Download Slide 7- 1 Homework, Page 575

Homework, Page 575
Determine whether the ordered pair is a solution to the system.
1. 5 x  2 y  8
2x  3y  1
 a  0, 4   5  0   2  4   8  8  8  No
 b  2,1
 5  2   2 1  8  2  2   3 1  1  Yes
 c  2, 9   5  2   2  9   8 2  2   3 9   23  23  1  No
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 1
Homework, Page 575
Solve the system by substitution.
5. 3 x  y  20
x  2 y  10
3 x  y  20
x  2 y  10  x  2 y  10
3  2 y  10   y  20  6 y  30  y  20  7 y  10  y  
10
7
 50 10 
20 70 50
 10 

x  2     10    
 , 
7
7
7
7
 7
 7
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 2
Homework, Page 575
Solve the system by substitution.
9.
x  3y  6
2 x  6 y  4
2 x  6 y  4
x  3 y  6  x  3y  6
2  3 y  6   6 y  4  6 y  12  6 y  4  12  4
No solution
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 3
Homework, Page 575
Solve the system algebraically. Support your answer graphically.
13.
y  6x2
7x  y  3
y  6 x2
2
  3x  1 2 x  3  0
2

6
x

7
x

3

0

7
x

6
x

3
7x  y  3
1 3 
 2 27    1 , 2  ,   3 , 27 
x   ,    y  3  7x  y   , 



3
3
2
2




3
2
3
2




 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 4
Homework, Page 575
Solve the system algebraically. Support your answer graphically.
17. x 2  y 2  9
x  3 y  1  x  3 y  1
 3 y  1
2
 y 2  9  9 y 2  6 y  1  y 2  9  10 y 2  6 y  8  0
5y  3y  4  0  y 
2
  3 
 3  4  5 4 
2
2  5

3  9  80  1.243, 0.643
10
x  3 y  1  x  2.730, 2.930   2.730,1.243 ,  2.930, 0.643
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 5
Homework, Page 575
Solve the system by elimination.
21. 3x  2 y  8
5 x  4 y  28
3x  2 y  8  6 x  4 y  16
5 x  4 y  28  5x  4 y  28
11x
 44  x  4
3  4   2 y  8  12  2 y  8  y  2   4,2 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 6
Homework, Page 575
Solve the system by elimination.
25. 2 x  3 y  5
6 x  9 y  15
6 x  9 y  15
2x  3 y  5 
6 x  9 y  15  6 x  9 y  15
00
Infinitely many solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 7
Homework, Page 575
Use the graph to estimate any solutions of the system.
29.
x  2y  0
0.5 x  y  2
No solution, parallel lines
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 8
Homework, Page 575
Use graphs to determine the number of solutions the system has.
33. 2x  4 y  6
3x  6 y  9
Infinitely many solutions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 9
Homework, Page 575
Solve the system graphically. Support numerically.
37.
y  x3  4 x
4  x  2y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 10
Homework, Page 575
Solve the system graphically. Support numerically.
41.
x2  y 2  9
y  x2  2
x2  y 2  9 
x2  y 2  9
2
2


x
 y  2
y  x 2
y2  y  7
y  y7  0 y 
2
 1 
 1  4 1 7   1 
2 1
2
1  28
2
2
y  3.193,2.193  x  y  2  x   2.193  2  2.048
 2.048,2.193 ,  2.048,2.193
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 11
Homework, Page 575
45. The table shows expenditures, in billions, from federal hospital and
medical insurance trust funds.
A. Find the quadratic regression equation and superimpose its graph on a
scatter plot of the data.
B. Find the logistic regression equation and superimpose its graph on the
scatter plot of the data.
C. When will the two models predict expenditures of 300 billion dollars?
D. Explain the long range implications of using the quadratic regression
model to predict future expenditures.
E. Explain the long range implications of using the logistic regression
model to predict future expenditures.
Year
Amount
Year
Amount
1990
110.2
1999
213.5
1995
183.2
2000
225.3
1997
209.5
2001
246.5
1998
210.2
2002
267.1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 12
Homework, Page 575
45.
A. Find the quadratic regression equation and superimpose its
graph on a scatter plot of the data.
B. Find the logistic regression equation and superimpose its graph
on the scatter plot of the data.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 13
Homework, Page 575
45. C. When will the two models predict expenditures of 300 billion
dollars?
The quadratic model predicts reaching $300-billion in 2006 and the logistic
model predicts reaching $300-billion in 2007.
D. Explain the long range implications of using the quadratic
regression model to predict future expenditures.
The quadratic model predicts expenditures reaching a maximum level of about
$575-billion and then decreasing, eventually reaching zero, which is not
realistic.
E. Explain the long range implications of using the logistic
regression model to predict future expenditures.
The logistic model predicts expenditures leveling out at about $354-billion,
which is also not realistic.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 14
Homework, Page 575
49.
Find the dimensions of a rectangle with a perimeter of 200 m and an
area of 500m2.
P  2l  2w  200; A  lw  500  l  w  100  l  100  w
w 100  w  500  w2  100w  500  0
w
  100  
w  94.721  l 
 100   4 1 500 
2
2 1

