Master Math: Algebra

Master Math: Algebra Debra

CAREER PRESS 3 Tice Road P.O. Box 687 Franklin Lakes, NJ 07417 1 -800-CAREER-1 20 1-848-03 10 (NJ outside U.S.) 201-848-1727

0 1996 by

book by by

MASTER MATH: ALGEBRA by by

VISA

on

Library of Congress Cataloging-in-PublicationData

/ by

:

p. )

1. 1996 5

Acknowledgments

Dr.

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Table of Contents

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Introduction

1

Chapter 1 Translating Problems into Algebraic Equations

5

1.1. 1.2.

5

1.3. 1.4.

6 8 12

Chapter 2 Simplifying Algebraic Equations

22

2.1. 22 2.2. 2.3.

23 26

2.4.

by

28 2.5. 30

Chapter 3 Solving Simple Algebraic Equations

31

3.1. 31 3.2. 39 3.3. 43

Chapter 4 Algebraic Inequalities

46

4.1. 46

Chapter 5 Polynomials

49

5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

49 51 52 53 55 61

5.7. 62

Chapter 6 Algebraic Fractions with Polynomial Expressions 6.1. 6.2. 6.3. 6.4.

72 72 73 74 76

Chapter 7 Solving Quadratic Polynomial Equations with One Unknown Variable

81

7.1 82

7.2 83 7.3 87 7.4 90

7.5. 92

Chapter 8 Solving Systems of Linear Equations with Two or Three Unknown Variables 8.1.

95

of

96 8.2.

98

8.3. 103 8.4.

105 8.5. 110 8.6.

111 8.7.

115 8.8.

119 8.9.

126

Chapter 9 Working with Coordinate Systems and Graphing Equations 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.

133 134 139 145 150 152 155 160

9.8. on

164

Index

169

Appendix Tables of Contents of First and Third Books in the Mwter Math Series

172

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Introduction

Algebra ter Math Basic Math and Re-Algebra metry. Master Math

MasPre-Calculus and Geo-

book

Algebra

Algebra

book

Algebra

Algebra

1

Algebra

book on on book

Master Math

Algebra

Master

Math no

Muster Math book

A

2

Introduction

“I do

up.

3

This page intentionally left blank

Chapter 1

Translating Problems intoAlgebraic Equations 1.1 1.2 1.3 1.4

1.1. Introduction to algebra

you you

$1.00 you

5

$20.00 you

buy

Algebra

“X.”

$1.00 $20.00?

you $20.00. $20.00.

x

$1.00 by $1.00

=

1.2. Translating English into algebraic equations

by added to, plus, sum, in all, altogether, increased by, the sum of, more, total. by difference,how many less, subtracted from, how many more, reduced by, decreased by, minus, less, take away. 6

Translating Problems into Algebraic Equations

by multiplied by, of, times, product, multiply, twice by 2), double by 2 ) . by divided by, quotient, divides, ratio. 2nd

squared

cubed

by equals, is equal, is, are, were, was, the sum is, the difference is, the quotient is, the product is, the same as, results in, the result is, makes, leaves, yields, gives. by a number, what.

38

8

an unknown, a variable,

3 x

3x + 8 = 38. 11.

5 y

(2y)2 - 5 = 11.

5 n

of 5 + n3 = 13. 7

13.

Algebra

1.3. Algebra terminology

A

Letters by

b,

A,

x, y, z,

A variable

by x, y, z,

A d,

b,

n,

constants. by or by

coeficient. 3y,

3

y -4

8

x

Translating Problems into Algebraic Equations

=3 * n

3n =

2*3#23

As expression

numerical

x4y

algebraic expression.

An equation

up

An

9

Algebra

+, -

=

by

+

terms. no 3x

2x

3

2.2.

9x, 6

2x, 2x.

by

Inequalities

on on >, <, 2

I,

.

10

Translating Problems into Algebraic Equations

factors

= 18x,

x

x

=

x2,

x

18

x

Polynomials no “1”

4x4 - 2x2 + 4 45x2 3x + 9 -8

+

5.1.

Linear equations do

1. 2x

+ 1= 5 x

A,

Ax y

+

=C ,

y= b y

on

9 on

11

+ b,

Algebra

Non-Linear equations 1. 1= 5

1.4. Simple word problems

Important note: 2,

3,

Perimeter Word Problems

1, book Master Math and Geometry

Be-CaZcuZus 2

3 2

4

+3

+4

=9

4

6 2(4

+

+ 12

=8

12

= 20

Translating Problems into Algebraic Equations

Area Word Problems

1, book Master Math CaZcuZus and Geometry

Pre-

=

=

3 2

(3

=6

Volume Word Problems

book Pre-CaZcuZus and Geometry

=

13

1, Master Math

Algebra

3

2

4 = 24

(3

Price and Profit Word Problems

=

- (%

=

$24

$24 = $X - (0.25)($~) x= $24 = $x(l $24 = $x(0.75) = $X

$x = $32 $32 i s 2

=

3

-

14

Translating Problems into Algebraic Equations

$16, gross

-

=

$S - $16 = $S - $16 = $S - $16 = $8 $S = $8 $16 $S = $24

+-

$24. Percent Word Problems o

(

17 +

20

= 85%

by 100,

by

$350.00

(40%)/(

= 0.40

15

Algebra

by = $140

$140. $350 - $140 = $210

Distance Word Problems

=

60

30

2 2

4

60

2 = 120

30

2 = 60

4 120

+ 60

= 180 =

up 16

Translating Problems into Algebraic Equations

Work Word Problems

1/ n

+ 1 / m = 1/ h ,

n do h do

1

by n

do Mary

1,000 1,000

4 6 1,000

1/4+ 1/6= lk

A

12.

+

=

=l k

by =1

17

Algebra

by h= h = 1 2 6 = 2.4 2.4

Mixture Word Problems

you $1.20 pound pound,

6 2

$1.60 pound

+

$1 =

$7.20 + $3.20 = $10.40 =

6

1.20

2

by 8 =

18

Translating Problems into Algebraic Equations

15 pound 15 pound

y 15 =

(15 =

by

+

+

+y = + 3 + y = 5.25 + 0 . 3 5 ~ y-

+y

15

+

= 5.25 -

2

3

0 . 6 5= ~ 2.25 y = 2.2510.65 y = 3.46 3.46

so

oat

19

3

Algebra

Interest Word Problems simple interest

x

=

x

$4,000

3% 3

1

3%

3/100 12

(0.03)($4,000)(1

= $120

12

3 ($120)(3 mo./12

= $30

$4,000 + $30 = $4,030 =

3 3

compound interest, by by 4, by 2,

20

Translating Problems into Algebraic Equations

$5,000

5%

1 5%

6 = $125

$5,000

+

6 = $5,125

6 5%

1

on = $128.12

6

$5,125 + 1

= $5,253.12

21

Chapter 2

Simplifying Algebraic Equations 2.1 2.2 2.3 2.4

by

2.5

2.1. Commutative, associative and distributive properties of addition and multiplication

22

Simplifying Algebraic Equations

commutative

b

associative

+ + x

+ +

= (b x = x x

b

distributive x

+

=

x

+

x

+c)= + b

2.2. Using associative and distributive properties The

Using the Associative Property

23

Algebra

+ +3=6

+ + 1= 6 3+ + 2

+ +

1 =6

+ + + = 10 + 2 + + 4 = 10 3 + +4)+ 1 = 10 4 + +2 + 10 + + + = 10 2 x 3 x 5 = 30 3x 2 x 5 5 x 3x 2 5 x 2 x 3 = 30

