Homework, Page 604

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Homework, Page 604Use substitution to solve the system of equations.

1. 3 0

2 3 1

2

x y z

y z

z

2 3 2 1y 2 6 1y 2 7y 7

2y

73 2 0

2x

212 0

2x

25

2x

25 7, , 2

2 2

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Homework, Page 604Use Gaussian elimination to solve the system of equations.

5. 3

4 5

3 2 4

x y z

x y

x y z

3

4 4 7

5 4 5

x y z

y z

y z

3

4 4 7

2

x y z

y z

y

4 2 4 7z 4 15z 15

4z

152 3

4x

8 15 12

4 4 4x

5

4x

5 15,2,

4 4

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Homework, Page 604Perform the indicated elementary row operations on the matrix.

9.

2 6 4

0 2 3

3 1 2

1 33 2 R R

2 6 4

0 2 3

3 1 2

1 33 2 R R

2 6 4

0 2 3

3 3 9 1 6 2

2 6 4

0 2 3

0 8 4

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Homework, Page 604What elementary row operations applied to the matrix will yield the given matrix.

13.

2 1 1 2

1 2 3 0

3 1 1 2

12R1 2 3 0

2 1 1 2

3 1 1 2

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Homework, Page 604Find a row echelon form for the matrix.

17. 1 3 1

2 1 4

3 0 1

1 2

1 3 1

2 0 7 6

3 0 1

R R

1 3

1 3 1

3 0 7 6

0 9 2

R R

3 2

1 3 1

3 0 20 0

0 9 2

R R

2

1 3 11

0 1 020

0 9 2

R

2 3

1 3 1

9 0 1 0

0 0 2

R R

3

1 3 11

0 1 02

0 0 1

R

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Homework, Page 604Find the reduced row echelon form for the matrix.

21. 1 0 2 1

3 2 4 7

2 1 3 4

1 0 2 1

0 1 1 2

0 0 0 0

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Homework, Page 604Write the augmented matrix corresponding to the system of equations.

25. 2 3 1

4 3

3 2

x y z

x y z

x z

2 3 1 1

1 1 4 3

3 0 1 2

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Homework, Page 604Write the system of equations corresponding to the augmented matrix.

29. 3 2 1

4 5 2

3 2 1

4 5 2

x y

x y

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Homework, Page 604Solve the system of equations by finding a row echelon form for the augmented matrix.

33. 2 8

2 3 9

3 3 5

x y z

x y z

x y z

1 2 1 8

2 1 3 9

3 1 3 5

1 0 0 2

0 1 0 1

0 0 1 4

2, 1,4

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Homework, Page 604Solve the system of equations by finding the reduced row echelon form for the augmented matrix.

37. 3 2

3 4 10 5

2 4 3

x y z

x y z

x y z

1 1 3 2

3 4 10 5

1 2 4 3

1 0 2 0

0 1 1 0

0 0 0 1

No solution

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Homework, Page 604Solve the system of equations by finding the reduced row echelon form for the augmented matrix.

41. 2 4

3 4 5

2 3 4

x y

x y

x y

1 2 4

3 4 5

2 3 4

1 0 0

0 1 0

0 0 1

No solution

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Homework, Page 604Write the system of equations as a matrix equation AX = B, with A as the coefficient matrix of the system.

45. 2 5 3

2 1

x y

x y

2 5

1 2

x

y

3

1

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Homework, Page 604Solve the system of equations by using an inverse matrix.

49. 2 3 13

4 5

x y

x y

AX B2 3 13

4 1 5

x

y

1

1 3

14 142 1

7 7

A

1X A B

1 31314 14

2 1 5

7 7

x

y

2

3

2,3

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Homework, Page 604Solve the system of equations by using an inverse matrix.

53. 2 3

2 3 12

3 2 3

2 3 3 3

x y z w

x y z w

x y z w

x y z w

1,2, 2,3

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Homework, Page 60453.

AX B

2 1 1 1 3

1 2 3 1 12

3 1 1 2 3

2 3 1 3 3

x

y

z

w

1

5 1 51

12 3 1213 2 1

112 3 127 1 1

16 3 67 3

1 24 4

A

1X A B

5 1 51

12 3 12313 2 1

1 1212 3 1237 1 1

16 3 6 37 3

1 24 4

x

y

z

w

1

2

2

3

1,2, 2,3

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Homework, Page 604Use the method of your choice to solve the system of equations.

57. 2 2 5

2 2 5

3 3 3 2 12

1

x y z w

x y z

x y z w

x z w

1 2 2 1 5

2 1 2 0 5

3 3 3 2 12

1 1 1 1 1

1 0 0 0 3

0 1 0 0 3

0 0 1 0 2

0 0 0 1 0

3,3, 2,0

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Homework, Page 604Use the method of your choice to solve the system of equations.

61. 2 4 1

2 1

2 0

x y z w

x y z w

x y z w

2 1 1 4 1

1 2 1 1 1

1 1 1 2 0

0 0 0 0 0

1 0 0 2 1

0 1 0 1 1

0 0 1 1 0

0 0 0 0 0

1 2 ,1 , ,w w w w

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Homework, Page 604Use the method of your choice to solve the system of equations.

65. 2 0

2 1

3 3

2 2 5 4

x y z w

y z w

x y w

x y z w

1 1 1 2 0

0 1 1 2 1

1 1 0 3 3

2 2 1 5 4

1 0 0 0 0

0 1 0 3 0

0 0 1 1 0

0 0 0 0 1

No solution

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Homework, Page 604Determine f so that its graph contains the given points.