100  10000  2000
2
500
 l  5.279  5.279m  94.271m
w
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 15
Homework, Page 575
53.The total cost of one medium and one large soda is $1.74. The large soda
costs $0.16 more than the medium soda. Find the cost of each soda.
l  m  1.74
l  m  0.16
2l  1.90  l  0.95 m  0.95  0.16  0.79
Large soda  $0.95
Medium soda  $0.79
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 16
57.
Homework, Page 575
Pedro has two plans to choose from to rent a van.
Company A: a flat fee of $40 plus $0.10 per mile
Company B: a flat fee of $25 plus $0.15 per mile
(a) How many miles can Pedro drive in order to be charged the same amount
by the two companies.
A  40  0.10m ; B  25  0.15m
15
40  0.10m  25  0.15m  0.05m  15  m 
 300
0.05
Both companies ch arg e the same for 300 miles of travel.
(b) Give reasons why Pedro might choose one plan over the other.
If Pedro is planning on driving more than 300 miles, Company A’s plan would
be less expensive. If he is planning to drive less than 300 miles, Company B’s
plan is less expensive.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 17
61.
A.
B.
C.
D.
E.
Homework, Page 575
Which of the following is a solution of the system
 3,1
 1,0 
 3, 2 
 3, 2 
 6,0 
2 x  3 y  12
x  2 y  1
2 x  3 y  12  2 x  3 y  12
x  2 y  1  2 x  4 y  2
7 y  14  y  2
x  2  2   1  x  4  1  x  3
 3, 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 18
Homework, Page 575
65. Consider the system of equations:
x2 y 2

1
4
9
x  y 1
(a) Solve the first equation in terms of x to determine the two implicit
functions determined by the equation.
2
2
2
2
2
2
9
4

x
x
y
y
x
9x


 1
 1  y2  9 
4
4
9
9
4
4
9 4  x2
y
3 4  x2 ; y   3 4  x2

4
2
2
(b) Solve the system of equations graphically.




Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 19
65. (c)
Homework, Page 575
Use substitution to confirm the solutions found in part (b).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 20
7.2
Matrix Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
The points (a) (1,  3) and (b) ( x, y) are reflected across the given line.
Find the coordinates of the reflected points.
1. The x-axis
2. The line y  x
3. The line y   x
Expand the expression,
4. sin( x  y )
5. cos( x  y )
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 22
Quick Review Solutions
The points (a) (1,  3) and (b) ( x, y ) are reflected across the given line.
Find the coordinates of the reflected points.
1. The x-axis (a) (1,3) (b) ( x,  y )
2. The line y  x (a) (  3,1) (b) ( y, x)
3. The line y   x (a) (  3,  1) (b) ( y,  x)
Expand the expression,
4. sin( x  y ) sin x cos y  sin y cos x
5. cos( x  y ) cos x cos y  sin x sin y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 23
What you’ll learn about






Matrices
Matrix Addition and Subtraction
Matrix Multiplication
Identity and Inverse Matrices
Determinant of a Square Matrix
Applications
… and why
Matrix algebra provides a powerful technique to manipulate large
data sets and solve the related problems that are modeled by the
matrices.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 24
Matrix
Let m and n be positive integers. An m × n matrix
(read "m by n matrix") is a rectangular array of m
rows and n columns of real numbers.
 a11
a
 21


 am1
a12
a22
am 2
a1n 
a2 n 



amn 
We also use the shorthand notation  aij  for this matrix.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 25
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 26
Example Determining the Order of a
Matrix
What is the order of the following matrix?
1 4 5
3 5 6 