+ + = 1x3 + 1x4 + 2x3 + 2x4 =3 + + 1+ = 3x1 + 3x2 + 4x1 + 4x2 = 3 + 6 + 4 + 8 =21 5.4,

24

Simplifying Algebraic Equations

Using the Distributive Property

+

2(3 + 4) = (2)(3) (2)(4) = 6 + 8 = 14 2(4 - 3) = (2)(4) - (2)(3) = 8 - 6 2 =I

for

on

undistributing unfactoring. 2(3

+

+

= (2)(3) (2)(5)

2

+

(2)(3)+ (2)(5)= 2(3 5) 2

2(3 + (2)(3)+ (2)(2)

25

Algebra

2.3. Combining like terms in algebraic equations

Like terms

b, c, d, x,y, z, x2,

&,&, G,

b2, c3, d4, x2b,

do 2x2, x2

26

2

Simplifying Algebraic Equations

3-h

2b

& 2b

2b2 1 0 6

no =

‘‘1” xl/n

=

&

only

27

Algebra

-

-

=

+

0 =

+

-

+

=

2.4. Simplifying algebraic equations by removing parentheses and combining like terms by

3y -

3y - 2y + 3y - 2y - 6 = 6

-

=6

-

28

+

= 6.

Simplifying Algebraic Equations

y by

6

"="

)

3y - 2(y 36 36 -

+

+

=6

=6

+ 2

+ +

+ -

=4

=4

=4 =4

29

-

= 4.

Algebra

+

-

= 2.

no

+

-

+

-

-

=2

=2

=2

2.5. The general order to perform operations in algebra

by

30

"+"

"-"

Chanter 3

Solving Simple Algebraic Equations 3.1

3.2 3.3

3.1. Solving algebraic equations that have one unknown variable

on

=

31

Algebra

y

y

y=3+2 y=5

on

y, y + 3 - 3 ~- 63 y+0=3 y=3

+ 3 = 6.

y

y

z

42 = 32.

3

by y 3 = 6.

+

y = 3.

3+3=6 6=6

by 4.

z,

4214 = 3214

2=8 32

So 1v ing Simple Algebraic Equations

by 42 = 32. z = 8.

(4)(8) = 32 32 = 32

- 6 = 25. - 6 +6 =25 +6 a+O=31 a=31

6

by

- 6 = 25. = 31.

31 - 6 ~ 2 5 25 = 25 x x,

x/4 = 2. by 4.

(x/4)(4)= (2)(4) (4x/4) =8

33

Algebra

by x/4 = 2. x = 8. 814 = 2 2=2

x

+

x,

4

x/4 4 - 4 = 2 - 4 x/4+0=-2 x/4 = -2

by 4. (d4)(4)= (-2)(4) (4x/4)= =

x=

34

x/4

+ 4 = 2.

Solving Simple Algebraic Equations

by

+

x/4 + 4 = 2.

-8/4 4 = 2 -2+4=2 2=2 z z,

-Z

- 4 +4 =2

-z- 4 = 2.

4

+4

- ~ = 6

by -1. -z,

z,

z = lz

-z = lz.)

(-z)/( - 1) = (6)/(- 1) = 1,

z = -6

by -z - 4 = 2.

-(-6) - 4 = 2

x

=

6-4=2

by

by

35

Algebra

y 8y - 3y

8y

+ 5 - 3y = 10.

by

+ 5 = 10 8y - 3y =

- 3)y = 5y,

5

5y

+ 5 - 5 = 10 - 5 by 5. = 515

by

8y

+5 -

+ 5 - 3y = 10. = 10

13 - 3 = 10 10 = 10

on

by by

36

or

Solving Simple Algebraic Equations

+

3x - 10 = -4 x.

x

by -x

x

3x - x - 10 = 2x - 10 = -4 10 2x - 10 10 = -4+ 10

+

by 2. = 612

by 3x - 10 = -4+ x. - 10 = -4+ 3 9-10=-4+3 -1= -1

by by x 10 +

- 20 =

10

+

-

-

by

5x 37

+

= -5(4

Algebra

+

10 + 10 + - 20 = 10 - 20 + =

+

+

-

=

10 + 10 +

+ 10

=

=

by 11. =

x=

by

+

10 + 10

-

+

10 +

+

-

10 x=

- 20 =

= -5(4

+ = -5(4 + = -5(4

+

- 10 = + 20 - 10 + 20 = -20 + 20 + + 10 = 60111 + 60111 + 10 =

10 = 10 = 10 = 10

+

38

Solving Simple Algebraic Equations

3.2. Solving simple algebraic equations containing fractions

by

by

1. 2. by

3.

4. by

5.

2 d 3 - 2d4 = 4

2x-2x- - = 3

4

lx -2x- - = 3 2

4 1 4 1 39

x.

Algebra

3 2

3 , 6 , 9 , 12, ... 2 , 4 , 6 , 8 , ...

6

by -(6)2x - - 3

(6)lx - -(6)4 2 1

( 2 ) 2 -~(3)lx= (6)4

4x - 3x = lx. l x = 24 x=24

by 2(24)/3 - 2(24)/4 = 4 4813 - 2412 = 411 6.

by 6.

40

Sol u ing Simple Algebraic Equations

-(2)48 - - 1

(3)24 - - 4 1 1

24 = 24 = Zd(2xi-4)

(2x+4), 2x -6 -10 2(x+2)

by

- x (5)(x+ 2) 5

(x+ 2)

+ 2)= x(5) 3x + 6 = 5x 3x

41

x.

Algebra

by 2.

612 = X 3=x

by

6 10

(x = 3)

6 6+4

6 - -6 10

10

on

6 _.

10

2x 2x+4 (6)*(2x+4)

(10)*(2x).

(6)(2x+4)= (10)(2x)

12x+24=20x 12x 24 = 2 0 -~1 2 ~ 42

Solving Simple Algebraic Equations

24 = 8x by 8.

24/8 = x 3 =x - = -

b

= bx.

y

3.3. Solving simple algebraic equations containing radicals

1.

on

2.

3. 4.

by

2

43

Algebra

2 d z = 8

on

by 2.

d Z = =

8 2

-

4

=

42

(dKT)(dX-l) =

16

1

by = 8

2 d E 2&

=

8

44

Solving Simple Algebraic Equations

by 2.

f i =8 2

16 = 16

&

=

4x4=

4

=

(x1/2)2= x

(G3=

(x113)3= x

(G4=

(x1/4)4= x

45

Chapter 4

Algebraic Inequalities

4.1

4.1. Solving algebraic inequalities with one unknown variable >,

by

<, 2

1.19., Basic Math and Pre-Algebra,

5.

book

by

Inequalities (>) (2)

(<),

or

on on

46

Algebraic Inequalities

x 3 -(x/3) s - 5 - 3 -(x/3) <

by -3.

x 2 (-8)(-3) x 2 24 47

3 - ( d 3 )< -5.

