Chapter 2 Linear Functions
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Section 2.1 Systems of Two Equations
Section 2.2 Systems with Three Variables
Section 2.3 Gauss-Jordan Method for General Systems
of 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


Systems with ___________________________________
Systems that have ________________________________
Application: Supply-and-Demand Analysis
Example 3 ways
Solve By Graphing
2x  y  3
x  2y  4
Example Solve by Intersect
2 x  3 y  1
Solve: 
 x  y  3
Inconsistent Systems
It 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.
4
For example, solve the system
3x  2 y  5
6 x  4 y  6
6 x  4 y  10
6 x  4 y  6
0 x  0 y  16
0  16
2
-4
-2
0
-2
-4
___________________.
2
4
Consistent or Not?
2 x  5 y  1
Solve: 
4 x  10 y  5
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 system
24 x  18 y  48
12 x  9 y  24
24 x  18 y  48
8 x  6 y  16
00
____________number of solutions possible.
In slope-intercept form
4
3
4
 x
3
12 x  9 y  24  y   x 
8 x  6 y  16  y
8
3
8
3
Same
equations
Infinitely Many?
3x  6 y  15
Solve: 
 x  2 y  5
CONCEPT
NUMBER OF SOLUTIONS OF A LINEAR SYSTEM
SUMMARY
y
y
x
_______________
______________
y
x
_______________
______________
x
_______________
_______________
Application
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.
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?
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?
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?
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?
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?
Your turn 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.
Supply and Demand

Supply:

Demand:

Equilibrium Price:
Example Equilibrium Price
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 Linear Function
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
Your Turn
The 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
Example
Cutter 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.
Matrices
DEFINITION
A ____________ is a __________ array of
_____________. The number in the array are called
the elements of the matrix. The array is enclosed
with _______________-.
A matrix is a rectangular array of
numbers. We subscript entries to tell
their location in the array
rows
a
a
a

11
12
13
row
a
a
a
21
22
23

A  a31 a32 a33


 
am1 am 2 am3
m n
 a1n 

 a2 n 
 a3n 

 
 amn 
Matrices
are
identified
by their
size.
Matrices
The ___________ of each element in a matrix is
described by the ______ and _________ in which
it lies.
 2  1  2 4
 1 3

5
7


 2 5  8 9 


7
9 0
4
4 4
Matrices
An array composed of a single ________ of
numbers is called a __________ matrix.
An array composed of a singe ________ of numbers
is called a column ____________.
3
1 5 0 2
 2
6 
 
1 
 
  3
Example
 2 6 5 1 0 
1 7 6 4 4 
For the matrix 

9 5 8 3 2 
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
•
A matrix that has the same number of rows as
columns is called a ___________________.
 a11 a12
a
a
21
22

A
a31 a32

a41 a42
a13
a23
a33
a43
a14 

a24 
a34 

a44 
3x  2 y  5 z  3
 2 x  y  4 z  2
x  4 y  7z  1
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.
 3 2 5 


1
4
Coefficient matrix A   2


 1
4  7
3x  2 y  5 z  3
 2 x  y  4 z  2
x  4 y  7z  1
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.
3
 3 2 5


#
1
4

2
Augmented matrix A   2


 1
4  7 1 
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
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).
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.
1
0

0
#
#
1
0
#
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.
Systems of Equations
Matrices can be used to represent
systems of equations. Consider the
following system of equations:
2 x1  x2  x3  5
3x1  5 x2  2 x3  11
x1  2 x2  x3  1
A coefficient matrix is formed
by using the coefficients of the
system.
 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 Method
A system of linear equations can be solved using the augmented
matrix and row operations.
Row Operations
1. 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
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.
1
0

0
#
#
1
0
#
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.
To obtain reduced row echelon form, you continue to do
more row operations to obtain the goal below.
1
0

0
0
1
0
0
0
1
#

#
#
This method requires no back substitution.
When you put the variables back in, you have
the solutions.
Example
Solve the system of equations
x  3 y  11
2 x  5 y  22
SOLUTION
Sequence of Equivalent
Systems of Equations
x  3 y  11
2 x  5 y  22
Multiply first equation
by –2 and add to second:
x  3 y  11
11y  44
Corresponding Equivalent
Augmented Matrices
Original
system
 1 3 11 
 2 5 22 


