14

5

13

25

y

x

1

3

y

x

Practice ProblemsP. 48 # 1, 4, 7, 11, 14, 17

p. 48 Check your answers!!

12/1

11#1

21

31#4

8/18/1

6/16/1#7

#11 det=0 so no unique

solution#14 det=-1

#17 det=-29

Determinates for 3x3Determinate of a 3x3

On the calculatorEnter the matrix2nd => Matrix => MATH => det( => Matrix

=> Choose the matrix

)()( 123312231213132321

333

222

111

cbacbacbacbacbacba

cba

cba

cba

Verifying InversesMultiply the matrices to ensure result is I

If not then the two matrices are not inverses

A= B=

AB= =

351

202

143

010

101

010

351

202

143

010

101

010

050301050

000202000

040103040

545

000

444

AB=

Solving a System of Equations with Matrices

59

825

132

zyx

zyx

zyx

(4, -10, 1)

Practice Problems1.

2

053

yx

yx 2.

zyx

zy

zx

6

12

4 3.

93

454

4

zy

yx

zyx

(5,-3)(5,0,1) (1,0,3)

Practice Solving Systems with MatricesSuppose you want to fill nine 1-lb tins with a

snack mix. You plan to buy almonds for $2.45/lb, peanuts for $1.85/lb, and raisins for $.80/lb. You have $15 and want the mix to contain twice as much of the nuts as of the raisins by weight. How much of each ingredient should you buy?

Let x represent almondsLet y represent peanutsLet z represent raisins

zyx

zyx

zyx

2

158.85.145.2

9

Calculator How To!!To input a matrix:

2nd, Matrix, EditBe sure to define the size of your matrix!!

To find the inverse of a matrix2nd, Matrix, 1, x-1, enter

HomeworkP. 50 # 1, 2, 6, 9, 10, 11, 13, 14