01 - College Algebra 01 [Compatibility Mode]

8/14/2019 01 - College Algebra 01 [Compatibility Mode]

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Cour se ou t l i ne

ST

Cou r se t i t l e : COLLEGE ALGEBRA

Cou r se Ob j ect i ve : On successfulcompletion of this course the students will

like solving algebraic equations, solvingsystem of equations, the formation of

quadratic equations, finding the solution ofuadratic e uation and the a lications of

1

all these.


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Algebra

The part of mathematics in whichletters and other general symbolsare used to represent numbers and

equations.

Example:

x + y = z

=

2


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ALGEBRA

Some Basic Definitions

Variable

A characteristic () whichchanges from one individual to theother, e.g. the height of a student

in your class, the temperature ofdifferent cities in Af hanistan

3

attitude of a person, etc.

Variable is denoted by the lower caseletters, e.g. x, y, z, etc.


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Constant ()

A characteristic () whichdoes not change, e.g. the length ofyour class room, the height of a

,1st semester exam, etc.

A constant is denoted by thealphabets like a, b, c, d, etc.

4


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Expression()

It is the combination of operandsand operators, e.g. x+y, 4-17, etc.

, ,operands, while the symbols +, -are called operators. The other

operators are *and .

5

algebraic expression and 4-17 iscalled the arithmetical expression.


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Equation ()

An expression which involves thesign of equality is called equation.

Examples

2x - 5y = 12

x2 + 3x- 5 = 0

x2 + y2 = r2(Equation of circle centered at the

6

or g n av ng ra us r

1/x + 7 = x/3


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Operations on expressions

Addition()

Add x + 2y and 3x 7y

x + 2y

3x - 7y

____________

4x 5y

7


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Continue.

Subtraction()Subtract 9x 2.5y from 4x + 5y

+9x 2.5y

__________

-5x + 7.5yIn addition and subtraction we combine

+_

8

the similar terms of the expressions.


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Continue.

Ex: Add 7x + 3y + z, -3x + y, and 5x 4z

7x + 3y + z

-3x + y + 0z

5x + 0y 4z

_____________

9


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1

Distributive () property ()

For any real numbers a, b, c

a(b+c)=ab+ac

Examples

1. 2(3+7)=2*3+2*7=20

2. 4(2x-4y)=4*2x-4*4y=8x-16y3. -7(3p-5q)=-7(3p)-7(-5q)=-21p+35q

10

. . . = . .

5. 3(x+y+z)=3x+3y+3z


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1

Group Activity

Simplify

1. 2x2 4x +8x2 3(x+2) x2 -2

Sol: 2x2 -4x +8x2 -3x-6-x2 -2

=(2x2 +8x2 x2)+(-4x-3x)+(-6-2)

= 9x2 -7x-8

2. x2 +2y-y2 +3x+5x2+6y2 +5y

11

3. 2 3+4 x-5 - 2- x-3


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Linear Equation

An equation of the form

ax+b=c (1)

where a,b and c, a0 are constants (realnumbers) is called a linear equation.

1. x+4=7 (a=1, b=4, c=7)

2. 2x-3=5.4 (a=2, b=-3, c=5.4)

3. -(x+3)-(x-6)=3x-4.5

Though this equation is not of the form (1), butit is still a linear e uation since it can be

12

rewritten as

5x-4.5=3 (a=5, b=-4.5, c=3)


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1

Solution of a linear equation

By the solution of a linear equationwe mean to find that value of theunknown x which satisfies the

.

Ex

1. x=3, is the solution of x-2=12. x=-6 is the solution of 2x+1=x-5

13

3. x=11/2 is the solution of3(x+2)=5(x-1)

4. y=9 is not a solution of 2y-3=5


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How to solve a linear equation

We can solve a linear equation by1. Substitution method2. Graphical method3. Algebraic method

ac me o s exp a ne y means oexamples.

The substitution method

In this method we make a guess for thesolution of the linear equation, we then

14

pu e guesse va ue n e equa onand check whether the guess is corrector not?


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1

Continue.

Examples

Let us try to solve the equation 4x+2=4 An accurate uess forthe solution is x=1/2, wesubstitute this value in the given

equation

15

+ =

2 + 2 =4

4 = 4


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1

Continue.

The substitution method is generallya time consuming method, i.e.sometimes it takes very long timeto find an accurate uess.

The graphical method

In this method we sketch the graph

of the linear equation, and then findthe oint where the ra h cuts the

16

x-axis, such a point is called the x-intercept . This method is also nota very good method.