Algebra

by 3-

3-

I

5 8-

s0

8

by 3. 24 I x

48

ChaDter 5

Polynomials

5.1 5.2 5.3

5.4 5.5 5.6 5.7 ax2

+ bx + c

5.1. Definitions

A monomiaZ

2x, 4 x 5 8

binomid 2~ + 8 , 3 - ~5x2

49

Algebra

A trinomiaZ 4x2 + 3x + 9, 4x4 - 2x2 + 4x 2x2

degree

2.

degree by 4x4 - 2x2 + 4x 5x2 + 3x + 9 4x2

4. 2. 2. 2.

0.

8

2x

1.

“1”

=

2

50

Polynomials

5.2. Addition of polynomials by

(y2

- 3y

+ 4)

+

(y - 3yz y3).

(y2-3y+4)+(y-3y2+y3) =y3+y2-3y2-3y+y+4 =y3

(y2

- 2y2 - 2y + 4

- 3y + 4) + (y - 3y2+y3) =y3 - 2y2 - 2y + 4 2y(y2 - 2y

2y(y2 -

+ 2)

3(y - 4y2

+ 2~3).

zY+ 2) + 3(y - 4y2 + 2y3). 2y

3

+ 4y + 3y - 1 2 ~ +2 6y3 = 2y3 + 6y3 - 4y2 - 12y2+ 4y + 3y

= 2y3 - 4y2

Add = By3 -

16y2

2yk2 - 2y

+ 7y

+ 2) + 3k - 4y2 + 2y3) = By3 - 16y2 + 7y 51

Algebra

5.3. Subtraction of polynomials by

- 3y +

(y - 3y2+

by

- 3y + 4 - y + 3y2 - y3

= =

+ 3y2 - 3y - y + 4 + 4y2 - 4y + 4 - 3y +4) - (y -

=

52

- 4y + 4

Polynomials

5.4. Multiplication of polynomials

muZtipZy monomials,

=

=

=

=

=

multiply a monomial with apolynomial,

+

-

= =

+

=

-

+ + 60x2 + 20x

= 24x

+ 53

+ 48

Algebra

multiply two binomials,

First, Outer, Inner

Last

multiply polynomials with polynomials,

on.

54

POlynomials

+ 2x - 5) = + + + + + = 2x3 + 4x2 + + + + 20 = 2x3 + 4x2 + + + + 20 = 2x3 - 18x + 20 (2x -

5.5. Division of polynomials by by

divide monomials,

+

- 6x =-

-3x2

2 2 6x - - - - --3xx -X

divide polynomials by monomials,

55

Algebra

(4x +

+

+

+6 =

+

=

+

= 4d6

2 3

+

= -x

- 6x -

-

-

+

+

= --

2x

+ -6x + 2x

4 -

3

+

-

=

+

-

+ 2 - -- -x 3 2x

+ + l/x

divide polynomials by polynomials, do by

no

56

Polynomials

+ +

(-1Ox 8 8x2) -+ (2x - 4):

Example:

?

8x2 + 2x = 4x. 4x Z X -) 4 82 ~ -10~+8 4x(2x - 4) = 8x2 - 16x 4x J

2 x - 4 8x2 - l O x + 8

8x2 -16x 4x Z X - 4J 8~ 2 - 1 0 +~ 8 8x2 - 16x 6x+8 by

6x, 6x +- 2x = 3.

57

2x

Algebra

+ 3

4x )

2

2 ~ - 8~ 4 OX+ 8 8x2 - 16x 6x+8 3(2x - 4) = 6x - 12

4x

+ 3

2x - 4 8x2 -1Ox + 8 8x2 - 16x 6x+ 8 6~ - 12

no

4x

+ 3

2~ - 4) 8~ 2 - 1 0 +~8 8x2 - 16x 6x+ 8 6~ - 12 20 20 (-IOx+ 8 + 8x2) + (2x - 4) = 4x + 3 + 2x- 4 20 20 - - 10 -2 ~ - 4 Z(X-2) x-2 4x + 3 + (lO/(x-2)).

final

58

Polynomials

+

+

(x2 4) +- (x 4):

?

x+4’x2+0x+4 x2 + x = x. X

x + 4 ’ x 2 +Ox+4

+

+

x(x 4) = x2 4x X

x + 4 ) x 2+ o x + 4 x2+ 4x

X

x + 4 ) x 2+ O x + 4 x2+ 4x - 4x+4

by

-4x,-4x + x = -4. 59

x

Algebra

x - 4 x+4)x2+0x+4 x 2 +4x - 4x+4

-4(x+ 4) = -4x x -4 x + 4Ix2 + ox + 4 x2 +4x -4x +4 - 4 - 16 ~

no

x -4 x+4’x2+ Ox + 4 x2+ 4x - 4x+4 - 4 ~1620

20 (x2 + 4) + (x + 4) = x - 4 + x+4’

60

Polynomials

5.6. Factoring polynomials with a common monomial factor

6= 2

6.

3

2a + 2b = 2(a + 2

+

2x + 5x = x 2x2 + 4x2y = 2x2

+

= 7x

1+

12x4 - 6x3 3x2 = 3x2

- 2x + 61

Algebra

5.7. Factoring polynomial expressions with the form ax2+bx+c ax2

ax2

(x + 2)(x + 3).

+

+

(x 2)(x + 3) = x2 + 3x 2x + (2)(3)

+ ( 2 + 3 ) +~ (2)(3)= x2 + 5x + 6 (4x + 2)(5x + 3). (4x + 2)(5x + 3)

= x2

+ (4x)(3)+ (2)(5x)+ (2)(3)

= (4x)(5x) = 20x2

+ 12x + 10x + 6 = 20x2 + 22x + 6

62

+ bx + c,

+ bx + c

Polynomials

Compare the factored binomial form toith the trinomial form n

+

+ = x2

= x2

+

+ nx +

+ + bx +

p, q,

+

= 1, b =

= pqx2

+ pnx +

+

+

ax2

b=

=

n

+

+ = pqx2

+

=

63

+ bx +

=qp,

Algebra

1.

(

)(

).

2.

3. 4.

.)

5.

x2

(

x

only

)(

+ 5x + 6.

).

x.

1

(x

2

1: ( (

3

3) 6) 64

1

6

Polynomials

5x.

+ 5x.

3x + 2x = 5x 2: 6x + -1x = 5x no 2

(x + 2)(x + 3),

1

(x + 2)(x + 3) = x2 + 3x+ ZX+ 6 = x2+ 5~ + 6 x2

(x + 2)(x + 3).

+ 5x + 6

x2 + 3x - 18. (

x

)(

).

x.

(x >(x )

6 2,18

-3, -6 1.

65

3,9

Algebra

3x.

+ 3x. (x + 6)(x - 3) = 3x (x - 6)(x + 3) = -3x (x + 9)(x - 2) = 7x (x - 9)(x + 2) = -7x (x 18)(x - 1) = 17x (x - 18)(x+ 1) = -17x

= -3x,

= 6x,

= 3x,

= -6x,

= -2x,

= 9x,

= ZX,

+

3x

66

= -9x,

= -lx,

= 18x,

= lx,

= -18x,

(x + 6)(x - 3).

Polynomials (X

+ 6 ) ( -~3) = X'

(X

+ 6 ) ( -~3).

