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236 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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
Copyright © 2016 Pearson Education, Inc.
(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:
238 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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
Copyright © 2016 Pearson Education, Inc.
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:____________________________________________________________________ ____________________________________________________________________________________________________________________________________________________
240 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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
Copyright © 2016 Pearson Education, Inc.
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 _____.
242 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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.
Section 8.3 Systems of Linear Equations: Determinants 243
Copyright © 2016 Pearson Education, Inc.
Example 3*: Evaluate 3 by 3 Determinants
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
Example 4*: Evaluate 3 by 3 Determinants
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
244 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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:
Section 8.3 Systems of Linear Equations: Determinants 245
Copyright © 2016 Pearson Education, Inc.
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
−→ =
− −_____.
246 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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
Copyright © 2016 Pearson Education, Inc.
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 .
Example 4*: Find the Product of Two Matrices
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 =
248 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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…
Example 7*: Find the Product of Two Matrices
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
Copyright © 2016 Pearson Education, Inc.
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.
250 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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
Copyright © 2016 Pearson Education, Inc.
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− +
.
252 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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
Copyright © 2016 Pearson Education, Inc.
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
− −
+
254 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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
Copyright © 2016 Pearson Education, Inc.
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):
256 Chapter 8 Systems of Equations and Inequalities
Copyright © 2016 Pearson Education, Inc.
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?
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