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The algebraic method

In this method we use the algebraicoperations to find the solution. Themethod is described in the followingexamples.

Exam les (Exercise Set 2.3, p#103)Solve the following equations and check

your solution.1. 2x=6Sol: Given the equation

17

Dividing both sides by 2x=2

26

2

3


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1

Continue.

Thus the solution is x=3.

Q16. x/3 = -2

Sol: Multiplying both sides by 3, we get

x = -6 which is the desired solution.

Q19. -32x = -96

Sol: Dividing both sides by -32, we get

x=3 as the required solution.Q24. -x = 9

18

Sol: Multiplying both sides by -1, we getthe solution x = -9.


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Continue.

Q25. -2 = -y

Sol: Multiplying both sides by -1, we get

y = 2 as solution.

13x = 10.Sol: Dividing both sides by 13, we get

x = 10/13 the required solution.

Q52. -2x = 3/5Sol: Dividing both sides by -2, we get x =

19

-3/10, which is the required equation.


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A review Session

What is a Number?

Even no.

Odd no.

Real no.

Rational no.

Irrational no.

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Numbers ...

Real Nos.

Rationalintegers

Negativeintegers

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Numbers...

Integers:

{....-3, -2, -1, 0 1, 2, 3, ....}

Natural no.

{1,2,3...}

ega ve n eger os ve n eger

.

{0, 1, 2, 3....}

22


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Exercise

Divide the following into theirrespective set of number.

a) Natural Nos.

.

c) Integers.

d) Rational Nos.

e) Irrational Nos.

23

}

5,7,9.2,7

4,9.0,3,96,

2

14,5.0,6


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Variable

Constant

Expression

Equation

Operands

Operators

A ge raic Expression

Linear Expression

24


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Review...

25


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Continue.

Q66. When solving the equation 3x = 5. Would youdivide () both sides of the equation by 3or by 5? Explain.

Q67. When solving the equation -2x = 5. Wouldou add 2 to both sides of the e uation or divide

both sides of the equation by -2? Explain.

Q69. Consider the equation 4x = 3/5. Would it beeasier to solve this equation by dividing both

sides of the equation by 4 or by multiplying bothsides of the equation by , the reciprocal of 4?

26

.problem.


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Exercise Set 2.4 (9-65), p# 110

Solve each equation.11. -2x-5=7Sol: Given

-2x-5=7ng o o s es

-2x-5+5=7+5-2x=12

Dividing both sides by -2-2x 2=12 -2 => x=-6

27

The desired solution.


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Continue.

25. -4.2 = 2x + 1.6

Sol: We have

-4.2 = 2x + 1.6

Subtractin 1.6 from both sides

-4.2 - 1.6 = 2x + 1.6 1.6

-5.8 = 2x

Now dividing both sides by 2-5.8/2 = 2x/2 => -2.9 = x (Using calculator)

28

Or x = -2.9 as desired.


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Continue.

Q33. x + 0.05x = 21

Sol: 1.05x = 21

Dividing both sides by 1.05 (the coefficient of x)

x = 21/1.05 => x = 20 as required.

- -36. . . .

Sol: Adding 1.4 to both sides

0.6x = -2.3 + 1.4

0.6x = -0.9Dividing through out by 0.6

29

x = -0.9/0.6 => x = -1.5


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Continue.

Q38. 32.76 = 2.45x 8.75x

Sol: Given

32.76 = 2.45x 8.75x

=> 32.76 = -6.30x

Dividing both sides by -6.30

x = -32.76/6.30

30

x = -5.2 t e answer.


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Continue.

Q46. -2(x+4) + 5 = 1Sol: Using distributive property

-2x 8 + 5 = 1or -2x 3 = 1

ng o o s es-2x -3 + 3 = 1 + 3

i.e. -2x = 4

Now dividing both sides by -2x = -4 2

31

or x = -2 as desired.


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Continue.

Q58. 0.1(2.4x + 5) = 1.7Sol: given the equation

0.1(2.4x + 5) = 1.7Using distributive property0.24x + 0.5 = 1.7Subtracting 0.5 from both sides0.24x = 1.7 0.5

0.24x = 1.2

Finally dividing both sides by 1.2x = 1.2/0.24

32

=the required answer.


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Continue.

Q65. 5.76 4.24x 1.9x = 27.864

Sol: Given that

5.76 4.24x 1.9x = 27.864 5.76 6.14x = 27.864

.

5.76 - 5.76 6.14x = 27.864 5.76

-6.14x = 22.104

Finally dividing both sides by -6.14

-6.14x/-6.14 = -22.104/6.14

33

.

