Homework, Page 604

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 1

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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



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



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


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


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


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


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


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.


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


Example Decomposing a Fraction with Distinct Linear Factors

3 3

Find the partial fraction decomposition of .1 2

x

x x


Example Decomposing a Fraction with Repeated Linear Factors

2

3 3Find the partial fraction decomposition of .

1 2

x

x x


Example Decomposing a Fraction with an Irreducible Quadratic Factor

2

2

3 1Find the partial fraction decomposition of .

2 1

x x

x x


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


Homework

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

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

7.5

Systems of Inequalities in Two Variables


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


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


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.


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.


Example Graphing a Linear Inequality

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


Example Solving a System of Inequalities Graphically

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


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?

Documents

Homework, Page 604