Get a 0 in the second row, first
column by multiplying first row
by –2 and adding to second row:
11 
1 3
0 11 44 


Example continued
Simplify the second
equation by dividing by –11:
x  3 y  11
y 4
Eliminate y from the first
equation by multiplying the
second equation by –3 and
adding it to the first:
x  1
y4
Simplify the second row by
dividing by –11:
1 3 11
0 1 4 


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.
Example
Use the Gauss-Jordan method to solve the system
Example
Use the Gauss-Jordan method to solve the system
Example
Your Turn
Use the Gauss-Jordan method to solve the system
Your Turn
Use the Gauss-Jordan method to solve the system
Application
Application
Homework Section 2.2
Pg 90-95 1, 5, 9, 12, 13, 16-24 even, 2528, 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 Form
A 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.
Examples
The following matrices _______ in reduced echelon form.
1 0 0 5 
0 1 0 3


0 0 1 7 
1 0 3 0 8 
0 1 1 0 2 


0 0 0 1 7 
The following matrices ________ in reduced echelon form.
1 0 0 6 
0 1 0 5 


0 0 4 7 
1
0

0

0
0
1
0
0
0
0
0
1
2
2
0
0
1
3 
0

2 
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.
1
0

0
#
#
1
0
#
1
#

#
#
To obtain reduced row echelon form, you continue to do
more row operations to obtain the goal below.
1
0

0
0
1
0
0
0
1
#

#
#
This method requires no _____________________.
Example
Find the reduced echelon form of the matrix:
 0 0 2 2 2 
 3 3 3 9 12 


 4 4 2 11 12 
R1  R2
 3 3 3 9 12 
 0 0 2 2 2 


 4 4 2 11 12 
1 1 1 3 4   1 
R2  R2

1
2 2 2   2 
  R1  R1  0 0
 3
 4 4 2 11 12  4R1  R3  R3
R2  R1  R1
2R2  R3  R3
1 1 1 3 4 
0 0 1 1 1 


0 0 2 1 4 
1 1 0 2 5 
0 0 1 1 1  2R3  R1  R1

 R3  R2  R2
0 0 0 1 6 
1 1 0 0 17 
0 0 1 0 5


0 0 0 1 6 
Summary
1. ___________. 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.
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 0 3 
0 1 0 2 


0 0 0 5 
1 0 5 
 0 1 2 


1 0 0 2 3 
0 1 0 5 2 


0 0 1 3 4 
1 0 1 2 
0 1 2 4 


0 0 0 0 
Example
2x  2 y  4z  8

Solve the System:  x  y  2 z  2
 x  5 y  2 z  2.

Example
Solve the System:
 x  y  z  3

 x  y  z  5
2 x  4 y  4 z  1.

Example
4 x  8 y  12 z  28

Solve the System: 1x  2 y  3z  7
3 x  6 y  9 z  15

Example
 x  2 y  z  13
Solve the System: 
2 x  5 y  3z  3
Example
 x  4 y  10

Solve the System: 2 x  3 y  13
5 x  2 y  16

Your Turn
Solve the system
 y  2z  7
 x  2 y  6 z  18


 x  y  2 z  5
2 x  5 y  15z  46
 x  3 y  4z  6

2 x  5 y  6 z  11
Applications
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.
62 , 65
Applications
Celia 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?
Homework Section 2.3
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 Matrices
EXAMPLE
 3 4x  3 9

Find the value of x such that 


2.1
7
2.1
7

 

SOLUTION
Matrix Addition
The ____ 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 Addition
EXAMPLE
Determine the sums A + B and B + C for the following matrices.
 2 1 1
A

0
5
2


SOLUTION
1 3 1 
B

2
1
4


 4 1
C


1
2


Matrix Addition
 2 1 3   1 4 7 
 4 0 5   8 3 2 

 