Which is the desired result.


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Challenging problems

Solve each equation.

1. 3(x-2) (x+5) 2(3-2x) = 18

2. -6 = -(x-5) 3(5+2x) 4(2x-4)

3. 4[3 2(x+4)] (x + 3) = 13

34


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Solving linear equations with the

variable on both sides of the

equation

Hints:1. Use the distributive property to remove the

parentheses.2. Combine like terms on the same side of the

equal sign..

the variable on one side of the equation and allthe terms not containing the variables on theother side of the equation.

4. Isolate the variable using the multiplicationproperty, this gives the solution.

35

. ,the original equation.


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Exercise Set 2.5(9-54), p#118/119

Solve each equation.

9. 4x = 3x + 5

Sol: Given the equation

Combining the terms involvingvariables on one side if the equation

4x 3x = 5

36

x =

Which is the required answer.


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Continue.

15. 15 3x = 4x 2xSol: Given that

15 3x = 4x 2xShifting 3x to the right side of the

.15 = 4x 2x + 3x

15 = 5x

Isolating x using the multiplicationproperty

37

15/5 = xor x = 3 as required.


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Continue.

25. x 25 = 12x + 9 + 3x

Sol: We have

x 25 = 12x + 9 + 3x

Shifting x to the right & 9 to left of theequat on

-25 9 = 12x + 3x x

-34 = 14x

Using multiplicative property

-34 14 = x

38

or x = 2.428 as required.


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Continue.

28. 4r = 10 2(r-4)Sol: Given that

4r = 10 2(r-4)Using distributive property4r = 10 2r + 8Shifting 2r to the left of the equation4r + 2r = 10 + 8

6r = 18

Dividing both sides by 6r = 18/6

39

=as required.


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Continue.

34. 3y 6y + 2 = 8y + 6 5y

Sol: Given the equation

3y 6y + 2 = 8y + 6 5y

-3y + 2 = 3y + 6

-equation

2 6 = 3y + 3y

-4 = 6y

Dividing both sides by 6

- =

40

or y = -2/3 or y = -0.667


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Continue.

33. 0.1(x + 10) = 0.3x -4Sol: Given

0.1(x + 10) = 0.3x -4Using the distributive property0.1x + 1 = 0.3x - 4

side of the equation & constant on the other side

0.3x 0.1x = 4 + 10.2x = 5

Using multiplicative property to isolate xx = 5/0.2

=

41

Which is the required solution.


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Continue.

36. 5(2.9x - 3) = 2(x +4)Sol:We are given that

5(2.9x - 3) = 2(x +4)By the use of distributive property, we have14.5x 15 = 2x + 8Shifting 2x to the left & -15 to the right of the

equation14.5x 2x = 8 + 15

12.5x = 23Isolating x=

42

.or x = 1.84 as desired.


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Continue.

37. 9(-y + 3) = -6y + 15 3y + 12

Sol: Given

9(-y + 3) = -6y + 15 3y + 12

Using the distributive property

- -

Since the same expressionappears on both sides of the equation,therefore the statement is true for all real

values of y. If we continue to solve thisequation further, we arrive at

43

0 = 0.


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A difference

Note that an equation is true on l y fo r a spec i f i c va lu e( s ) of thevariable, while the identity is true

.previous question the givenequation is an identity as it is truefor all values of x. Further, every

44

n oeve ry equation is an identity.


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Continue.

39. -(3 - p) = -(2p + 3)

Sol: Given that

-(3 - p) = -(2p + 3)

Using distributive property

- -

Shifting 2p to the left(???) & -3 to the rightof the equation

p + 2p = -3 + 3

3p = 0

= =

45

Which was required.


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Continue.

51. 5 + 2x = 6(x + 1) 5(x - 3)Sol: We have

5 + 2x = 6(x + 1) 5(x - 3)Using distributive property, we have

+ x = x + x +Or 5 + 2x = x + 21

Shifting x to the left & 5 to the right

2x x = 21 5x = 16

46

as required.


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Challenging Problems

1. Solve

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

2. Solve

4(2x-3)-(x+7)-4x+6=5(x-2)-3x+7(2x+2)

3. Solve

4 - [5 - 3(x + 3)]= x - 3

47


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Ratio and Proportion

Definition

A Ratio is a quotient of twoquantities. Ratios provide a way of

quantities. The ratio of a and b iswritten as

a to b, a:b or a/b

48

ratio.


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EXAMPLES

1. If an algebra class consists of 32males and 13 females, find

(a) The ratio of males to females.

