Chapter 2 Linear Functions Section 2.1 Systems of Two Equations Section 2.2 Systems with Three Variables Section 2.3 Gauss-Jordan Method for General Systems

Chapter 2 Linear Functions Section 2.1 Systems of Two Equations Section 2.2 Systems with Three Variables Section 2.3 Gauss-Jordan Method for General Systems o

f Equations Section 2.4 Matrix Operations Section 2.5 Multiplication of Matrices Section 2.6 The Inverse of a Matrix Section 2.7 Leontief Input-Output Model in Economics Section 2.8 Linear Regression

2.1 Systems of Two Equations

Methods & Types of Systems

Solution by _________________ ________________ Method ________________ Method ________________ Systems

Systems with ___________________________________

________________Systems Systems that have ________________________________

Application: Supply-and-Demand Analysis

Example 3 waysSolve By Graphing

2 32 4

x yx y

Example Solve by Intersect2 3 1

Solve: 3

x y

x y

Inconsistent SystemsIt is not always the case that a system has exactly ____________. If the equations represent _________ lines, they will have no ___________ in common and a solution to the system __________. This is called an inconsistent system.

For example, solve the system

3 2 56 4 6

x yx y

6 4 106 4 6

0 0 160 16

x yx y

x y

___________________.-4

-2

0

2

4

-4 -2 2 4

Consistent or Not?2 5 1

Solve: 4 10 5

x y

x y

Systems with Many Solutions

When the equations in a system represent the same ________, the graphs will be _________ and every point on the line is a ________ to the given system.

For example, solve the system12 9 248 6 16

x yx y

24 18 4824 18 4

0 0

8x yx y

____________number of solutions possible.

In slope-intercept form

4 8

3 34 8

3 3

12 9 24

8 6 16

x y y x

x y y x

Same equations

Infinitely Many?

3 6 15Solve:

2 5

x y

x y

NUMBER OF SOLUTIONS OF A LINEAR SYSTEMCONCEPT

SUMMARY

y

x

y

x

_______________

______________

_____________________________

y

x

______________________________

Summary

A movie theater sells tickets for $8.00 each, with seniors receiving a discount of $2.00. One evening the theater took in $3580 in revenue. If x represents the number of tickets sold at $8.00 and y the number of tickets sold at the discounted price

of $6.00, write an equation that relates these variables.

Suppose we also know that 525 tickets were sold. Write another equation relating the variables x and y.

Application

Pet Products has two production lines, I and II. Line I can produce 5 tons of regular dog food per hour and 3 tons of premium per hour. Line II can produce 3 tons of regular dog food per hour and 6 tons of premium. How many hours of production should be scheduled in order to produce 360 tons of premium and 460 tons of regular dog food?

Application

Zest Fruit juices makes two kinds of fruit punch from apple juice and pineapple juice. The company has 142 gallons of apple juice and 108 gallons of pineapple juice. Each case of Golden Punch requires 4 gallons of apple juice and 3 gallons of pineapple juice. Each case of Light Punch requires 7 gallons of apple juice and 3 gallons of pineapple juice. How many cases of each punch should be made in order to use all the apple and pineapple juice?

Application

George invested $5000 in securities. Part of the money was invested at 8% and part at 9%. The total annual income was $415. How much did he invest at each rate?

Application

A gardener has two solutions of weedkiller and water. One is 5% weedkiller and the other is 15% weedkiller. The gardener needs 100 L of a solution that is 12% weedkiller. How much of each solution should she use?

Application

A plant supervisor must apportion her 40 hour work week between hours working on the assembly line and hours supervising the work of others. She is paid $12 per hour for working and $15 per hour supervising. If her earnings for a certain week are $504, how much time does she spend on each task?

Application

5 The band boosters are organizing a trip to a national competition for the 226-member marching band. A bus will hold 70 students and their instruments. A van will hold 8 students and their instruments. A bus costs $280 to rent for the trip. A van costs $70 to rent for the trip. The boosters have $980 to use for transportation. Write a system of equations whose solution is how many buses and vans should be rented. Solve the system.

Your turn Application

Supply:

Demand:

Equilibrium Price:

Supply and Demand

Suppose that the quantity supplied, S, and the quantity demanded, D, of cellular telephones each month are given by the following functions:

S(p) = 60p – 900 & D(p) = -15p + 2850

Where p is the price in dollars of the telephone.

a. Find the equilibrium price

•

b. Determine the prices for which quantity supplied is greater that quantity demanded.