 2 1 3   1 4 7 
 4 0 5   8 3 2 

 

 1 8 
 2 1 3  

 4 0 5   4 3 

 7 2


Your Turn
For the following matrices
Find
A+B
A+C
B-A
Scalar Multiplication
Scalar multiplication is the operation of multiplying a
matrix by a _______ (________). Each entry in the
matrix is ___________ by the scalar.
 5 2 1
EXAMPLE


Multiply the following matrix by –3,  0 1 4 
 1 3 6 
1 2 2 
A

0

1
3


0 2 3
B

1
2
1


3 1 
C

0

4


Your Turn
For the following matrices
Find
2A + B
3A - B
-3C
Application
Application
Use 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
Application
The 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.
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 Matrices
DOT PRODUCT
The __________ 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
 a1
a2
 b1 
b 
an    2   a1b1  a2b2 
 
 
bn 
 anbn
The dot product of a row and column is a _________________.
Example
 3
2
2

1
3


 
 5 
 3
2
4
0
2

1


 
 5 
Finding the Product of Two Matrices
Find AB if
–2 3
A = 1 –4
6
0
and
B=
–1
–2
SOLUTION
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.
Use a similar procedure to write the other entries of the
product.
3
4
Finding the Product of Two Matrices
A

B

AB
3X2

2X2

3X2
–2
3
1
–4
6
0
–1
3
–2
4
(– 2)(– 1) + (3)(– 2)
(– 2)(3) + (3)(4)
(1)(–1) + (– 4)(–2)
(1)(3) + (– 4)(4)
(6)(– 1) + (0)(– 2)
(6)(3) + (0)(4)
Finding the Product of Two Matrices
A

B

AB
3X2

2X2

3X2
–2
3
1
–4
6
0
–1
3
–2
4
(– 2)(– 1) + (3)(– 2)
(– 2)(3) + (3)(4)
(1)(–1) + (– 4)(–2)
(1)(3) + (– 4)(4)
(6)(– 1) + (0)(– 2)
(6)(3) + (0)(4)
Finding the Product of Two Matrices
A

B

AB
3X2

2X2

3X2
–2
3
1
–4
6
0
–1
3
–2
4
(– 2)(– 1) + (3)(– 2)
(– 2)(3) + (3)(4)
(1)(–1) + (– 4)(–2)
(1)(3) + (– 4)(4)
(6)(– 1) + (0)(– 2)
(6)(3) + (0)(4)
Finding the Product of Two Matrices
A

B

AB
3X2

2X2

3X2
(– 2)(– 1) + (3)(– 2)
(– 2)(3) + (3)(4)
(1)(– 1) + (– 4)(– 2)
(1)(3) + (– 4)(4)
(6)(– 1) + (0)(– 2)
(6)(3) + (0)(4)

–4
6
7
– 13
–6
18
Example
 3 2 0 
 2 1 3  


2
1
2
 3 0 2 


  5 3 1


 7 12 -5 

2 
-19 0


 3 2 0 
 2 1 2   2 1 3  is not defined.

  3 0 2 

 5 3 1 
Example
Find the product AB of the two matrices given below:
1 3 
 4 5
A
and B  


2

1
1
6




SOLUTION
Identity Matrix


The ___________________ of size n is the nxn square matrix
with all zeros except for ones down the upper-left-to-lowerright diagonal.
Here are the identity matrix of sizes 2 and 3:
Example
1 0   2 1
0 1   3 0


1
 2 1 3  
 3 0 2  0

 0

3

2
0 0

1 0
0 1 
For all ___________________________________
Multiplication of Matrices
Given 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.
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
Determine if B is the inverse of A if
1
2 5 4 
 1 2
A  1 4 3  and B   5 8
2 
1 3 2 
 7 11 3
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
1 3 2 
Find the inverse of A   2 4 2 
1 2 1
 __ __ __ __ __ __ 
SOLUTION
 __ __ __ __ __ __ 
Adjoin I to A to obtain 