+ 3x - 18 x2 + 3x - 18

- 3~ + 6~ - 18 = X'

20x2 + 22x (

)(

+ 6.

).

2x lOx, 20x (2x )(10x ) (20x )(lx ) (5x )(4x 1

lx, 5x

4x.

2 ( (

3,l 6. 2)( 3) I)( 6)

22x. 22x. (2x ) ( l O X ) (20x >(lx ) (5x )(4x 1 ( 2)( 3) ( 6)

67

Algebra

+ 20x,

+

2x,

= 6x,

=

= 26x

+

= 60x,

=

= 62x

+ 8x,

+

+

lOx,

+

+ 4x,

+

= 15x,

=

= 23x

+

=

=

= 22x

+

= 120x,

=

= 121x

+

= 30x,

=

= 34x

no

22x

+

+

+

+

= 20x2

+

+

+ 12x + 10x + 6 = 20x2 + 22x + 6 20x2 + 22x + 6

68

Polynomials

Special Binomial Products to Remember

+

differenceof two squares, x2 - y2, -

+

-

= x2

-

= x2

- xy + xy - y2 = x2 - y2 - 3x + 3x - 32

- 32

-9

4x2 - 25. = 4x2

25?

(5)(5)= 25 4x2 - 25 =

+ 5)(2x - 5).

by

+ = 4x2 +

-

+ - 25 = 4x2 +

- 25

= 4x2 -

- 25 = 4x2 - 25

+

sum of two squares, x2 y2, binomiaZ squared

+ 2xy + y2 + = x2 + xy + xy + y2 =x2 + 2xy + y2 = x2 - 2xy + y2 - y)(x - = x2 - xy - xy + y2 = x2 - 2xy + y2

+ +

= x2

69

Algebra

x4 - 81.

~4

- 81 = (x' + 9 ) ( ~ -29) (x2 - 9)

+

x2 - 9 = (x 3)(x - 3) x4 - 81

+

(x2 + 9)(x 3)(x - 3).

+

+

(x2 9)(x 3)(x - 3). (x 3)(x - 3) = x2 - 3x 3x - 32 = x2 - 9 (x2 9)(x2 - 9)= x4 - 9x2 9x2 - 92 = x4 - 81

+ +

ax2

+

+

+ bx + c 2x2

+ 1Ox + 12. 2.

+

2x2 + 1 0 ~12 = 2(x2 + 5x + 6)

+ + 6).

(x2 5x

70

Polynomials

no

+ +

(x + 2)(x 3) (x + 6)(x 1)

)

(x + 2)(x+ 3) = 5x (x + 6)(x+ 1) = 7x

= 3x,

= 2x,

= lx,

= 6x,

5x,

(x + 2)(x + 3). 2

(2)(x+ 2)(x + 3).

+

+ + +

(2)(x+ 2)(x 3) = (2)(x2 3x 2x 6) = (2)(x2+ 5x + 6) = 2x2 + 1Ox + 12

2x2 + 1Ox + 12

(2)(x+ 2)(x + 3).

71

Chapter 6

Algebraic Fractions with Polynomial Expressions 6.1 6.2 6.3 6.4

6.1. Factoring and reducing algebraic fractions

by

72

Algebraic Fractions with Polynomial Expressions

--

-

-

-

+ 1

=

-

=

+

-

+ 5x + x2 x2

+ 4x +

-

--

-

6.2. Multiplication of algebraic fractions

by

by

73

Algebra

+

+

(x2

+ 6x + x2

+

+ 5x +

24x2 + 6x +

6x

(x+ 6x

-

+

-

+ 2)

24x2 + +

+ +

+

+

-

- 4x2 + 8 x -

(x+

6.3. Division of algebraic fractions

by by

74

Algebraic Fractions with Polynomial Expressions

by by

by

(x' + 4~ + 4)/(2~2 - 8) + (x' + 2 ~ ) / ( -48)~ x 2 + 4 x + 4 -. ( x 2 + 2 x ) 2x2 - 8 4 ~8 -

up

by

24)( ~ - 2 ) - x 2 + 4 x + 4X 4 x - 8 - ( ~ + 2 ) ( ~ + X 2x2 - 8 (x2+2x) 2(x2- 4 ) x ( x +2)

- ( x + 2 ) ( x + 2 )X 4 ( x - 2 ) - (x + 2) 2(x+2)(x-2) x(x+2) 2 ( x - 2 )

+

(x2 - 4) = (x 2)(x - 2).

75

X

4(x - 2 ) x(x+2)

Algebra

6.4. Addition and subtraction of algebraic fractions

Fractions with Common Denominators

by

2x/(3(x - 2)) - 4/(3(x - 2)).

Fractions With Different Denominators

1.

76

Algebraic Fractions with Polynomial Expressions

if

+

you

6, 6

4 12, 16...,

4 4, 8, 6, 12, 24, 30...,

12.

2.

by

=

3.

by

77

Algebra

+

(x2 + 4x + 4)/(2x2 - 8) + (4x - 8)/(x2 2x).

x2+4x+4 + 4 x 4 -- ( X + 2) ( X + 2) + 4 ( x - 2) 2x2 - 8 x2+2x 2(x+2)(x-2) x(x+2) - ( ~ + 2 ) 4(~-2) +

2(x-2)

(2)(x- 2)

x(x+2)

+

(x)(x 2). (2)(x)(x 2)(x - 2).

+

by

(x+ 2) 2(x-2)

+

(x)(x 2). x(x + 2) x ( x + 2) 78

Algebraic Fractions with Polynomial Expressions

4(x - 2) x(x+2) (2)(x - 2). 2(x - 2 ) 2(x-2)

by

( X + 2) (x)( X + 2)

+ (4) ( X - 2) (2)( X - 2)

79

Algebra

x3 + 12x2 - 28x + 32 - x3 + 12x2 - 28x + 32 2x3 - 8x (2x)(x2- 4 )

+

(x2 + 4x + 4)/(2x2 - 8) + (4x - 8)/(x2 2x) x 3 + 12x2 - 28x + 32 2x3 - 8x

80

Chapter 7

Solving Quadratic Polynomial Equations with One UnknownVariable 7.1 7.2

7.3 7.4

7.5

81

Algebra

7.1. Defining and solving quadratic (polynomial) equations

Quadratic equations x 2

x2. ax2

a

b by

b

+

bx c = 0 3x2 + 2x + 5 = 0 3x2 + 2x = 0

82

+ bx +

= 0.

Quadratic Polynomial Equations, One Unknown Variable

1. 2.

3. b zero,

4. b

7.2. Using factoring to solve quadratic equations with one unknown variable

:

1. ax2

+ bx + c = 0. 83

Algebra

+ bx +

2.

3. by

4.

x2 - 2x = 3.

+ bx + -

- 3=0

= 0.

+ bx +

- 2x -

+

-

+

-

+

=

-

=

+

= 3x,

= 2x

+

84

= =

Quadratic Polynomial Equations, One Unknown Variable

by ( x + l)(x - 3) = x 2 - 3x+ l x - 3 =x2 - 2x - 3 (x + (x - 3). (x+ 1 ) = 0 (x - 3) = 0 x+1=0 x=-1 ~ - 3 ~ x=3

0 x2 - 2x = 3,

x=-landx=3. by x2 - 2x = 3 For x = -1, (-1)2- 2(-1)= 3 1+2=3 3=3 x = 3, 32 - 2(3) = 3 9-6=3 3=3

85

Algebra

x2 = 2x.

