chapter 8: systems of equations and inequalities section 8 ... · chapter 8: systems of equations...

236 Chapter 8 Systems of Equations and Inequalities

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Chapter 8: Systems of Equations and Inequalities Section 8.2: Systems of Linear Equations: Matrices

Exploration 1*: Write the Augmented Matrix of a System of Linear Equations A streamlined version of the elimination method is a matrix. Definition: A matrix is defined as a rectangular array of numbers,

Each of the values in the matrix is called an __________, and each has a subscript. The subscript helps to locate the __________ and _______________. For example, 35a indicates:

_________________________________________________________________________. A matrix used to represent a system of linear equations is called an ___________________. Example 1*: Write the Augmented Matrix of a System of Linear Equations Write the augmented matrix of each system of equations.

(a) 3 2 3

2 2

x yx y− =

− + = −

(b)

3 2 5 0

2 4 2 0

4 7 0

x yx z

x y z

− + =− + + = + − =

Example 2*: Write the System of Equations from the Augmented Matrix Write the system of linear equations corresponding to each augmented matrix.

(a) 2 1 31 1 2

− −

Section 8.2 Systems of Linear Equations: Matrices 237


(b) 3 2 5 32 1 4 2

1 4 7 1

− − − −

Exploration 2*: Perform Row Operations on a Matrix Row Operations on an augmented matrix are used to solve the corresponding system of equations. There are three basic row operations: Row Operations 1: Interchange any _____ rows. 2: Replace a row by _________________________ of that row. 3: Replace a row by the _____ of that row and a constant nonzero multiple of __________

Use words or an example to explain each of the above operations. 1. Interchange any _____ rows: 2. Replace a row by _________________________ of that row:

3. Replace a row by the _____ of that row and a constant nonzero multiple of ________:

Example 3*: Perform Row Operations on a Matrix

Apply the row operation 2 1 23R r r= − + to the augmented matrix: 1 2 23 5 9

− −

(a) What is meant by 2R ?

(b) What is meant by 1 23r r− + ?

(c) Apply the operation to get the augmented matrix:



Example 4*: Perform Row Operations on a Matrix

(a) Find a row operation that will result in the augmented matrix 1 2 3 72 5 1 03 6 2 5

− − − −

having a

zero in row 2, column 1.

1 2 3 72 5 1 03 6 2 5

− − → − −

(b) Write your result from part (a) in the space next to the original matrix and then find a row

operation that will result in this new augmented matrix having a zero in row 3, column 1.

1 2 3 72 5 1 03 6 2 5

− − → → − −

Exploration 3*: Solve a System of Linear Equations Using Matrices Using augmented matrices and row operations, systems of linear equations can be written in row echelon form, which will allow them to be solved. A matrix is in row echelon form when: 1: The entry in row _____, column _____, is a 1 and 0’s appear below it. 2: The first nonzero entry in each row after the first row is _____, 0’s appear below it, and it

appears to the __________ of the first nonzero entry in any row above. 3: Any rows that contain all _____ to the left of the vertical bar appear at the bottom.

1. An example of a matrix in row echelon form is:

2. Use words to explain how the above matrix in row echelon form will help to solve the

system of linear equations.

Section 8.2 Systems of Linear Equations: Matrices 239


Example 5*: Solve a System of Linear Equations Using Matrices

Solve: 3 11 13

5 7

x yx y

+ = + =

(a) Write the augmented matrix: (b) Perform row operations that result in the entry in row 1, column 1 becoming 1: (c) Which row operation did you use? (d) Perform row operations that result in the entries in column 1 below row 1 to become 0’s: (e) Perform row operations that result in the entry in row 2, column 2 becoming 1: (f) The augmented matrix is now in row echelon form. Rewrite as a system of linear

equations to solve: General Matrix Method for Solving a System of Linear Equations(Row Echelon Form) Step 1:____________________________________________________________________ Step 2:____________________________________________________________________ Step 3:____________________________________________________________________ __________________________________________________________________________ Step 4:____________________________________________________________________ ____________________________________________________________________________________________________________________________________________________ __________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________ Step 5:____________________________________________________________________ __________________________________________________________________________ Step 6:____________________________________________________________________ ____________________________________________________________________________________________________________________________________________________



Example 6*: Solve a System of Linear Equations Using Matrices

Solve:

1

4 3 2 16

2 2 3 5

x y zx y zx y z

+ − = − − + = − − =

Example 7*: Solving a Dependent or Inconsistent System Using Matrices

Solve:

3 2 6

2 3 10

8 3 28

x y zx y z

x y z

− + =− + − = − + =

Example 8*: Solving a Dependent or Inconsistent System Using Matrices

Solve:

