7.1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

7.1

Solving Systems of Two Equations

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 2

What you’ll learn about

The Method of Substitution Solving Systems Graphically The Method of Elimination Applications

… and whyMany applications in business and science can be modeled using systems of equations.


Solution of a System

A solution of a system of two equations in two

variables is an ordered pair of real numbers that

is a solution of each equation. A system is solved when all of its solutions are found.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-WesleySlide 7- 4

Example Using the Substitution Method

Solve the system using the substitution method.

2 10

6 4 1

x y

x y


Example Solving a Nonlinear System by Substitution

Solve the system by substitution.

38

361

x y

xy


Example Solving a Nonlinear System Algebraically

2

Solve the system algebraically.

6

8

y x x

y x


Example Solving a Nonlinear System Graphically

2

Solve the system:

ln

8

y x

y x x


Example Using the Elimination Method

Solve the system using the elimination method.

5 3 21

3 2 5

x y

x y


Example Finding No Solution

Solve the system:

3 2 5

6 4 10

x y

x y


Example Finding Infinitely Many Solutions

Solve the system.

3 6 10

9 18 30

x y

x y


Example Solving Word Problems with Systems

Find the dimensions of a rectangular cornfield with a perimeter of 220 yd and an area of 3000 yd2.


Homework

Homework Assignment #9 Read Section 7.2 Page 575, Exercises: 1 – 65 (EOO)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

7.2

Matrix Algebra


Quick Review

The points (a) (1, 3) and (b) ( , ) are reflected across the given line.

Find the coordinates of the reflected points.

1. The -axis

2. The line

3. The line

Expand the expression,

4. sin( )

5. cos

x y

x

y x

y x

x y

( )x y


Quick Review Solutions

The points (a) (1, 3) and (b) ( , ) are reflected across the given line.

Find the coordinates of the r

(a) (1,3) (b) ( ,

eflected points.

1. The -axis

2. The line

3.

)

(a) ( 3,

Th

1) (b) ( , )

e

x yx

x x

x

y y

y

line

Expand the expression,

4.

(a) ( 3, 1) (b) ( , )

sin cos sin cossin( )

5. cos( cos cos sin s n) i

y x

x y y x

x y x

y

y

x

x y

x y


What you’ll learn about

Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications

… and whyMatrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.


Matrix

11 12 1

21 22 2

1 2

Let and be positive integers. An

(read " by matrix") is a rectangular array of

rows and columns of real numbers.

We also use the short

n

n

m m mn

m n

m n m

n

a a a

a a a

a a a

matrix

m × n

hand notation for this matrix.ija


Matrix Vocabulary

Each element, or entry, aij, of the matrix uses

double subscript notation. The row subscript is

the first subscript i, and the column subscript is

j. The element aij is in the ith row and the jth

column. In general, the order of an m × n

matrix is m×n.


Example Determining the Order of a Matrix

What is the order of the following matrix?

1 4 5

3 5 6


Matrix Addition and Matrix Subtraction

Let and both be matrices of order .

1. The is the matrix .

2. The is the matrix .

ij ij

ij ij

ij ij

A a B b m n

m n A B a b

m n A B a b

sum +

difference

A B

A B


Example Matrix Addition

1 2 3 2 3 4

4 5 6 5 6 7


Example Using Scalar Multiplication

1 2 3

34 5 6


The Zero Matrix

The matrix 0 [0] consisting entirely of zeros is

the .

m n zero matrix


Additive Inverse

Let be any matrix.

The matrix consisting of the additive

inverses of the entries of is the

because 0.

ij

ij

A a m n

m n B a

A

A B

additive inverse

of A


Matrix Multiplication

1 1 2 2

Let be any matrix and be any matrix.

The product is the matrix where

+ ... .

ij ij

ij

ij i j i j ir rj

A a m r B b r n

AB c m n

c a b a b a b


Example Matrix Multiplication

Find the product if possible.

1 01 2 3

and 2 1 0 1 1

0 1

AB

A B


Identity Matrix

The matrix with 1's on the main diagonal and 0's elsewhere

is the .

1 0 0 0

0 1 0 0

0 0 1 0

0

0 0 0 0 1

n

n

n n I

I

identity matrix of order

n n


Inverse of a Square Matrix

1

Let be an matrix. If there is a matrix

such that , then is the of .

We write .

ij

n

A a n n B

AB BA I B A

B A

inverse


Inverse of a 2 × 2 Matrix

11

If 0, then .

The number is the determinant of the 2 2 matrix

and is denoted det .

a b d bad bc

c d c aad bc

ad bc

a b a bA A ad cb

c d c d


Minors and Cofactors of an n × n Matrix

If is an matrix where 2, the minor corresponding

to the element is the determinant of the 1 1 matrix

obtained by deleting the row and column containing .

The cofactor correspondin

ij

ij

ij

A n n n M

a n n

a

g to is 1 .i j

ij ij ija A M


Determinant of a Square Matrix

Let be a matrix of order ( 2). The

determinant of , denoted by det or | | , is the

sum of the entries in any row or any column multiplied

by their respective cofactors. For example, exp

ijA a n n n

A A A

th1 1 2 2

anding

by the row gives det | | ... .i i i i in ini A A a A a A a A


Transpose of a Matrix

Let be a matrix of order . The transpose

of , denoted by is the matrix in which the rows in

become the columns in and the columns in

become the rows in or .

ij

T

T

T Tji

A a n m

A A

A A A

A A a


Example Using the Transpose of a Matrix

If pizza sizes are given by the matrix

Size Pers Sm Med Larg , pizza sales are given by

the matrix Sales 55 25 15 10 , and pizza prices are

given by the matrix Price $2.50 $3.50 $7.50 $11.50 ,

what are the total sa

les for the day?


Inverses of n × n Matrices

An n × n matrix A has an inverse if and only if

det A ≠ 0.


Example Finding Inverse Matrices

1 3Find the inverse matrix if possible.

2 5A


Properties of MatricesLet A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.1. Community propertyAddition: A + B = B + AMultiplication: Does not hold in general2. Associative propertyAddition: (A + B) + C = A + (B + C)Multiplication: (AB)C = A(BC)3. Identity propertyAddition: A + 0 = AMultiplication: A·In = In·A = A4. Inverse propertyAddition: A + (-A) = 0Multiplication: AA-1 = A-1A = In |A|≠05. Distributive propertyMultiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BCMultiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC


Reflecting Points About a Coordinate Axis

To reflect a point about the -axis, express the point as a

1 01 2 matrix and multiply by to obtain the 1 2

0 1

matrix of the reflected point.

To reflect a point about the -axis, express the point

x

y

as a

1 01 2 matrix and multiply by to obtain the 1 2

0 1

matrix of the reflected point.


Example Using Matrix Multiplication

46. A company has two factories, each manufacturing three products. The number

of products made in factory in one week is given by in the matrix

120 70

150 110 . If production i

80 160

iji j a

A

s increased by 10%, write the new production levels

as a matrix . How is related to ? B B A

7.1

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