+ bx +

= 0.

+ bx + -

=0

x

x2 = 2x, by x2 = 2x x = 0, 02 = x = 2, 22 = 4=4

86

x =0

x = 2.

Quadratic Polynomial Equations, One Unknown Variable

7.3. Using the quadratic formula to solve quadratic equations with one unknown variable quadratic formula

discriminant, b2 b2 b2 -

1.

> 0, < 0, = 0,

2 2 1

+ bx + c = 0.

2.

b 87

Algebra

b

3.

by

4.

. by

5.

3x2 + 2 =

+ bx +

3x2

+ 5x + 2 = 0

= 0.

b b

k by

k

J25-24 6

k

fi

6 88

Quadratic Polynomial Equations, One Unknown Variable

so, 4

+

f = 1.

-5 + l x = 6

-

-5 - 1 x = 6

3x2 + 2 = -5x x = -2/3

x = -1.

by 3x2 2 = -5x. x = -2/3, 3(-2/3)2 2 = -5(-2/3) 3(4/9) 2 = 10/3 12/9 2 = 10/3 4/3 + 2 = 10/3

+

+ + +

2/1= 6/3,

4/3 + 613 = 10/3 10/3 = 10/3 For x = -1, 3(-1)2+ 2 = - 5(-1) 3(1) + 2 = 5 5=5

89

Algebra

7.4. Using the square root method to solve quadratic equations with one unknown variable square root method b b ax2

+ bx +

=0

ax2+c=o.

1.

x2

on

2

x2

x,

3. by

4.

3x2 + 4 = 31. 4

3x2 = 27 x2 by 3.

90

dx2 = k x.

Quadratic Polynomial Equations, One Unknown Variable x2 = 2713 x2 = 9

x = +3 3x2 + 4 = 31

x=3

x = -3. by 3x2 + 4 = 31.

x = 3, 3(3)2 4 = 31 3(9) + 4 = 31 27 + 4 = 31 31 = 31 x = -3, 3(-3)2 4 = 31 3(9) 4 = 31 27+4=31 31 = 31

+

+ +

91

Algebra

7.5. Using the method of completing the square to solve quadratic equations with one unknown variable method of completing the square

b

x2

1.

by

2.

+ bx = + =

by b,

b,

3.

by

4.

x2

5.

on

by

92

Quadratic Polynomial Equations, One Unknown Variable

x2 - 2x - 10

=

-7.

x2

+ bx =

10 =

3 by b,

= (-2/2)2 =

b, = 1.

= =4

=4

-

-

= 4 =

-

-

-

=

= 4

= 4

x2

-

on

=4

x2

by

93

-

Algebra

x-l=&2

@ = kx . x

+2

-2.

+2, x-l=2 x=3 -2, x - l = -2 x = -1 x2

x=3

x=

by x2 - 2x - 10 = x = 3, (3)' - 2(3) - 10 = -7 7 9 - 6 - 3= 0

x= (-1)2 - 2(-1)- 10

=

-7 7

1+2-3= 0

o=o 94

- 2x 10 = 7

Chanter 8

Solving Systems of Linear Equations with Two or Three Unknown Variables 8.1

8.2 8.3 8.4

8.5 8.6

8.7

8.8

95

Algebra

8.9

8.1. Solving systerils of linear equations with two or more unknown variables

n

n

96

Solving Systems of Linear Equations

Setting up word problems using two variables

20, 10? x

y x + y = 20.

20,

x - y = 10.

10,

50

you you

10% 14% 14% 50

10%

x y 14% 50

x

+y

= 50

(8%)(x

(10%)(50 are

97

+ (14%)(y

=

Algebra

Methods for two equations and two unknowns

Methods for three equations and three unknowns

8.2. Using the elimination method to solve systems of linear equations with two unknown variables e h i n a t i o n method by by by

98

Solving Systems of Linear Equations

1.

2. by

3.

4.

by

5.

x - y =4

x

x y

x

+

y 3x = 4x

+ 2y = y 99

Algebra

x by

y x - y = 4 by 2 =

=

y 8

- 2y = 8

+ 2y 5x + 0

=

7

= 15

5x = 15 x = x =3 y. 4 = 4

= = =

y

1

=

x y =4 x =3

y=

100

3x + 2y = 7

Solving Systems of Linear Equations

by x - y =4 x-y = 4 3= 4 3+1 = 4 4 = 4

3x = =

3x

+ 2y = 7

7

7

20

10. x y 20,

x + y = 20.

10,

x - y = 10.

x + y = 2 0 + x - y = 10 2x

= 30

101

+ 2y = 7.

Algebra

2x = 30 x = 3012 x = 15 x = 15 y. = 10

15 - y = 10 15 - 10 = y

x + y = 20 by =

x +y

=

20

20

20 = 20

x-y

=

15 - 5 = 10 10 = 10

102

10

x - y = 10

Solving Systems of Linear Equations

8.3. Using the substitution method to solve systems of linear equations with two unknown variables substitution method by

1. by on 2.

3. 4.

5.

by

103

on

Algebra

2x - y = 4

2x + 4y = 4.

by

on

2x-y = 4 2 ~ - =4 y y

2x+4y = 4 2 ~ + 4 ( 2 ~ -=4 4)

x 16

4

ZX + 4 ( 2 -~4)

=

4

2 ~ + 8 16 ~ = - 4 1ox = 20 by 10. x = 20/10 x = 2

x =2 y.

2x-y = 4 2(2) - y = 4 4-y = 4 4

y

4-4

104

Solving Systems of Linear Equations

2x - y = 4

2x + 4y = 4

by 2x - y = 4

-0

= 4

2x = =

+ 4y = 4

4

4

8.4. Using the method of determinants to solve systems of two linear equations with two unknown variables method of determinants by

105

Algebra

A matrix.

2 by 2

determinant

3

4

-

2 by 2

(1)(4)- (3)(2)= 4 - 6 = -2

method ofdeterminants Cramer’s ruZe x

106

Solving Systems of Linear Equations

1.

x+ y= x + b2 y = x

b

y

(D, D,

2.

Dy),

up

Also, 3.

x

y,

4.

x

107

y

Algebra

50

you you

50 x y

50

x

+y

= 50

+

=

(10%)(50

x

+

+

y=

x b2 y =

+

=

by 100

8x

+ 14y = 500 1,

1,

50, b2

8, 14

500, 108

Solving Systems of Linear Equations

x=IMD

b2

I

1 1

D= 8

Dx=

14

500

= (1)(14) - (8)(1) = 14 - 8 = 6

14

1 1 8

50 500 x

x=

= (50)(14)

- (500)(1) = 700 - 500 = 200

= (1)(500) - (8)(50) = 500 - 400 = 100

y

= 200/6 = 10013

y = Dy/D = 10016 = 50/3

109

Algebra

x

+

+ y = 50

= (0.1)(50)

y = 5013

x=

14%

by

+

= 50

= 50

50 = 50 (O.O8)(x)+

8/3

+

+

= (0.1)(50) =

=5

=5

5=5

8.5. Solving systems of three linear equations with three unknown variables

110

Solving Systems of Linear Equations

8.6. Using the elimination method to solve systems of three linear equations with three unknown variables ezimination method

111

Algebra

1. 2. by

3. 4.