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 27
Matrix Addition and Matrix Subtraction
Let A   aij  and B  bij  both be matrices of order m  n.
1. The sum A + B is the m  n matrix A  B   aij  bij  .
2. The difference A  B is the m  n matrix A  B   aij  bij  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 28
Example Matrix Addition
1 2 3  2 3 4 
4 5 6  5 6 7 

 

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 29
Example Using Scalar Multiplication
1 2 3
3

4
5
6


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 30
The Zero Matrix
The m  n matrix 0  [0] consisting entirely of zeros is
the zero matrix.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 31
Additive Inverse
Let A   aij  be any m  n matrix.
The m  n matrix B  bij  consisting of the additive
inverses of the entries of A is the additive inverse
of A because A  B  0.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 32
Matrix Multiplication
Let A   aij  be any m  r matrix and B  bij  be any r  n matrix.
The product AB  cij  is the m  n matrix where
cij  ai1b1 j +ai 2b2 j  ...  air brj .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 33
Example Matrix Multiplication
Find the product AB if possible.
1 2 3 
A

0
1

1


1 0 
and B   2 1 


 0 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 34
Identity Matrix
The n  n matrix I n with 1's on the main diagonal and 0's elsewhere
is the identity matrix of order n  n.
1
0

I n  0


0
0 0
1 0
0 1
0 0
0
0

0


1 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 35
Inverse of a Square Matrix
Let A   aij  be an n  n matrix. If there is a matrix B
such that AB  BA  I n , then B is the inverse of A.
We write B  A1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 36
Inverse of a 2 × 2 Matrix
1
a b 
1  d b 
If ad  bc  0, then 

.



ad  bc  c a 
c d 
The number ad  bc is the determinant of the 2  2 matrix
a b
a b 
A
and is denoted det A 
 ad  cb.

c d
c d 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 37
Minors and Cofactors of an n × n Matrix
If A is an n  n matrix where n  2, the minor M ij corresponding
to the element aij is the determinant of the  n  1   n  1 matrix
obtained by deleting the row and column containing aij .
The cofactor corresponding to aij is Aij   1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
i j
M ij .
Slide 7- 38
Determinant of a Square Matrix
Let A   aij  be a matrix of order n  n (n  2). The
determinant of A, denoted by det A or | A | , is the
sum of the entries in any row or any column multiplied
by their respective cofactors. For example, expanding
by the i th row gives det A | A | ai1 Ai1  ai 2 Ai 2  ...  ain Ain .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 39
Transpose of a Matrix
Let A   aij  be a matrix of order n  m. The transpose
of A, denoted by AT is the matrix in which the rows in
A become the columns in AT and the columns in A
become the rows in AT or AT   a ji  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 40
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 sales for the day?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 41
Inverses of n × n Matrices
An n × n matrix A has an inverse if and only if
det A ≠ 0.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 42
Example Finding Inverse Matrices
 1 3
Find the inverse matrix if possible. A  

2
5


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 43
Properties of Matrices
Let A, B, and C be matrices whose orders are such that the following sums, differences,
and products are defined.
1. Community property
Addition: A + B = B + A
Multiplication: Does not hold in general
2. Associative property
Addition: (A + B) + C = A + (B + C)
Multiplication: (AB)C = A(BC)
3. Identity property
Addition: A + 0 = A
Multiplication: A·In = In·A = A
4. Inverse property
Addition: A + (-A) = 0
Multiplication: AA-1 = A-1A = In |A|≠0
5. Distributive property
Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC
Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 44
Reflecting Points About a Coordinate
Axis
To reflect a point about the x-axis, express the point as a
1 0 
1 2 matrix and multiply by 
to obtain the 1 2

0 1
matrix of the reflected point.
To reflect a point about the y -axis, express the point as a
 1 0 
1 2 matrix and multiply by 
to obtain the 1 2

 0 1
matrix of the reflected point.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 45
Example Using Matrix Multiplication
46. A company has two factories, each manufacturing three products. The number
of products i made in factory j in one week is given by aij in the matrix
120 70 
A  150 110  . If production is increased by 10%, write the new production levels


 80 160 
as a matrix B. How is B related to A?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 46
Homework



Homework Assignment #10
Read Section 7.3
Page 590, Exercises: 1 – 65 (EOO), skip 53
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 47
7.3
Multivariate Linear Systems and Row
Operations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
1. Find the amount of pure acid in 45L of a 58%
acid solution.
2. Find the amount of water in 30 L of a 28%
acid solution.
3. Is the point (0,  1) on the graph of the function
f ( x)  x  4 x  1?
3
4. Solve for x in terms of the other variables:
x  z  w  2
 2 1
5. Find the inverse of the matrix 
.