(b) The ratio of females to the entireclass.

Sol: (a) 32:13

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Exercise Set 2.6, p#127

The results of an English examination are 5As, 6 Bs, 8 Cs, 4 Ds and 2 Fs. Writethe following ratios.

1. As to Cs.. .

3. Ds to Fs.4. Grades better than C to total grades.

5. Total grades to Ds.6. Grades better than C to grades less than

50

.(Try these!)


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Continue.

Determine the following ratios. Write each ratioin lowest term.

7. 5 feet to 3 feet.8. 60 dollars to 80 dollars.9. 20 hours to 60 hours.

10. peop e to peop e.11. 4 hours to 40 minutes.12. 6 feet to 4 yards. (1yard=3feet)

13. 26 ounces to 4 pounds. (1pound=16 ounces)14. 7 dimes to 12 nickels. (1dime=10cents,

1nickle=5cents

51

1dime=2nickle)


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Gear Ratio

Gear Ratio= (Number of teeth on the drivinggear)/

(Number of teeth on the driven gear).(1)

15. Driving gear, 40 teeth; driven gear, 5 teeth

Sol: Using eq. (1)

Gear ratio = 40/5= 8/1 i.e. 8:1

52

16. r ven gear, ee ; r v ng gear, ee .

(Try this)


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Proportion

A proportion is a special type of equation.It is a statement of equality between tworatios.

How to denote a proportion?

A proportion is denoted as a :b = c:d (readas a is to b as c is to d). We can alsodenote a proportion as

a/b = c/d, a and d are referred as theextremes and the b and c are referred as

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the means of the proportion.


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Cross Multiplication

If a/b = c/d then ad = bcNote that the product of means equal to the product ofextremes.If any three of the four quantities of a proportion areknown, the fourth can easily be found.

ExampleSolve for x using cross multiplying x/3 = 25/15.

SolutionGiven that

x/3 = 25/15Using cross multiplicationx.15 = 3*25

54

x = 75/15x = 5 (Check the answer!!!!!!!!)


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Exercise Set 2.6 (21-32) p#128

Solve for the variable by cross multiplying.21. 4/x = 5/2022. x/4 = 12/4823. 5/3 = 75/x24. x/32 = -5/4

-. 26. -3/8 = x/4027. 1/9 = x/4528. y/6 = 7/42

29. 3/z = 2/-2030. 3/12 = -14/z

55

. 32. 12/3 = x/-100


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To Solve Problem Using

Proportion

1. Represent the unknown quantity by avariable (a letter).

2. Set up the proportion by listing thegiven ratio on the left side of the equals gn, an e un nown an o er g venquantities on the right side of the equalsign. When setting up the right side of

the proportion, the same respectivequantities should occupy the same

56

respec ve pos ons on e e anright.


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Exercise Set 2.6, P#127

18. In 1970 in the United States, 72,700metric tons of aluminum was used forsoft-drink and beer containers. In 1990this amount had increased to 1,251,900metric tons. Find the ratio of the amountof aluminum used for beer and soft-drinkcontainers in 1990 to the amount used in1970.

57

o : ry .


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Continue.

33. A car can travel 32 miles on 1 gallon of gasoline.How far it can travel on 12 gallon of gasoline?

Ans: Let the distance covered in 12 gallon gasolinebe x miles.

Now

miles/gallon = miles/gallon

Or 32/1 = x/12

Using cross multiplication

x . 1 = 32 * 12

Or x = 384 as re uired.

58


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Inequalities in One Variable

An inequality in one variable is theMathematical statement in which one ormore of the following symbol are used

, greater than symbol

, less than or equal to symbol

, greater than or equal to symbol

The direction of the symbol is sometimes

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inequality.


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Examples of inequalities in one

variable

1. x+3-x+3 (4 is greater than-x+3)4. 2x+6-2 (2x+6 is less than or equal to -2)Prope r t i es Used t o So lve I nequa l i t i es

For real numbers a b and c1. Ifa>b, then a+c>b+c.2. Ifa>b, then a-c>b-c.3. Ifa>b and c>0, then ac>bc4. Ifa>b and c>0, then a/c>b/c5. Ifa>b and c


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Exercise Set 2.7 (1-40), p#137

Solve the inequality and graph thesolution on the real line.

8. -4-x-3Sol: Given

- - -Adding 3 to both sides-4+3-x-3+3

-1 -xMultiplying both sides by -1

61

or x 1 (as -1


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Continue.