5.

by

6.

x, y

+ 4x + +

3y 32 = 4 3y - 42 = 5 2y 52 = 4

x

112

Solving Systems of Linear Equations

+ 3y + 32 = 4 + 3y - 42 = 5

2x

+

x =-8

- 6y -

+

4

- 4y -

+

2x + 3y 32 = 4 6y - = by z.

6y -

= 9

+

= 4 = 13 = 13

by 10.

y.

y= y = 1.3 z by

y = 1.3

- z =9 113

Algebra

7.8 -

=I

9 9

=

x by

y = 1.3

2x + 3y + 32 = 4

+ + + 3.9 - 3.6 2x + 0.3 = 4

=4

4

0.3 2x = 3.7 by 2.

x = 1.85 2x

+ 3y + 32 = 4 + 3y - 42 = 5

x = 1.85, y = 1.3

=

by 2x + 3y + 32 = 4

+

+

3.7 + 3.9 - 3.6 = 4 7.6 - 3.6 = 4

=4

114

=

Solving Systems of Linear Equations

- 2+ ~3y - 42 = 5 -2(1.85)+ 3(1.3) - 4(-1.2) 5 -3.7+ 3.9+ 4.8= 5 0.2 + 4.8= 5 5=5 =I

4x + 2y + 5z = 4 4(1.85)+ Z(1.3)+ 5(-1.2) =4 7.4+ 2.6 - 6 = 4 10 - 6 = 4 4=4

8.7. Using the substitution method to solve systems of three linear equations with three unknown variables substitution method

two

115

Algebra

1.

on

on

2.

or

3.

by

+ 2y - 22 = 2~ - 6y + = 6 4x

y.

+

4x 2y - 22 = -2 2y = 22 - 4x - 2

by 2. y = (1/2)(2z - 4x - 2)

116

4x

2z

Solving Systems of Linear Equations

y

3x+2(z-2x-l)+z=-l 3x+ 22 - 4x - 2 z = -1

+

- x + 32 = -1 2 - ~ + 3 ~ = 1 y

x. 2~ - 6(z - 2~ - 1) + 6~ = 6 2~ - 6~ + 1 2 ~6 + 6~ = 6

+

14x + 6 = 6 6 14x = 0

by 14. x=o

z,

-x + 32 = 1. - ~ + 3 ~ = 1 -0+ 32 = 1 32=1 by 3. z = 1/3

x =0

117

Algebra

x =0

z=

y

-1 -

y= y= y= 4x

y.

-1

+ 2y - 22 = - 6y + 6~ 0, y = -

z=

by

4x

+ 2y - 22 = + -

=

=

= =

3x

+ 2y + z = + + + 113 =

=

=

-1= 118

Solving Systems of Linear Equations

+

2~ - 6y 62 = 6 2(0) - 6(-2/3)+ 6(1/3) = 6 12/3 613 = 6 4+2=6 6=6

+

8.8. Using the matrix method to solve systems of three linear equations with three unknown variables matrix method

on

on

119

Algebra

1.

by on

b

2.

by

3. by 4.

for

5.

by

+

x b2 y b3

+

z= = d2 z = d3

120

Solving Systems of Linear Equations

upper triangular matrix,

1 0

1

c2

d2

by

-

= -2

121

b3 b2

Algebra

1 1 1 - 1 2 - 1 1 0 -1 1 -1 - 2 1

1by -2

2 -2 + 2

0

1

+

-1 0

1 1 0 -3 0 2

-2 -1

-2

-3

-1

1

row 3

-2 x 2

1 . 0

1 1 -1 1 -1 - 2

2

2 2

1 2

row3

0

-1 2 0 - 3

1 -1

122

Solving Systems of Linear Equations

1 0

1

1

-

1

2

0

2

-1 2

+

0 0

2

0

-1

3.

0 -1

2

-1

-1 2

(-1)

1.

1 0 0

1 1

1-1 1 1 -1 2 b3

3

3. 0 3 3 + 0 -3 -1

0

1 0 0

0

1 1 0

2

3 2

3

5

3

1-1 1 1 2 5 123

2

Algebra

1,

c3

1 0 0

1 -1 1 1 1 5/2

1 1 0

+ ly + 12 = -1 ly+lz=l lz = 5/2 z = 5/2.

z = 5/2 ly+ lz= 1 y + 5/2 = 1 5/2

y = 1- (5/2) y=2/2 - 5/2 y = -312 y = -3/2

x + y + z = -1 x + (-3/2)+ (5/2)= -1 x + 2/2 = -1 x+l=-1 1 x = - 1 -1

z = 5/2

124

3 by 2.

Solving Systems of Linear Equations

x=

+ y - z = -2 x=

y=

z=

by

+ z= + + y

=

-2+ 1= =

2x z =o 2(-2)- (-312)+ 512 = 0 -4+ 812 = 0 -4+4=0

-x + y - z = -(-2) + (-3/2)- 512 = -2 2 - 812 = -2 2-4=-2 -2 =

125

Algebra

8.9. Using the method of determinants of a square matrix to solve systems of three linear equations with three unknown variables A square matrix

3 by 3

126

Solving Systems of Linear Equations

2

by 2

method of determinants of a square matrix, Cramer’s rule

1. x+ x b2 y + x + b3 y +

+

z= z = d2 z = d3

3 by 3

2.

Dy

D,),

b2 b3

127

Algebra

Dx=

Dy=

=

128

Solving Systems of Linear Equations

x, y

3.

z

x=DX/D y=DYD z=DJD by

4.

x, y

z

3 ~ + 2 ~ - 2 ~ = 8 - 6y 6~ = 2 1 0 ~8y 1 0 =~ -8

+

+ +

b, d 3 4 10

2 -6 8

-2 6 10

3(-60 - 48) - 4(20 - -16) + lO(12 - 12) = 3(-108)-

4(36) + lO(0) = -324 - 144 = -468 129

Algebra

8

2 -8

3 4 10

2 -6 8

8 2 -8

-2 6 10

-2 6 10

130

Solving Systems of Linear Equations

3 Dz= 4 10

2 -6

8 2

8

-8

+ lO(4 - -48) = 3(32) - 4(-80)+ lO(52) = 96 + 320 + 520 = 3(48 - 16) - 4(-16 - 64)

=

936

Calculate x, y and z. = -9361-468= 2 y = DylD = 4681-468 = -1 z = DzlD = 9361-468= -2

x=

Therefore the solutions to: 3 ~ + 2 ~ - 2 ~ = 8 4 ~ - 6 ~ + 6 ~ = 2 1 0 ~8y + 1 0 =~ -8 are x = 2, y = -1 and z = -2.