•

c. Graph S(p) and D(p) and label the equilibrium price

Example Equilibrium Price

Graphic

Jim has $440 in his bank account and adds $14 dollars a week. At the same time, Rhoda $260 in her bank account and adds $18 a week. How long until they have the same amount? How much will Jim have?

a. Find the equilibrium amount

•

b. Determine the time from which Rhoda has more money.

•

•

•

c. Graph R(x) and J(x) and label the equilibrium price

Example Linear Function

Your TurnThe Bike Shop held an annual sale. The consumer price demand relationship is given by. D(p) = -2p + 179. The Bike Shop manufactures its own ten-speed bicycles the relationship between supply and price are given by S(p) = 1.5p + 53

a. Find the equilibrium amount

b. Determine the prices for which quantity supplied is greater that quantity demanded.

c. Graph S(p) and D(p) and label the equilibrium price

Homework Section 2.1

Pg 71-75 1-63 odd, 64, 68, 74

2.2 Systems of Three Variables

Elimination Method

Matrices

Matrices and Systems of Equations

Gauss-Jordan Method

Application

ExampleCutter was allowed to pick some books for his cousin’s birthday from an on-line store that had a $1 basket, $2 basket and a $3 basket. Based on the following information, determine how many books Cutter selected from each basket.

• He selected 5 books at a total cost of $10.

• Shipping costs were $2.00 for each $1 book and $1.00 for each $2 and $3 book.

• The total shipping cost was $6.00.

MatricesDEFINITIONA ____________ is a __________ array of _____________. The number in the array are called the elements of the matrix. The array is enclosed with _______________-.

row

nmrows

mnmmm

n

n

n

aaaa

a

a

a

aaa

aaa

aaa

A

321

3

2

1

333231

232221

131211

A matrix is a rectangular array of numbers. We subscript entries to tell

their location in the array

Matrices are

identified by their

size.

Graphic

MatricesThe ___________ of each element in a matrix is described by the ______ and _________ in which it lies.

0974

9852

7531

4212

44

MatricesAn array composed of a single ________ of numbers is called a __________ matrix.

An array composed of a singe ________ of numbers is called a column ____________.

20513

3

1

6

2

ExampleFor the matrix

Find the following:a) The (1,1) element (a11)

• b) The (2,5) element (a25)

• c) The location of –4

• d) The location of 0

•

2 6 5 1 0

1 7 6 4 4

9 5 8 3 2

A matrix that has the same number of rows as columns is called a ___________________.

main diagonal

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A

Def Square Matrix

174

242

3523

zyx

zyx

zyx

741

412

523

ACoefficient matrix

If you have a system of equations and just pick off the coefficients and put them in a matrix it is called a coefficient matrix.

Coefficient Matrix

174

242

3523

zyx

zyx

zyx If you take the coefficient matrix and then add a last column with the constants, it is called the augmented matrix. Often the constants are separated with a line.

1741

2412

3523#AAugmented matrix

Augmented Matrix

We are going to work with our augmented matrix to get it in a form that will tell us the solutions to the system of equations. The three things above are the only things we can do to the matrix but we can do them together (i.e. we can multiply a row by something and add it to another row).

Operations that can be performed without altering the solution set of a linear system

1. Interchange any two rows

2. Multiply every element in a row by a nonzero constant

3. Add elements of one row to corresponding elements of another row

ERO’s

#100

##10

###1

After we get the matrix to look like our goal, we put the variables back in and use back substitution to get the solutions.

We use elementary row operations to make the matrix look like the one below. The # signs just mean there can be any number here---we don’t care what.

Ref

Systems of EquationsMatrices can be used to represent systems of equations. Consider the following system of equations:

A coefficient matrix is formed by using the coefficients of the system.

1 2 3

1 2 3

1 2 3

2 53 5 2 11

2 1

x x xx x xx x x

2 1 1

3 5 2

1 2 1

An augmented matrix also includes the numbers on the right-hand side of the equation. It gives complete information about the system of equations.

2 1 1 5

3 5 2 11

1 2 1 1

Gauss-Jordan MethodA system of linear equations can be solved using the augmented matrix and row operations.

Row Operations1. Interchange two rows.2. Multiply or divide a row by a nonzero constant.3. Multiply a row by a constant and add it to or subtract it

from another row.