69. 2 1, 4 , 1, 2Family of Curves f x ax bx c

2 1, 4 , 1, 2f x ax bx c

24 1 1a b c 4a b c

22 1 1a b c 2a b c

1 1 1 4

1 1 1 2

0 0 0 0

1 0 1 3

0 1 0 1

0 0 0 0

23 for any f x c x x c c

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Homework, Page 60473. Children ride a train for 25 cents, adults pay $1.00, and seniors pay 75 cents. On a given day, 1400 passengers paid $740 for their rides. There were 250 more children than all other passengers. Find the number of children, adults, and senior riders.

a adults

s seniors

0.25 0.75 740a c s 1400a c s

250a s c 250a c s 1 0.25 0.75 740

1 1 1 1400

1 1 1 250

1 0 0 410

0 1 0 825

0 0 1 165

410 adults; 825 children; and 165 seniors

c children

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 21

Homework, Page 60477. Morgan has $50,000 to invest and wants to receive $5,000 interest the first year. He puts part in CDs earning 5.75% APY, part in bonds earning 8.7% APY, and the rest in a growth fund earning 14.6% APY. How much should he put in each fund if he puts the least amount possible in the growth fund.

c CDsb bonds

0.087 0.146 5000b g

g growth fund50000b g

$0.00

$38,983.05

$11,016.95

CDs

Bonds

Growth fund

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Homework, Page 604Use inverse matrices to find the equilibrium point for the supply and demand curves.

81.100 5 Demand curve

20 10 Supply curve

p x

p x

1 15 ,733 3

173 315 3

2 1 1003 31 1 2015 15

p

x

1

2 13 3

1 115 15

A

1 5 100

1 10 20

p

x

AX B10 20p x 5 100p x 100 5

20 10

p x

p x

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

7.4

Partial Fractions

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What you’ll learn about

Partial Fraction Decomposition Denominators with Linear Factors Denominators with Irreducible Quadratic Factors Applications

… and whyPartial fraction decompositions are used in calculus in integration and can be used to guide the sketch of the graph of a rational function.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 25

Partial Fraction Decomposition of f(x)/d(x)

1. Degree of degree of : Use the division algorithm to divide by to obtain the

( ) ( )quotient and remainder and write ( ) .

( ) ( )

f d f d

f x r xq r q x

d x d x

2

2

2. Factor ( ) into a product of factors of the form ( ) or ( ) , where

is irreducible.

u vd x mx n ax bx c

ax bx c

1 2

1 22

3. For each factor ( ) : The partial fraction decomposition of ( ) / ( ) must

include the sum ... , where , ,..., are real numbers.

u

uuu

mx n r x d x

AA AA A A

mx n mx n mx n

2

1 1 2 21 22 22 2

1 2

4. For each factor ( ) : The partial fraction decomposition of ( ) / ( ) must

include the sum ... , where , ,...,

and , ,..., are real nu

v

v vvv

v

ax bx c r x d x

B x CB x C B x CB B B

ax bx c ax bx c ax bx c

C C C

mbers.

The partial fraction decomposition of the original rational function is the sum of ( ) and

the fractions in parts 3 and 4.

q x

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Example Decomposing a Fraction with Distinct Linear Factors

3 3

Find the partial fraction decomposition of .1 2

x

x x

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Example Decomposing a Fraction with Repeated Linear Factors

2

3 3Find the partial fraction decomposition of .

1 2

x

x x

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Example Decomposing a Fraction with an Irreducible Quadratic Factor

2

2

3 1Find the partial fraction decomposition of .

2 1

x x

x x

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Example Reversing a Decomposed Fraction to Identify the Parent Function

Find the function that yields the partial fraction decomposition:

2 13 .

1 3x

x x

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Homework

Homework Assignment #12 Read Section 7.5 Page 614, Exercises: 1 – 49 (EOO), skip 45

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7.5

Systems of Inequalities in Two Variables

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 32

Quick Review

Find the - and -intercepts of the line.

1. 3 4 24

2. 120 30

Find the point of intersection of the two lines.

3. 3 and 2 5

4. 1 and 3 1

5. 7 3 10 and 1

x y

x y

x y

x y x y

x y y x

x y x y

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 33

Quick Review Solutions

(0,6) and (8,0)

(0,30) and

Find the - and -intercepts of the line.

1. 3 4 24

2. 1 20 30

Find the point of intersection of the two lines.

3. 3 and 2 5

(2

0,0)

(8/3

4. 1 a

,1/3)

x y

x y

x y

x y x y

x y

nd 3 1

5. 7 3 10 an

(

d

0,1)

(1.3,1 ) 0.3

y x

x y x y

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What you’ll learn about

Graph of an Inequality Systems of Inequalities Linear Programming

… and whyLinear programming is used in business and industry to maximize profits, minimize costs, and to help management make decisions.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 35

Steps for Drawing the Graph of an Inequality in Two Variables

1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is < or>. Use a solid line if the inequality is ≤ or ≥.

2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.

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Example Graphing a Linear Inequality

Draw the graph of 2 4. State the boundary of the region.y x

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Example Solving a System of Inequalities Graphically

2Solve the system 2 3 4 and .x y y x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 38

Example Solving a Word Problem38. Paul’s diet is to contain at least 24 units of carbohydrates and 16 units of protein. Food substance A costs $1.40 per unit and each unit contains 3 units of carbohydrates and 4 units of protein. Food substance B costs $0.90 per unit and each unit contains 2 units of carbohydrates and 1 units of protein. How many units of each food substance should be purchased to minimize cost? What is the minimum cost?