+

Check the results by substituting into the original equations. The first equation: 3x + 2y - 22 = 8 3(2) 2(-1) - 2(-2)= 8

+

131

Algebra

6 + -2 + 4 = 8 4+4=8

+ 6~= 2 4(2)- 6(-1)+ 6(-2)= 2 8 + 6 + -12=2 14 + -12= 2 4~ - 6y

IOX + 8y + 10 z = -8

lO(2)+ 8(-1)+ 10(-2) = -8 20 + -8+ -20= -8 12 + -20 = -8 -8 = -8

on

132

Chapter 9

Working with Coordinate Systems and Graphing Equations 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 on

133

Algebra

9.1. Introduction and definitions on on by

real number line. on coordinate on

by two-dimensional coordinate system. by three-dimensional coordinate system. on

on

by on

rectangular coordinate x

systems A x, y

cylindrical

spherical.

134

y,

Coordinate Systems and Graphing Equations

on

on on

Apoint on

-4

-3 -2 -1

0

1 2 3 4

A point on planar coordinate system

X axis

Y

on

on x = 1, y = 1.

135

Algebra

on

3on

x = 3, y = -3,

= 3.

on

-4 -3 -2 -1 0 1 2 3 4 -3

136

Coordinate Systems and Graphing Equations

on on 4

3 2

0

-3 -2 -1

'I

1 2

3

Xaxis

-31

Y

-3 on 2 on

on

(-3,

on

A A A A

x

x

on on

y y

137

Algebra

on on

+3 on

-3 on

+3 on

by

(x, y, z) = (3, -3,3).

A Line

by on on

axis

138

Coordinate Systems and Graphing Equations

Perpendicular Lines

(90")

ParaLZeL Lines do

9.2. Graphing linear equations

by on

A line

x

on

A Linear equation

139

by y

Algebra

A

by

1.

2. by

y

x,

x

y. by

3.

4. by x,

y

on

- 6y -

4~-

- 6y -

8.

+8 140

Coordinate Systems and Graphing Equations

- 4 - ~8y = 16 x = 0, -4(0) - 8y = 16 = 16 y = -16/8 -2

y.

y = 0, - 4 - ~8(0) = 16 - 4 =~16 x = -1614 x = -4

x.

-4.

4

3 2

by

x -4x - 8y = 16. on

x=1

y.

141

y,

Algebra - 8y = 16

-4 -

16 = 20

y = - 2018 = - 2 1/2 - 2 1/2>

An

Ax +

1.

on

=C

by

2.

x y. by

3.

by

4.

on

y =x

y.

x x = 1,

y.

y=3 3).

142

+ 2.

Coordinate Systems and Graphing Equations

by

y = x + 2.

3=1+2 3=3 x = -1, y=-1+2 y=l

y =x

by

2=0+2 2=2 x = 2,

y

y.

by 1=-1+2 1=1 x = 0, y=o+2 y=2

x

+ 2.

(-1,l). x y

y.

(0,2). x y y = x + 2.

y.

y=2+2 y=4

by

y = x + 2.

(2,4). x y

143

Algebra

x=

y.

by y =x

(1,3),

+ 2.

x

y

(0,2),

X axis

Y axis

b

To graph an inequality, on

144

(-2,O>,

Coordinate Systems and Graphing Equations

>

<,

on 2

I, on

on on on

9.3. Slope of a line

on

slope

by by by first

145

Algebra

second by

first second m. equation for the slope of a line

x2 - x1

4 3 2

(-4,0)

-2),

- y1 x2 - x1 XI,

y1, x2

y2.

Coordinate Systems and Graphing Equations

-2 - 0 0 - -4

-2 - - -1 4 2 -1/2.

= y2

-

x2 - x1

on

147

Algebra

- XO)

y - yo

(xo, on

(xo, yo), yo). (x

y - yo =m(x -

b

yo.

x x :

148

y,

y,

Coordinate Systems and Graphing Equations

find the equation of a line when the slope of the line and one point on the line are identified: y - yo = - xg) on

+

y - 4=

find the slope (m) and y-intercept (6) of a line when the equation known: 2x

+ 6y = 12, y=

y= y=

+

+2 y =m x b =2

=

+ b,

find the equation of a line when two points on the line are identified: - y1 x - x1 x2 - x1 _.

find the equation of a line when two intercept points (a, 0) on the X-axis and (0, b) on the Y-axis are identified:

149

Algebra

9.4. Graphs of the equations for the parabola

y =ax2

x=

+ bx + + by +

y=

150

+ bx + c ,

Coordinate Systems and Graphing Equations

horizontal-axis parabola

A

axis of symmetry

axip

x, =

x, = 2,

drawn 2 on x,

X-axis.

=2

x

y = a x 2 + bx + y (x,,

151

yv

Algebra

9.5. Graphing quadratic equations quadratic equation

x, =

+ bx, +

x,

yv =

yv

x on y

y,).

x,

An x y

2x2 - 4x + 4 = 2y. by 2.

y. y= y = x2 - 2x

- 4x + 4) +2

152

Coordinate Systems and Graphing Equations

x, =

(x,,y,)

y = ax2+ bx + y=x2-2x+2

b=

= 1.

x,, x, = -b/2a x, = 4-2)/2(1) x, = 2/2 = 1 yv

+

y = x2 - 2x 2 y = (1)2 - 2(1) 2 y=1-2+2 y=-1+2 y=l

+

(1, 1).

x =2 y=x2-2x+2 y = 2 2 - 2(2) 2 y=4-4+2 y=2 (2,2).

x on y. y.

+

153

x,

Algebra

Choose x = 0 and solve for y. y = x2 - 2x 2 y = 02 - 2(0) 2 y=o-0+2 y=2 The pair is (0,2). Choose x = -1 and solve for y. y = x2 - 2x 2 y = (-1)2- 2(-1)+ 2 y=1--2+2 y=3+2 y=5 The pair is (-1,5). Choose x = 3 and solve for y. y = x2 - 2x 2 y = 32 - 2(3) + 2 y=9-6+2 y=5 The pair is (3,5).

+

+

+

+

154

Coordinate Systems and Graphing Equations

3 -5-4 -3 -2 -1

x axis

1 2 3 4 5

-3 -4

-5 -6

Y axis

9.6. Using graphing to solve quadratic equations

ax2

+ bx +

=0

0 y =ax2

y.

+ bx + by

1.

y.

155

Algebra

2. by x, =

y,. x

3.

y x

4.

x2

by y = 0). 2x2 = 2x

by

0 = 2x2 - 2x - 2 y= y. y = 2x2 - 2x - 2 by x, =

x,

=

= 2/4 = 1/2

y,.

156

+ bx +

+ 2,

Coordinate Systems and Graphing Equations

yv=2x2-2x-2 Substitute xv for x. yv = 2(1/2)2 - 2(1/2) - 2 yv = 2(1/4) - 2/2 - 2 yv=1/2-1-2 Using a common denominator of 2, the equation becomes: yv= 1/2 - 2/2 - 4/2 yv=--

1 - 2-4 2

yv = -5/2 Therefore, the vertex point is (1/2, -5/2). Choose x values near the vertex point and solve for the corresponding y values. Choose x = 1 and solve for y. y = 2x2 - 2x - 2 y = 2(1)2 - 2(1) - 2 y=2 - 2-2 y = -2

The pair is (1, -2). Choose x = -1 and solve for y. y = 2x2 - 2x - 2 y = 2(-1)2- 2(-1) - 2 y = 2 - -2 - 2 y=2 The pair is (-1,2).