The technique used to reduce an augmented matrix to a simple matrix is called the Gauss-Jordan Method. It attempts to reduce the augmented matrix until there are 1’s in the diagonal locations and 0’s elsewhere (except the last column) so the solution to the system can easily be read from the matrix.

• Rowswap(Matrix, Row A, Row B)• Switches Row A and Row B

• Row+(Matrix, Row A, Row B)• Adds Row A to Row B and replaces Row B

• *Row(Value, Matrix, Row)• Multiplies Row by value and replaces the row

• *Row+(Value, Matrix, Row A, Row B)• Multiplies Row a by the value, adds the result

to Row B, and replaces Row B

#100

##10

###1

After we get the matrix to look like our goal, we put the variables back in and use back substitution to get the solutions.


Ref

#100

#010

#001

This method requires no back substitution. When you put the variables back in, you have the solutions.

To obtain reduced row echelon form, you continue to do more row operations to obtain the goal below.

Rref

ExampleSolve the system of equations 3 11

2 5 22x yx y

SOLUTIONSequence of Equivalent Systems of Equations

Corresponding Equivalent Augmented Matrices

3 112 5 22

x yx y

1 3 11

2 5 22

Original system

Multiply first equation by –2 and add to second:

3 1111 44

x yy

Get a 0 in the second row, first column by multiplying first row by –2 and adding to second row:

1 3 11

0 11 44

Example continued

Simplify the second equation by dividing by –11:

3 114

x yy

Simplify the second row by dividing by –11:

1 3 11

0 1 4

Eliminate y from the first equation by multiplying the second equation by –3 and adding it to the first:

14

xy

Get a 0 in the first row, second column by multiplying second row by –3 and adding to the first:

1 0 1

0 1 4

Read the solution from the augmented matrix. The first row gives x = –1, and the second row gives y = 4.

ExampleUse the Gauss-Jordan method to solve the system

ExampleUse the Gauss-Jordan method to solve the system

Example

Your TurnUse the Gauss-Jordan method to solve the system

Your TurnUse the Gauss-Jordan method to solve the system

Application

Application


Pg 90-95 1, 5, 9, 12, 13, 16-24 even, 25-28, 31-51 odd, 53, 54, 59, 62, 63, 65, 66, 75

2.3 Gauss-Jordan for General Systems

Reduced Echelon Form

Application – Augmented Matrices

Gauss-Jordan Method

General Systems of Equations

Reduced Echelon FormA matrix is in reduced echelon form if all the following are true:1. All rows consisting entirely of zeros are grouped at the

bottom of the matrix.2. The leftmost nonzero number in each row is 1. This

element is called the leading 1 of the row.3. The leading 1 of a row is to the right of the leading 1 of the

rows above.4. All entries above and below a leading 1 are zeros.

ExamplesThe following matrices _______ in reduced echelon form.

The following matrices ________ in reduced echelon form.

1 0 0 5 1 0 3 0 8

0 1 0 3 0 1 1 0 2

0 0 1 7 0 0 0 1 7

1 0 0 2 11 0 0 6

0 1 0 2 30 1 0 5

0 0 0 0 00 0 4 7

0 0 1 0 2

#100

##10

###1


Ref

#100

#010

#001

This method requires no _____________________.

To obtain reduced row echelon form, you continue to do more row operations to obtain the goal below.

Rref

ExampleFind the reduced echelon form of the matrix:

0 0 2 2 2

3 3 3 9 12

4 4 2 11 12

3 3 3 9 12

0 0 2 2 2

4 4 2 11 12

R1 R2

1 1 1 3 4

0 0 2 2 2

4 4 2 11 12

1

3R1 R1

1 1 1 3 4

0 0 1 1 1

0 0 2 1 4

1

2R2 R2

4R1 R3 R3

1 1 0 2 5

0 0 1 1 1

0 0 0 1 6

R2 R1 R1

2R2 R3 R3

2R3 R1 R1

R3 R2 R2

1 1 0 0 17

0 0 1 0 5

0 0 0 1 6

Summary1. ___________. At least one row has all zeros

in the coefficient portion of the matrix (the portion to the left of the vertical line) and a nonzero entry to the right of the vertical line.

1 0 0 3

0 1 0 2

0 0 0 5

2. _______________. The number of nonzero rows equals the number of variables in the system.

3. ____________________. The number of nonzero rows is less than the number of variables in the system.