6

2 3

2 2 0

x y zx y z

x y z

+ + = − − = + + =

Section 8.3 Systems of Linear Equations: Determinants 241


Chapter 8: Systems of Equations and Inequalities Section 8.3: Systems of Linear Equations: Determinants

From the previous sections, two methods have been used to solve systems of linear equations. The first method was to solve using ____________________________________. The second method was to solve using ___________________________________________. A third method utilizes determinants. Definition: If a, b, c, and d are four real numbers, the symbol

D =

is called a 2 by 2 determinant. Its value is the number __________; that is,

__________D = =

Example 1*: Evaluate 2 by 2 Determinants

Evaluate: 2 3

4 1

−−

(a) Algebraic Solution (b) Graphing Solution

By solving the system of linear equations ax by scx dy t

+ = + =

by elimination, Cramer’s Rule is

derived: Cramer’s Rule for Two Equations Containing Two Variables Theorem: The solution to

the system of equations: ax by scx dy t

+ = + =

is given by:

x y= =

provided that the denominator does not equal _____.



A way of remembering Cramer’s Rule is the following: Cramer’s Rule for Two Equations Containing Two Variables Theorem: The solution to

the system of equations: ax by scx dy t

+ = + =

if

, , and 0x y

a b b aD D D D

c d d c= = = ≠

s st t

is:

x y= =

Example 2*: Use Cramer’s Rule to Solve a 2 by 2 System

Use Cramer’s Rule, if applicable, to solve the system: 3 6 24

5 4 12

x yx y

− = + =

(a) Algebraic Solution (b) Graphing Solution

Evaluate 3 by 3 Determinants

To use Cramer’s Rule to solve a system of three equations, a 3 by 3 determinant is used and

symbolized by: 11 12 13

21 22 23

31 32 33

a a aa a aa a a

in which 11 12,a a , …, are real numbers. One method for

evaluating this determinant is as follows:

11 12 13

21 22 23 11 12 13

31 32 33

a a aa a a a a aa a a

= − +

The 2 by 2 determinants shown above are called minors of the 3 by 3 determinant where the minor ijM of entry ija is the determinant resulting from removing the _____ row and the

_____ column.




For the determinant

2 1 3

2 5 1

0 6 9

A−

= −−

, find 21M .

Definition: For an by n n determinant A , the cofactor of entry ija , denoted by ijA , is

given by:

ijA = _______________

where ijM is the minor of entry ija


Find the value of the 3 by 3 determinant

1 2 1

3 5 1

2 6 7

by expanding:

(a) across row 1: (b) down column 1: Cramer’s Rule for Three Equations Containing Three Variables Theorem: The solution

to the system of equations: 11 12 13 1

21 22 23 2

31 32 33 3

a x a y a z ca x a y a z ca x a y a z c

+ + = + + = + + =

where 11 12 13

21 22 23

31 32 33

0

a a aD a a a

a a a= ≠ is given

by:

zx y= = =

where:

1 12 13 11 1 13 11 12 1

2 22 23 21 2 23 21 22 2

3 32 33 31 3 33 31 32 3

, , x y z

a a a a a aD a a D a a D a a

a a a a a a= = =

c c cc c cc c c



Example 5*: Use Cramer’s Rule to Solve a 3 by 3 System

Use Cramer’s Rule, if applicable, to solve the system:

2 1

3 5 3

2 6 7 1

x y zx y zx y z

+ + = + + = + + =

Exploration 1: Know Properties of Determinants

Theorem: The value of a determinant changes sign if any two rows (or any two columns) are interchanged.

In Example 1, 2 3

4 1

−−

= _____. Demonstrate this theorem by interchanging the rows or

columns and finding the value of the determinant:



Theorem: If all the entries in any row (or any column) equal 0, the value of the determinant is 0. Theorem: If any two rows (or any two columns) of a determinant have corresponding entries that are equal, the value of the determinant is 0.

In Example 4,

1 2 1

3 5 1

2 6 7

, suppose row two was the same as row one, or

1 2 1

2 6 7

.

Demonstrate this theorem by finding the value of the determinant:

Theorem: If any row (or any column) of a determinant is multiplies by a nonzero number k , the value of the determinant is also changed by a factor of k .

In Example 1, 2 3

4 1

−−

= _____. Demonstrate this theorem by finding the value of

2 3

4 1

k k−−

:

Theorem: If the entries of any row (or column) of a determinant are multiplied by a nonzero number k and the result is added to the corresponding entries of another row (or column), the value of the determinant remains unchanged.