 0 3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 49
Quick Review Solutions
1. Find the amount of pure acid in 45L of a 58%
acid solution. 26.1 L
2. Find the amount of water in 30 L of a 28%
acid solution. 21.6 L
3. Is the point (0,  1) on the graph of the function
f ( x)  x  4 x  1? yes
3
4. Solve for x in terms of the other variables:
x  z  w  2 x  2 z w
 2 1
5. Find the inverse of the matrix 

 0 3
1/2 1/ 6 
.

0
1/ 3 

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 50
What you’ll learn about






Triangular Forms for Linear Systems
Gaussian Elimination
Elementary Row Operations and Row Echelon Form
Reduced Row Echelon Form
Solving Systems with Inverse Matrices
Applications
… and why
Many applications in business and science are modeled by
systems of linear equations in three or more variables.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 51
Triangular Form of a System of Equations
A system of equations is said to be in triangular
form, if it has as many equations as variables and
if the equations are arranged in such a manner
that the top equation has all variables, the next
lacks one variable, the next lacks the first
variable and a second and so on. For example,
5x  3 y  z  7
2 y  3z  8
6z  4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 52
Example Solving a System of Equations
in Triangular Form by Substitution
Solve the system.
5x  3 y  z  7
2 y  3z  8
6z  4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 53
Equivalent Systems of Linear Equations
The following operations produce an equivalent
system of linear equations.
1.Interchange any two equations of the system.
2.Multiply (or divide) one of the equations by
any nonzero real number.
3.Add a multiple of one equation to any other
equation in the system.
These operations, when used to reduce a system
to triangular form, are called Gaussian
elimination.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 54
Example Solving a System of Equations
Using Gaussian Elimination
Solve the system
2x  y  0
x  3 y  z  3
3y  z  8
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 55
Example Solving a System of Equations
Using Gaussian Elimination
Solve the system
x  y  3z  1
2x  3y  z  4
3x  7 y  5 z  4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 56
Augmented Matrix
An augmented matrix is one in which there is one more column than row and
where the first columns are the coefficients of a system of equations and the last
column contains the constants of the equations. For instance, the system
x  y  3z  1
2x  3y  z  4
3x  7 y  5 z  4
may be represented by the augmented matrix
1 1 3 1
 2 3 1 4 


 3 7 5 4 
Augmented matrices may be used to record the steps of the Gaussian elimination
process.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 57
Row Echelon Form of a Matrix
A matrix is in row echelon form if the following
conditions are satisfied.
1. Rows consisting entirely of 0’s (if there are
any) occur at the bottom of the matrix.
2. The first entry in any row with nonzero
entries is 1.
3. The column subscript of the leading 1 entries
increases as the row subscript increases.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 58
Elementary Row Operations on a Matrix
A combination of the following operations will
transform a matrix to row echelon form.
1. Interchange any two rows.
2. Multiply all elements of a row by a nonzero
real number.
3. Add a multiple of one row to any other row.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 59
Example Finding a Row Echelon Form
Solve the system:
x  2 y  z  2
2x  3y  2z  2
4 x  8 y  5 z  5
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 60
Reduced Row Echelon Form
If we continue to apply elementary row
operations to a row echelon form of a matrix, we
can obtain a matrix in which every column that
has a leading 1 has 0’s elsewhere. This is the
reduced echelon form.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 61
Example Solving a System Using Inverse
Matrices
Solve the system
2x  3y  0
2 x  2 y  10
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 62
Example Solving a System Using Inverse
Matrices
Solve the system.
3x  3 y  6 z  20
x  3 y  10 z  40
 x  3 y  5 z  30
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 63
Example Solving a Word Problem
74. Stewart’s Metals has three silver alloys on hand. One is 22% silver, one is
30%, and the third is 42%. How many grams of each alloy are required to
produce 80 grams of a new alloy that is 34% silver if the amount of the 30%
alloy is twice the amount of the 22% alloy used?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 64