1 0 5

0 1 2

1 0 1 2

0 1 2 4

0 0 0 0

1 0 0 2 3

0 1 0 5 2

0 0 1 3 4

2 2 4 8

2 2

5 2 2.

x y z

x y z

x y z

ExampleSolve the System:


3

5

2 4 4 1.

x y z

x y z

x y z


4 8 12 28

1 2 3 7

3 6 9 15

x y z

x y z

x y z


2 13

2 5 3 3

x y z

x y z


4 10

2 3 13

5 2 16

x y

x y

x y

Your TurnSolve the system

2 7

2 6 18

2 5

2 5 15 46

y z

x y z

x y z

x y z

3 4 6

2 5 6 11

x y z

x y z

Applications

62 , 65

A brokerage firm packaged blocks of common stocks, bonds, and preferred stocks into three different portfolios. They contained the following:

Portfolio I: 3 blocks of common stock, 2 blocks of bonds, and 1 block of preferred stock

Portfolio II: 1 block of common stock, 4 blocks of bonds, and 1 block of preferred stock

Portfolio III: 5 blocks of common stock, 10 blocks of bonds, and 3 blocks of preferred stock.

A customer wants to buy 50 blocks of common stock, 160 blocks of bonds, and 25 blocks of preferred stock. Show that it is impossible to fill this order with the portfolios described.

ApplicationsCelia had one hour to spend at the athletic club, where she will jog, play handball, and ride a bicycle. Jogging uses 13 calories per minute; handball, 11; and cycling, 7. She jogs twice as long as she rides the bicycle. How long should she participate in each of these activities in order to use 660 calories?


Pg 112-117 1, 9, 15, 21, 29, 32, 40, 44, 60, 62, 66, 67, 69, 73

2.4 Matrix Operations

Additional Uses of Matrices

Equal Matrices

Addition of Matrices

Scalar Multiplication

Matrix Operations_______________________Two matrices of the same size are _________ matrices if and only if their corresponding ____________ are equal. Matrices are the same size if they have the same ____________.

Equal MatricesEXAMPLE

Find the value of x such that3 4 3 9

2.1 7 2.1 7

x

SOLUTION

Matrix AdditionThe ____ of two matrices of the same size is obtained by _____ corresponding elements. If two matrices are ____ the same size, they cannot be added; we say that their sum _______________. _____________ is performed on matrices of the same size by subtracting corresponding elements.

Matrix AdditionEXAMPLEDetermine the sums A + B and B + C for the following matrices.

2 1 1 1 3 1 4 1

0 5 2 2 1 4 1 2A B C

SOLUTION

Matrix Addition

2 1 3 1 4 7

4 0 5 8 3 2

1 82 1 3

4 34 0 5

7 2

2 1 3 1 4 7

4 0 5 8 3 2

Your TurnFor the following matrices

Find

A + B A + C B - A

Scalar MultiplicationScalar multiplication is the operation of multiplying a matrix by a _______ (________). Each entry in the matrix is ___________ by the scalar.

EXAMPLEMultiply the following matrix by –3,

5 2 1

0 1 4

1 3 6

Work Page

1 2 2 0 2 3 3 1

0 1 3 1 2 1 0 4A B C

Your TurnFor the following matrices

Find

2A + B 3A - B -3C

Application

ApplicationUse a matrix to display the following information about students at City College.

645 freshmen had GPAs of 3.0 or higher. 982 freshmen had GPAs of less than 3.0.569 sophomores had GPAs of 3.0 or higher.722 sophomores had GPAs of less than 3.0.531 juniors had GPAs of 3.0 or higher.562 juniors had GPAs of less than 3.0.478.seniors had GPAs of 3.0 or higher.493 seniors had GPAs of less than 3.0

ApplicationThe Department of Veteran Affairs keeps records of surviving veterans, their surviving dependent children, and surviving spouses. The tables below show, as of July 1997 and May 2001, the number surviving for the Civil War, World War I, and World War II.

As of July 1997: As of May 2001:

Use matrices to represent the information in these tables and use a matrix operation to find the decrease, from 1997 to 2001, in each category.

Work Page

As of July 1997: As of May 2001:

HW 2.4

Pg 126-130 1-47 Odd, 49-57 every other odd

2.5 Matrix Multiplication

Dot Product

Matrix Multiplication

Identity Matrix

Row Operations using Multiplication

Multiplication of MatricesDOT PRODUCTThe __________ is defined only when the row and column matrices have the same number of entries. The general form of the dot product of a row and column is

1

21 2 1 1 2 2n n n

n

b

ba a a a b a b a b

b

The dot product of a row and column is a _________________.