In Example 1, 2 3

4 1

−−

= _____. Demonstrate this theorem by multiplying row 2 by -2 and

add it to row 1. This becomes your new row 1: 2 3

4 1 4 1

−→ =

− −_____.



Chapter 8: Systems of Equations and Inequalities Section 8.4: Matrix Algebra

Definition: Two by m n matrices A and B are said to be equal, written as

A B= provided that A and B have the _______ number of rows and the _______ number of columns and each entry ija in A is equal to the corresponding entry ijb in B

If A and B are ______ m n× matrices then the sum and difference of A and B , denotedA B+ and A B− , is a matrix obtained by adding or subtracting ____________________ entries of A and B .

Example 1*: Find the Sum and Difference of Two Matrices

Suppose that 1 2 2 3 0 4

and 0 1 3 2 1 4

A B− −

= = − − . Find:

(a) A B+ (b) A B− Like the algebraic properties of sums of real numbers, matrices are commutative, associative and have a zero matrix: Suppose that , , and A B C are by m n matrices, then: Commutative Property of Matrix Addition: A B+ = __________ Associative Property of Matrix Addition: ( )A B C+ + = __________ A matrix whose entries are all equal to 0 is called a zero matrix, where 0 0A A A+ = + = . The zero matrix is the _______________ identity in matrix algebra.

Example 2*: Find Scalar Multiples of a Matrix Multiply each matrix by the real number, or scalar indicated. Suppose that:

3 1 5 4 1 0 9 0 C

2 0 6 8 1 3 3 6A B

= = = − − −

Find (a) 4A (b) 1

3C (c) 3 2A B−

Properties of Scalar Multiplication

( ) ( )

( )

( )

k hA kh Ak h A kA hA

k A B kA kB

=+ = +

+ = +

Section 8.4 Matrix Algebra 247


Find the Product of Two Matrices Definition: A row vector R is a _____ by _____ matrix, [ ]R =

A column vector C is an _____ by _____ matrix, C

=

The product RC of R times C is defined as the number:

[ ]RC

= =

______________________________

Example 3*: Find the Product of Two Matrices

Find RC if [ ]1 2 4R = − and

2

1

3

C = −

:

Definition: Let A denote an by m r matrix, and let B denote an by r n matrix. The product AB is defined as the __________ matrix whose entry in row _____, column _____ is the product of the _____ row of A and the _____ column of B .


Find the product AB if

2 43 2 1

1 30 4 1

3 1

A B

− = = − − −

.

(a) Explain why the product AB is defined: (b) Explain why the product AB will be a _____ _____× matrix:

(c) AB =



Example 5*: Find the Product of Two Matrices Is matrix multiplication commutative? _____ Use Example 4 to find the product BA . (a) Explain why the product BA is defined: (b) Explain why the product BA will be a _____ _____× matrix:

(c) BA = Example 6*: Find the Product of Two Matrices

If 1 3 2 0

and 2 7 3 4

A B−

= = − − ,

Find (a) AB (b) BA Assuming the matrix multiplication is defined, then: Theorem: Matrix multiplication is _____ commutative. Associative Property of Matrix Multiplication: ( ) __________A BC = Distributive Property: ( ) __________A B C+ =

For an by n n square matrix, the entries located in row i , column i , 1 i n− ≤ ≤ , are called the diagonal entries. An by n n square matrix whose diagonal entries are _____, while all

other entries are _____, is called the identity matric nI . For example,

2 3 I I

= =

, and so on…


Let

2 43 2 1

and 1 30 4 1

3 1

A B

− = = − − −

,

Find (a) 3AI (b) 2I A (c) 2BI

Section 8.4 Matrix Algebra 249


Identity Property: If A is an by m n matrix, then

_____ and _____m nI A AI= =

If A is an by n n square matrix, then

_____n nAI I A= =

Find the Inverse of a Matrix

Definition: Let A be a square by n n matrix. If there exists an by n n matrix 1A− , read

“ A inverse,” for which 1 1 ___AA A A− −= = , then 1A− is called the inverse of the matrix A .

Example 8*: Find the Inverse of a Matrix

Show that the inverse of 1

113 1 2 is

4 2 32

2

A A−

− − − − = =

Procedure for Finding the Inverse of a Nonsingular Matrix To find the inverse of an by n n nonsingular matrix A , proceed as follows: Step 1: Form the matrix __________. Step 2: Transform the matrix into _________________________ form. Step 3: This form will contain the identity matrix on __________ of the vertical bar; the

by n n matrix on the __________ of the vertical bar is the inverse of A .

Example 9*: Finding the Inverse of a Matrix

The matrix

1 1 2

0 1 3

2 2 1

A−

= −

is nonsingular. Find its inverse.