157

Algebra

Choose x = 0 and solve for y. y = 2x2 - 2x - 2 y = 2(0)2 - 2(0) - 2 y=o - 0 - 2 y = -2 The pair is (0, -2). Choose x = 2 and solve for y. y = 2x2 - 2x - 2 y = 2(2)2 - 2(2) - 2 y=8 - 4 - 2 y=2 The pair is (2,2). Plot points (1/2, -5/2), (1, -2), (-1,2),(0, -2), (2,2),and sketch the parabola. 6 5 4 3 '2

41

1 n

A

158

Coordinate Systems and Graphing Equations

x by y=

x = +1.62 x = -0.62 by 0 = 2x2 - 2x - 2. x = +1.62, 0 = 2(1.62)2- 2(1.62)- 2 0=2(2.62)-3.24 - 2 0 = 5.24- 3.24- 2 0=2-2

x = -0.62, 0 = 2(-0.62)' - 2(-0.62)- 2 0 = 2(0.38) 1.24- 2 0 = 0.76 1.24- 2 0=2-2

+

+

o=o x = 1.62

0 = 2x2 - 2x - 2 x = -0.62.

159

Algebra

9.7. Using graphing to solve two linear equations with two unknown variables by by x

1.

by y. 2.

by x =0 x.

y,

y =0

by y

x

3. 4.

x

y by

5.

by

- By - 8 = 0 by by 4

160

by 2

Coordinate Systems and Graphing Equations

x+y-4=0 x - 2y - 2 = 0

x = 0, O+y-4=0 y=4

x + y - 4 = 0. y.

(0,4). y = 0,

x.

~+0-4=0 x=4 (4,O). (0,4) x = 0, - 2y - 2 = o -2y = 2

y = 0,

x - 2y - 2 = 0. y.

x.

x - 2(0) - 2 = 0 x-2=0 x=2 (2,

161

(4,O).

Algebra

-1)

(2,O). by

2

1

I Y

(3 1/3, 2/3). 2x + 2y - 8 = 0 4x - 8y - 8 = 0 x = 3 1/3 = 10/3

y = 2/3 by

10/3 + 2/3 = 4 12/3 = 4

162

Coordinate Systems and Graphing Equations

10/3- 2(2/3)= 2 -10 -3

4 =2 3

2=2

A by

A no by

do

A for by

163

Algebra

9.8. Examples of other equation forms that graph to shapes on a coordinate system

Ellipses

(0,O) r)

r)

-

A*

+-

b2

- 1

A,

An by

x y y x

164

Or

x =0 y =0

Coordinate Systems and Graphing Equations

(0,-3):

(2,0),(-2,O),(0,3) 6

5 4

-4

-5 -6

Y axis

circle

(0,O)

A=

A,

x2 + y2 =

=

A

(p, q)

by (x -

+ (y -

=

165

Algebra

x y

x =0 y =0

y x

x

(2, O), (-2,O>, (0,2)

(0, -2):

-3 -4

-5 -6

Y

166

Coordinate Systems and Graphing Equations

hyperbola

(0,O) 4 x 2 X2

+

=C

+

=

- = 1 b2

A

(p,

by

(x-pl2 b2

y

x

167

Algebra

A A

asymptotes.

+

+

=0

=0

xy=k k

k

Y

Y

L

J

X

3

168

6

25 22 76 51

20 8

8 5, 28

134 106, 127 42

39

43

50

43

105, 127 37 3 by 3

126

31

9 72 46

69, 73 76 87

13

16 22, 23

22,

168

23, 37

151

7

49

74 69

55

62

55 by

164

55

by

9 26

55 39, 76

169

98, 111 164 7 9

10, 46 10, 46 7

8 146

26, 28 138, 139

166 167 164 145

11, 139, 140

39

9 106 119 72 25

87, 92

105 61 39

127

ax2 4 bx 11 54 43

4

83 18

62

19 7 76

73

76

53 144

53 139

53

152

54 164

54

150

12, 82 9

10, 46

10, 46 150, 152, 156 139 15 12 139 on 136

164 10, 46 144 20

170

on

72 126

138 on a

90 43 137

103, 115 52

11, 49

14

6

14, 15

69 82, 150,

163

152, 155

10, 49 115, 127

92

110 96, 110, 111, 115, 119, 127 50, 62 96 unknown 96, 103, 105

83 87 90 87

43

by

160

7 134 87

120

74

26 134 87

8 13 14

145

7 6

155

12

96

6

by

17

160

171

Math series, Basic Math and Pre-Algebra Introduction

1

Chapter 1 Numbers and Their Operations

5

1.1.

6

-

1.2. 1.3. 1.4. 1.5.

1

13

1

15 22 35 43 45 46 47

1.6. 1.7. 1.8. 1.9. 1.10.

odd

1.11. 48

1.12. 51 52 53 54

1.13.

1.14. 1.15.

172

1.16. 1.17. 1.18. 1.19. 1.20.

>, <,

>, I

54 55 59 62 66

Chapter 2 Fractions

67

2.1. 2.2. 2.3,

68 69

2.4.

10

76

2.5. 2.6. 2.7.

c c

1 1

2.8 2.9

78 82 83

Chapter 3 Decimals

81

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

87 90 93 95 97 98 99

Chapter 4 Percentages

101

4.1. 4.2. 4.3.

101 102

105 107 109

4.4. 4.5.

173

Chapter 5 Converting Percentages, Fractions and Decimals 5.1, 5.2. 5.3. 5.4.

5.5. 5.6.

to

111 111 112 113 113 115 117

Chapter 6 Ratios, Proportions and Variation

118

6.1. 6.2. 6.3.

118 120 121

Chapter 7 Powers and Exponents

123

7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11.

123 124 125 126 126 128 128 129 130 131 131

Chapter 8 Logarithms

133

8.1. 8.2. 8.3. 8.4. 8.5.

133 135 136 138 139

8.6.

140

174

Chapter 9 Roots and Radicals

145

9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7. 9.8. 9.9. 9.10. 9.11. 9.12.

146 147 149 152 153 154 155 155 156 156 157 158

by

Chapter 10 Important Statistical Quantities

159

10.1. 10.2. 10.3. 10.4. 10.5.

159 16 1 163 164

Index

170

Appendix

173

164

175

Table of Contents for the third book in the Master Math series, Pre-Calculus and Geometry Introduction

1

Chapter 1 Geometry

3

1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

4 10 13 18 22

29 1.7.

1.8.

34 40

Chapter 2 Trigonometry

43

2.1. 2.2. 2.3.

43 44

2.4. 2.5.

52 53 of 54

2.6. 2.7.

58 60

Chapter 3 Sets and Functions

62

3.1. 3.2.

62 65

176

Chapter 4 Sequences, Progressions and Series

70

4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.

70 71 73 74 77 80 86 88 92

Chapter 5 Limits

94

5.1. 5.2.

94 98

Chapter 6 Introduction to the Derivative

104

6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10.

105 110 113 114 114 118 119 120 125

p

128 131

6.11. 6.12. (minimum

134 142

6.13. 6.14. An

143

177

Chapter 7 Introduction to the Integral

146

7.1. 147 7.2. 150 152 153

7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9.

156 160 162 164

odd

165

by

Index

169

Appendix

175

178