Example

3

2 1 3 2

5

3

4 0 2 1 2

5

Finding the Product of Two Matrices

– 2 3 1 – 4 6 0

– 1 3– 2 4

Find AB if A = and B =

Use a similar procedure to write the other entries of the product.

Because A is a 3 X 2 matrix and B is a 2 X 2 matrix, the product AB is defined and is a 3 X 2 matrix.

To write the entry in the first row and first column of AB, multiply corresponding entries in the first row of A and the first column of B. Then add.

SOLUTION

(– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4)

(1)(– 1) + (– 4)(– 2) (1)(3) + (– 4)(4)

(6)(– 1) + (0)(– 2) (6)(3) + (0)(4)

3 X 2 2 X 2 3 X 2

A B AB

– 2 3

1 – 4

6 0

– 1 3

– 2 4


3 X 2 2 X 2 3 X 2

A B AB


(– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4)

(1)(– 1) + (– 4)(– 2) (1)(3) + (– 4)(4)

(6)(– 1) + (0)(– 2) (6)(3) + (0)(4)

– 2 3

1 – 4

6 0

– 1 3

– 2 4

3 X 2 2 X 2 3 X 2

A B AB


(– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4)

(1)(– 1) + (– 4)(– 2) (1)(3) + (– 4)(4)

(6)(– 1) + (0)(– 2) (6)(3) + (0)(4)

– 2 3

1 – 4

6 0

– 1 3

– 2 4

3 X 2 2 X 2 3 X 2

A B AB


– 4 6

7 – 13

– 6 18

(– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4)

(1)(– 1) + (– 4)(– 2) (1)(3) + (– 4)(4)

(6)(– 1) + (0)(– 2) (6)(3) + (0)(4)

Workpage

Example

7 12 -5

-19 0 2

3 2 02 1 3

2 1 23 0 2

5 3 1

is not defined.

3 2 02 1 3

2 1 23 0 2

5 3 1

ExampleFind the product AB of the two matrices given below:

1 3 4 5 and

2 1 1 6A B

SOLUTION

Identity Matrix The ___________________ of size n is the nxn square matrix

with all zeros except for ones down the upper-left-to-lower-right diagonal.

Here are the identity matrix of sizes 2 and 3:

Example

1 0 2 1 3

0 1 3 0 2

1 0 02 1 3

0 1 03 0 2

0 0 1

For all ___________________________________

Multiplication of MatricesGiven matrices A and B, to find AB = C (matrix multiplication):

1. Check the number of columns of A and the number of rows of B. If they are equal, the product is possible. If they are not equal, no product is possible.

2. Form all possible dot products using a row from A and a column from B. The dot product of row i with column j gives the entry for the (i,j) position in C.

3. The number of rows in C is the same as the number of rows in A. The number of columns in C is the same as the number of columns in B.

Note: It is not necessarily true that AB will equal BA. The order of multiplication does matter.

Application

Application

Two stores sell the exact same brand and style of a dresser, a night stand, and a bookcase. Matrix A gives the retail prices (in dollars) for the items. Matrix B gives the number of each item sold at each store in one month.

Calculate AB and interpret the entries of AB

Your Turn

HW 2.5

Pg 142-149 3-60 Every Third Problem, 63-94

2.6 The Inverse of a Matrix

Inverse of a Square Matrix

Matrix Equations

Using A-1 to Solve a System

Identity Matrix

For _____________ matrices, there exists a matrix (I) such that _______________ for all matrices A. The matrix, I, is called the identity matrix. An identity matrix has the _______dimensions of A with ones on the diagonals and zeros everywhere else.

__ __ __ 1 3 2 1 3 2 __ __ __ __ __ __

__ __ __ 5 2 1 5 2 1 __ __ __ __ __ __

__ __ __ 3 3 3 3 3 3 __ __ __ __ __ __

I A A I A

Inverse of a Matrix A

If A and B are square matrices such that ___________, then B is the __________ matrix of A. The inverse of A is denoted _______. If B is found so that AB = I, then a theorem from linear algebra states that BA = I, so it is sufficient to just check ______________. Only square matrices have _____________.

Example

2 5 4 1 2 1

1 4 3 and 5 8 2

1 3 2 7 11 3

A B

Determine if B is the inverse of A if

SOLUTION

Method to Find Inverses

The method to find the inverse of a square matrix is

1. To find the inverse of a matrix A, form an augmented matrix [A|I] by writing down the matrix A and then writing the identity matrix to the right of A.