Example 10*: Finding the Inverse of a Matrix

Show that the matrix 2 1

4 2A

− = −

is singular and has no inverse.

Example 11*: Solve a System of Linear Equations Using an Inverse Matrix

Solve the system of equations

2 1

3 2

2 2 1

x y zy zx y z

− + =− + = − + + = −

Section 8.5 Partial Fraction Decomposition 251


Chapter 8: Systems of Equations and Inequalities Section 8.5: Partial Fraction Decomposition

We know how to add rational expressions – in other words to add to a single rational expression. Let’s explore this.

1. Add : 2 4

2 4x x+

+ +

2. Partial Fraction Decomposition __________ this process, and there are __________

cases to study.

Case 1*: Decompose PQ

, Where Q Has Only Nonrepeated Linear Factors.

Under the assumption that Q has only nonrepeated linear factors, the polynomial Q has the form:

( ) ________________________________________Q x =

where none of the numbers ia are equal. In this case, the partial fraction decomposition of

PQ

is of the form:

( )

...( )

P xQ x

= + + +

where the numbers iA are to be determined.

Example 1*: Decompose PQ

, Where Q Has Only Nonrepeated Linear Factors

Write the partial fraction decomposition of 2 5 6

xx x− +

.



Case 2*: Decompose PQ

, Where Q Has Repeated Linear Factors.

If the polynomial Q has a repeated linear factor, say ( ) , 2nx a n− ≥ an integer, then, in the

partial fraction decomposition of PQ

, we allow for the terms:

...+ + +

where the numbers iA are to be determined.

Example 2*: Decompose PQ

, Where Q Has Repeated Linear Factors

Write the partial fraction decomposition of 3 2

2

2

xx x x

+− +

Case 3: Decompose PQ

, Where Q Has Nonrepeated Irreducible Quadratic Factor.

If Q contains a nonrepeated irreducible quadratic factor of the form 2ax bx c+ + , then, in

the partial fraction decomposition of PQ

, allow for the term:

where the numbers and A B are to be determined.

Section 8.5 Partial Fraction Decomposition 253


Example 3: Decompose PQ

, Where Q Has Nonrepeated Irreducible Quadratic Factor

Write the partial fraction decomposition of ( ) ( )2

1

1 4x x+ +

Case 4: Decompose PQ

, Where Q Has a Repeated Irreducible Quadratic Factor.

If the polynomial Q contains a repeated irreducible quadratic factor

( )2 , 2, n

ax bx c n n+ + ≥ is an integer, then, in the partial fraction decomposition of PQ

, we

allow for the terms:

...+ + +

where the numbers 1 1 2 2, , , ,..., ,n nB A B A BA are to be determined.

Example 4: Decompose PQ

, Where Q Has a Repeated Irreducible Quadratic Factor

Write the partial fraction decomposition of ( )2

22

5 2

4

x x

x

− −

+



Chapter 8: Systems of Equations and Inequalities Section 8.6: Systems of Nonlinear Equations

The methods of solving systems of nonlinear equations mirrors the methods used to solve systems of linear equations. As a review, those methods are: 1. Use a graphing utility to find _____________________________________________ 2. The ____________________ method 2. The ____________________ method

Example 1*: Solve a System of Nonlinear Equations Using Substitution

Solve the following system of equations: 2

3 2

2 0

x yx y

− = −

− =

(a) Algebraic Solution using Substitution: (b) Graphing Solution (use a graphing device, then sketch graph below):

Section 8.6 Systems of Nonlinear Equations 255


Example 2*: Solve a System of Nonlinear Equations Using Elimination

Solve the following system of equations: 2 2

2

13

7

x yx y

+ =

− = −

(a) Algebraic Solution using Elimination: (b) Graphing Solution (use a graphing device, then sketch graph below): Example 3: Solve a System of Nonlinear Equations

Solve the following system of equations: 2 2

2

49

6 49

x yy x

+ =

− =

(a) Algebraic Solution: (b) Graphing Solution (use a graphing device, then sketch graph below):



Example 4: Solve a System of Nonlinear Equations

Solve the following system of equations: 2

2 2

3

9

y xx y

= +

+ =

(a) Algebraic Solution: (b) Graphing Solution (use a graphing device, then sketch graph below):

Example 5: Running a Marathon In a marathon, or 26.2-mile race, the winner crosses the finish line 1 mile ahead of the second-place runner and 4 miles ahead of the third-place runner. Assuming that each runner maintains a constant speed throughout the race, by how many miles does the second-place runner beat the third-place runner?

chapter 8: systems of equations and inequalities section 8 ... · chapter 8: systems of equations...

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