Turn in your interims - PowerPoint Presentation

Slide1

Turn in your interimsSlide2

Unit 3

Linear Programming

Solving Systems of Equations with 3 Variables

Inverses & Determinants of Matrices

Cramer’s RuleSlide3

Linear Programming

What is it?

Technique that identifies the minimum or maximum value of a quantity

Objective function

Like the “parent function”

Constrains (restrictions)

Limits on the variables

Written as inequalities

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

Feasible region

Contains all of the points which satisfy the constraintsSlide4

Vertex Principle of Linear Programming

If there is a max or a min value of the linear objective function, it occurs at one or more vertices of the feasible regionSlide5

Testing Vertices

Find the values of x and y that maximize and minimize P?

What is the value

of P at each vertex?Slide6

1. Graph the constraints

2. Find coordinates of

each vertex

3. Evaluate P at each vertex

when x=4 and y=3 P has a max value of 18Slide7

Furniture Manufacturing

A 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 chairsSlide8
Slide9

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.

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

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?

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

Experienced

Teams

Training Teams

Total

#

of Teams

x

y

x+y

# of Rangers

2x

y

30

# of Trainees

0

2y

16

# of trees planted500x

200y

500x+200ySlide10

Ranger Problem

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

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?

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

15 experienced teams, 0 training teams

none

7500 trees

11 experienced teams, 8 training teams

7100 treesSlide11

Announcements

Homework due Wednesday

Unit 3 Test on Tuesday 10/8Slide12

Solving Systems of Equations with 3 Variables

We are going to focus on solving in two ways

Solving by Elimination

Solving by SubstitutionSlide13

Elimination

Ensure all variables in all equations are written in the same order

Steps:

Pair the equations to eliminate a variable (ex: y)

Write the two new equations as a system and solve for final two variables (ex: x and z)

Substitute values for x and z into an original equation and solve for y

Always write solutions as: (

x,y,z

)Slide14

ExampleSlide15

PracticeSlide16

Substitution

Choose one equation and solve for the variable

Substitute the expression for x into each of the other two equations

Write the two new equations as a system. Solve for y and x

Substitute the values for y and z into one of the original equations. Solve for xSlide17

ExampleSlide18

PracticeSlide19

Unit 4

Working with MatricesSlide20

Inverses and Determinates (2x2)

Square matrix

Same number of rows and columns

Identity Matrix (I)

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

Inverse Matrix

AA

-1

=I

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

To show they are inverses AB=ISlide21

Verifying Inverses for 2x2

A= B=

AB= =Slide22

Determinates for 2x2

Determinate of a 2x2 matrix is

ad-

bc

Symbols:

detA

Ex: Find the determinate of

= -3*-5-(4*2)

=15-8 =7Slide23

Inverse of a 2x2 Matrix

Let If

det

A≠0, then A has an inverse.

A

-1

=

If

det

A=0 then there is NOT a unique solutionSlide24

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

Since

det

M does not equal 0 an inverse exists!Slide25

Systems with Matrices

System of Equations Matrix equation

Coefficient matrix A

Variable matrix X

Constant matrix

BSlide26

Solving a System of Equations with Matrices

Write the system as a matrix equation

Find A

-1

Solve for the variable matrixSlide27

Practice Problems

P. 48 # 1, 4, 7, 11, 14,

17Slide28

p. 48

Check your answers!!

#

1

#

4

#

7

#11

det

=0 so no unique solution

#14

det

=-1

#

17

det

=-29Slide29

Determinates for 3x3

Determinate of a 3x3

On the calculator

Enter the matrix

2

nd

=> Matrix => MATH =>

det

( => Matrix => Choose the matrixSlide30

Verifying Inverses

Multiply the matrices to ensure result is I

If not then the two matrices are not inverses

A= B=

AB= =

AB=Slide31

Solving a System of Equations with Matrices

(4, -10, 1)Slide32

Practice Problems

2.

3

.

(5,-3)

(5,0,1)

(1,0,3)Slide33

Practice Solving Systems with Matrices

Suppose 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 th

e 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 almonds

Let y represent peanuts

Let z represent raisinsSlide34

Calculator How To!!

To input a matrix:

2

nd

, Matrix, Edit

Be sure to define the size of your matrix!!

To find the inverse of a matrix

2

nd

, Matrix, 1, x

-1, enterSlide35

Homework

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