2. Perform a sequence of row operations that reduces the A portion of this matrix to reduced echelon form.

3. If the A portion of the reduced echelon form is the identity matrix, then the matrix found in the I portion is A–1.

4. If the reduced echelon form produces a row in the A portion that is all zeros, then A has no inverse.

Example

Find the inverse of

1 3 2

2 4 2

1 2 1

A

SOLUTIONAdjoin I to A to obtain

__ __ __ __ __ __

__ __ __ __ __ __

__ __ __ __ __ __

1 3 2 1 0 0

0 2 2 2 1 0

0 1 3 1 0 1

Use row operations to get zeros in column 1.

Example continued

1

2

1 3 2 1 0 0

0 1 1 1 0

0 1 3 1 0 1

Divide row 2 by –2.

Use row operations to get zeros in the second column.

3

2

1

2

1

2

1 0 1 2 0

0 1 1 1 0

0 0 2 0 1

Divide row 3 by –2.

3

2

1

2

1 1

4 2

1 0 1 2 0

0 1 1 1 0

0 0 1 0

Example continued1724

3 1

4 2

1 1

4 2

1 0 0 2

0 1 0 1

0 0 1 0

Use row operations to get zeros in the third column.

Now that the left-hand portion of the augmented matrix has been reduced to the identity matrix, A–1 comes from the right-hand portion of the augmented matrix.

1

____ __

__ __ __

__ __ __

A

Find the Inverse

Your TurnFind the Inverse of each matrix

Writing Linear Systems as Matrix Equations

Consider the system

Let A = Let X = Let B =

Write the equation AX = B using the above matrices

2 1Solve the system of equations: 3 2

2 2 1

x y zy z

x y z

Your TurnSolve the following systems by writing them as a matrix equation

3 8

2 4

x y

x y

3 8

2 4

a b

a b

HW 2.6

Pg 161-165 2-50 Even, 67-69

2.7 Leontief Input-Output Model

The Leontief Input-Output Model

______________ analysis is used to analyze an economy in order to meet given ________________ and export _____________.



The economy is divided into a number of ________. Each industry produces a certain _________ using the outputs of other industries as ___________. This interdependence among the industries can be summarized in a matrix - an input-output matrix. There is one _____________ for each industry’s input requirements. The entries in the column reflect the _____________ of input required from each of the industries.



An economy is composed of three industries - coal, steel, and electricity. To make $1 of coal, it takes no coal, but $.02 of steel and $.01 of electricity; to make $1 of steel, it takes $.15 of coal, $.03 of steel, and $.08 of electricity; and to make $1 of electricity, it takes $.43 of coal, $.20 of steel, and $.05 of electricity. Consumer demand is projected to be $2 billion for coal, $1 billion for steel and $3 billion for electricity. Find the production levels that would meet the demand.

Input-Output Matrix

A typical input-output matrix looks like:

Industry 1 Industry 2 Industry 3

Industry 1

From .Industry 2

Industry 3

Input requirements of:

Each column gives the dollar values of the various inputs needed by an industry in order to produce $1 worth of output.


Coal Steel Electricity

Coal.

Steel

Electricity

Final DemandThe final demand on the economy is a column matrix with one entry for each industry indicating the amount of consumable output demanded from the industry not used by the other industries:

__________________

final demand __________________ .


Coal

Steel

Electricity

D

Leontief Input-Output ModelThe matrix equation for the Leontief input-output model that relates total production to the internal demands of the industries and to consumer demand is given by

_____________________

or the equivalent,

_____________________

where A is the input-output matrix giving information on internal demands, D represents consumer demands, and X represents the total goods produced.

The solution to (I – A)X = D is

________________________________________

Variable Definitions

X – AX = D

X: ___________________________________________

A: ___________________________________________

D: ___________________________________________

AX: __________________________________________

X = ___________________________________


ExampleAn input-output matrix for electricity and steel is

If the production capacity of electricity is $15 million and the production capacity for steel is $20 million, how much of each is consumed internally for capacity production?

0.25 0.20

0.50 0.20A

SOLUTION

Your turnA simplified economy consists of the three sectors Manufacturing, Energy, and Services has the input-output matrix

How many cents of energy are required to produce $1 worth of manufactured goods?

How many cents of energy are required to produce $1 worth of services?