 __
Use row operations to
get zeros in column 1.
__
__
__
__
__ 
1 3 2 1 0 0 
0 2 2 2 1 0 


0 1 3 1 0 1 
Example continued
Divide row 2 by –2.
0 0
1 3 2 1


1
0 1 1 1  2 0 


0 1
 0 1 3 1
1
Use row operations to get 
zeros in the second column. 0

0
1

Divide row 3 by –2.
0


0
0
1
1 2
1
1
0 2
0
0 1 2
1
1
1
0
1
0
3
2
1

2
1

2
0

0 

1

3
2
1

2
1
4
0

0 
1
 
2
Example continued
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.
7
 1 0 0 2
4


3
0
1
0
1


4

1
0
0
1
0

4
 __

1
A   __
 __
1
 
2
1
2
1
 
2
__
__
__
__ 

__ 
__ 
Find the Inverse
Your Turn
Find 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
x  y  2z  1

Solve the system of equations:   y  3z  2
2 x  2 y  z  1
Your Turn
Solve the following systems by writing them as a
matrix equation
 3x 

2 x 
y  8
y  4
 3a  b  8

2a  b  4
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 _____________.
2.7 Leontief Input-Output
Model
The Leontief Input-Output Model
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.
2.7 Leontief Input-Output
Model
The Leontief Input-Output Model
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:
Input requirements of:
Industry 1 Industry 2 Industry 3
Industry 1 
From Industry 2 

Industry 3 



.




Each column gives the dollar values of the various inputs needed
by an industry in order to produce $1 worth of output.
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.
Coal Steel Electricity
Coal 
Steel 
Electricity 

.


Final Demand
The 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


 



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.



Electricity 
Coal
Steel

D


Leontief Input-Output Model
The 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 = ___________________________________
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.
Example
An input-output matrix for electricity and steel is
 0.25 0.20 
A

0.50
0.20


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?
SOLUTION
Your turn
A 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 turn
A 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 needed
to produce 10 million dollars worth of goods from each
sector?
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 Turn
A 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
Section 2.8
Linear Regression
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.
6
5
4
negative
deviations
3
y
The method of least
squares finds the line
that minimizes the
distance from each data
point to the regression
line.
positive
deviations
2
1
0
-1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
minimize
d  d12  d22  ....  dn2
Your Turn
Consider the data points (1,2), (2,5), and (3, 11). Find
the straight line that provides the best linear regression line,
to these data.
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 Turn
The following table is an abbreviated life
expectancy table for U.S. males.
1.
2.
3.
Find the straight line that provides the
Linear Regression line for these data.
Use the straight line of part (1) to estimate
the life expectancy of a 30-year-old U.S.
male.
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.
A
B
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.
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?
U.S. Dept. of Education, National Center for Education Statistics, Digest of Education Statistics, 2001.
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.
1. Use the method of least
squares to obtain the straight
line that best fits these data.
2. What relationship between
independence of banks and
inflation is indicated by the
least- squares line?
T. Bradford DeLong (Harvard) and L. H. Summers (World Bank).
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.
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.
T. Bradford DeLong (Harvard) and L. H. Summers (World Bank).
Your Turn
The 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.)
1. Use the method of least squares to obtain
the straight line that best fits these data.
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 Regression
Given the points (x1,y1), (x2,y2), …, (xn,yn), the augmented
matrix M of the system
Am + Bb = C
Dm + Eb = F
whose solution gives the least squares line of best fit for
the given points is the product
 x1 1 y1 
x 1 y 
x
x
x
 1 2
 A B C
2
n 2
M 





1
1
1
1
D
E
F






 xn 1 yn 
Example
Consider the scatterplot
of the data in the table.
Use matrices to find the
regression line for this
data.
x
y
1
62.0
2
68.2
100
95
90
85
80
3
76.5
4
85.8
70
5
96.2
60
75
65
1
1
2
1 2 3 4 5 
3
M 

1 1 1 1 1 
4
 5
SOLUTION
1
1
1
1
1
2
3
4
62.0 
68.2 
55 15 1252.1
76.5  

 15 5 388.7 
85.8 
96.2  Solving 55m  15b  1252.1
5
15m  5b  388.7
gives y  8.6 x  51.94