Which sector of the economy requires the greatest amount of services in order to produce $1 worth of output?

Your turnA simplified economy consists of the three sectors Manufacturing, Energy, and Services has the input-output matrix

What is the dollar amount of the energy costs neededto produce 10 million dollars worth of goods from eachsector?

A conglomerate has three divisions, which produce computers, semiconductors, and business forms. For each $1 of output, the computer division needs $.02 worth of computers, $.20 worth of semiconductors, and $.10 worth of business forms. For each $1 of output, the semiconductor division needs $.02 worth of computers, $.01 worth of semiconductors, and $.02 worth of business forms. For each $1 of output, the business forms division requires $.10 worth of computers and $.01 worth of business forms. The conglomerate estimates the sales demand to be $300,000,000 for the computer division, $100,000,000 for the semiconductor division, and $200,000,000 for the business forms division. At what level should each division produce in order to satisfy this demand?

Suppose that the conglomerate of the previous example is faced with an increase of 50% in demand for computers, a doubling in demand for semiconductors, and a decrease of 50% in demand for business forms. At what levels should the various divisions produce in order to satisfy the new demand?

Suppose that the conglomerate experiences a doubling in the demand for business forms. At what levels should the computer and semiconductor divisions produce?

A multinational corporation does business in the United States, Canada, and England. Its branches in one country purchase goods from the branches in other countries according to the matrix

where the entries in the matrix represent proportions of total sales by the respective branch. The external sales by each of the offices are $800,000,000 for the U.S. branch, $300,000,000 for the Canadian branch, and $1,400,000,000 for the English branch. At what level should each of the branches produce in order to satisfy the total demand?

An economy consists of the three sectors agriculture, energy, and manufacturing. For each $1 worth of output, the agriculture sector requires $.08 worth of input from the agriculture sector, $.10 worth of input from the energy sector, and $.20 worth of input from the manufacturing sector. For each $1 worth of output, the energy sector requires $.15 worth of input from the agriculture sector, $.14 worth of input from the energy sector, and $.10 worth of input from the manufacturing sector. For each $1 worth of output, the manufacturing sector requires $.25 worth of input from the agriculture sector, $.12 worth of input from the energy sector, and $.05 worth of input from the manufacturing sector. At what level of output should each sector produce to meet a demand for $4 billion worth of agriculture, $3 billion worth of energy, and $2 billion worth of manufacturing?

Your TurnA town has a merchant, a baker, and a farmer. To produce $1 worth of output, the merchant requires $.30 worth of baked goods and $.40 worth of the farmer's products. To produce $1 worth of output, the baker requires $.50 worth of the merchant's goods, $.10 worth of his own goods, and $.30 worth of the farmer's goods. To produce $1 worth of output, the farmer requires $.30 worth of the merchant's goods, $.20 worth of baked goods, and $.30 worth of his own products. How much should the merchant, baker, and farmer produce to meet a demand for $20,000 worth of output from the merchant, $15,000 worth of output from the baker, and $18,000 worth of output from the farmer? What amounts would be consumed during production?

HW 2.7

Pg 175-179 1-21,26,27, 29

2.8 Linear Regression

Linear Regression

0.80.70.60.50.40.30.20.10.0

6

5

4

3

2

1

0

-1

x

y

The method of least squares finds the line that minimizes the distance from each data point to the regression line.

positive deviations

negative deviations

Section 2.8

2 2 21 2 .... nd d d d minimize

A plot of a set of data points is called a scatter plot. When the scatterplot resembles a straight line, a regression line y = mx + b can be computed from a system of two equations.

Consider the data points (1,2), (2,5), and (3, 11). Findthe straight line that provides the best linear regression line,to these data.

Your Turn

The table gives the U.S. per capita health care expenditures for several years.

1. Find the Linear Regression line for this data.2. Use the Linear Regression line to estimate the per capita health

care expenditures for the year 2005.3. Use the Linear Regression line to estimate when the per capita

health care expenditures will reach $7000.

The following table gives the percent of persons 25 years and over who have completed four or more years of college.

(a) Use the method of Linear Regression to obtain the straight line that best fits these data.

(b) Estimate the percent for the year 1993. (c) If the trend determined by the straight line in

part (a) continues, when will the percent reach 28%

Your TurnThe following table is an abbreviated life expectancy table for U.S. males.

1. Find the straight line that provides the Linear Regression line for these data.

2. Use the straight line of part (1) to estimate the life expectancy of a 30-year-old U.S. male.

3. Use the straight line of part (1) to estimate the life expectancy of a 50-year-old U.S. male.

Objective: Find a linear function and use the equation to make predictions

A scatter plot is a graph used to determine whether there is a relationship between paired data.

correlation

A B

Your Turn

The following table gives the crude male death rate for lung cancer in 1950 and the per capita consumption of cigarettes in 1930 in various countries.

1. Use the method of least-squares to obtain the straight line that best fits these data.

2. In 1930 the per capita cigarette consumption in Finland was 1100. Use the straight line found in part 1 to estimate the male lung cancer death rate in Finland in 1950.

These data were obtained from Smoking and Health, Report of the Advisory Committee to the Surgeon General of the Public Health Service, U.S. Department of Health, Education, and Welfare, Washington, D.C., Public Health Service Publication No. 1103, p. 176.

The accompanying table shows the 1999 price of a gallon (in U.S. dollars) of fuel and the average miles driven per automobile for several countries.

1. Find the straight line that provides the best least-squares fit to these data.

2. In 1999, the price of gas in Canada was $2.04 per gallon. Use the straight line of part 1 to estimate the average number of miles automobiles were driven in Canada.

U.S. Department of Transportation, Federal Highway Administration, Highway Statistics, 2000.

The table shows the 1999 price of a gallon (in U.S. dollars) of fuel and the average miles driven per automobile for several countries.

3. In 1999 the average miles driven in the United States was 11,868. Use the straight line of part 1 to estimate the 1999 price of a gallon of gasoline in the United States.

U.S. Department of Transportation, Federal Highway Administration, Highway Statistics, 2000.

The following table gives enrollment (in millions) in public colleges in the United States for certain years.

U.S. Dept. of Education, National Center for Education Statistics, Digest of Education Statistics, 2001.

1. Use the method of least squares to obtain the straight line that best fits these data.

2. Estimate the enrollment in 1988.3. If the trend determined by the straight line in part 1 continues, when

will the enrollment reach 13 million?

Two Harvard economists studied countries‘ relationships between the independence of banks and inflation rates from 1955 to 1990. The independence of banks was rated on a scale of -1.5 to 2.5, with -1.5, 0, and 2.5 corresponding to least, average, and most independence, respectively. The following table gives the values for various countries.

T. Bradford DeLong (Harvard) and L. H. Summers (World Bank).


2. What relationship between independence of banks and inflation is indicated by the least- squares line?

Two Harvard economists studied countries‘ relationships between the independence of banks and inflation rates from 1955 to 1990 The independence of banks was rated on a scale of -1.5 to 2.5, with -1.5, 0, and 2.5 corresponding to least, average, and most independence, respectively. The following table gives the values for various countries.

T. Bradford DeLong (Harvard) and L. H. Summers (World Bank).

3. Japan has a 0.6 independence rating. Use the least-squares line to estimate Japan's inflation rate.

4. The inflation rate for Britain is 6.8. Use the least-squares line to estimate Britain's independence rating.

Your TurnThe table gives the average price of a pound of potato chips in January of the given years. (Source: U.S. Bureau of Labor Statistics, Consumer Price Index.)


2. Estimate the average price of a pound of potato chips in January 1999

3. If this trend continues, when will the average price of a pound of potato chips be $3.63?

HW 2.8

Pg 186-188 1-19

More on Linear RegressionGiven the points (x1,y1), (x2,y2), …, (xn,yn), the augmented matrix M of the system

Am + Bb = CDm + Eb = F

whose solution gives the least squares line of best fit for the given points is the product

1 1

2 21 2

1

1

1 1 1 1

1

n

n n

x y

x yx x x A B CM

D E F

x y

ExampleConsider the scatterplot of the data in the table. Use matrices to find the regression line for this data.

60

65

70

75

80

85

90

95

100

1 2 3 4 5

x y

1 62.0

2 68.2

3 76.5

4 85.8

5 96.2

SOLUTION 1 1 62.0

2 1 68.21 2 3 4 5 55 15 1252.1

3 1 76.51 1 1 1 1 15 5 388.7

4 1 85.8

5 1 96.2

M

Solving 55 15 1252.1

15 5 388.7giv 8.6 51.94es y

m bm b

x

Documents

Chapter 2 Linear Functions Section 2.1 Systems of Two Equations Section 2.2 Systems with Three Variables Section 2.3 Gauss-Jordan Method for General Systems