12. 6-2x

Sol: This inequality can also bewritten as

-

Dividing both sides by -2

-2x/-2

6/-3Or x -3

62

Which is the desired answer.


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Continue.

15. 12x + 24 < -1217. 4 6x > -519. 15 > -9x + 5024. -2x - 4 -5x + 1228. 2(x - 3) < 4x + 1029. -3(2x - 4) > 2(6x - 12)32. x + 5 x 233. 6(3 - x) < 2x +12

35. -21(2-x) + 3x > 4x + 438. -2(-5-x) > 3(x+2) +4 x

63

39. -40. -3(-2x +12) < -4(x+2) - 6


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Think!!!!!!!!!!!!!

41. When solving an inequality, if you obtain theresult 3 < 5, what is the solution?

42. When solving an inequality, if you obtain theresult 4 2, what is the solution?

43. When solving an inequality, if you obtain theresult 5 < 2, what is the solution?

44. When solving an inequality, if you obtain theresult -4 -2, what is the solution?

45. When solving an inequality, under whatconditions will it be necessary to change the

64

Practice Test (1-20), p#141


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3.1 Formulas

Def i n i t i on :

A formula is an equation commonlyused to express a specific relationshipmathematically.

Examp les :

1. The formula for the area of a rectangle is

area = length . Width or A = lw

2. The formula for the perimeter of a square is

perimeter = 4. one side or P=4s

3. The formula for the area of a trian le is

65

area = .base . Height or A=bh


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Simple Interest Formula

A formula used in banking is the simpleinterest formula, which is given byinterest = principal . rate . Time

Or i = prt

invested or borrowed), r is the interestrate, and t is the amount of time of theinvestment or loan.This formula is used to determine thesim le interest i earned on some

66

savings accounts, or the simple interestan individual must pay on certain loan.


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How to use the simple interest

formula?

Examp le1 : Avery borrows $2000 from a bank for 3years. The bank charges 12% simple interest per yearfor the loan. How much the interest will Avery owe thebank?

So l: Given thatThe principal, p, is $2000The rate, r , is 12% = 12/100=0.12

and the time, t , is 3 yearsUsing the formula i = prtPutting the corresponding values

i = 2000(0.12)(3)i = 720

67

The simple interest is $720. Thus Avery will pay$2720, after 3 years. (the principal, $2000 + theinterest, $720).


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Continue.

Example : Amber invests $5000 in savings account whichearns simple interest for 2 years. If the interest earned fromthe account is $800, find the rate.

So l: HerePrincipal (investment) =p=$5000Time =t =2years

= =rate =r=?Using the simple interest formula

i=prtSolving for r, we get r = i/ptPutting the values

r = 800/[(5000)(2)]

68

r = 800/10000r = 0.08

Thus the simple interest rate is 0.08, or 8% per year.


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Exercise Set 3.1, p#153

In Exercise 73 through 76, use the simple interest formula.73. Mr. Thongsophapporr, borrowed $4000 for 3 years at 12%

simple interest rate per year. How much interest did hepay?

74. Ms. Rodriguez lent her brother $4000 for a period of 2years. At the end of the 2 years, her brother repaid the$4000 plus $640 interest. What simple interest rate did herbrother pay?

75. Ms. Levy invested a certain amount of money in a savingsaccount paying 7% interest per year. When she withdrewher money at the end of 3 years, she received $1050 ininterest. How much money did Ms. Levy place in the

savings account?76. Mr. OConnor borrowed $6000 at 7% simple interest per

ear. When he withdrew his mone he received 1800 in

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interest. How long had he left his money in the account?


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Geometric Formulas

Per ime te r : The perimeter, P, is the sum ofthe lengths of the side of a figure.Perimeters are measured in the samecommon units as the sides. For examplethe perimeters may be measured incen me ers, nc es, or ee .

Area : The area, A, is the total surfacewithin the figures boundaries. Areas are

measured in square units. For example,area may be measured in square

70

, ,feet.


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Formula for Areas and Perimeters of

Quadrilaterals and Triangles

Figure Sketch Area Perimeter

Square A = s2 P = 4s

s

Parallelogram A = lh P =2l+2w

w

l

lw

h

71

rapezo = + =a+ +c+

Triangle A =bh P=a+b+c

a h b

a

b

ch


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Formula for Circles

Circle Area Circumference

r

A = r2 C = 2r

72


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Formula for the Volumes of Three

Dimensional Figure

gure e c o ume

RectangularSolid

V=lwh

Right Circular V=r2h

h

wl

y n er

Right CircularCone V=(1/3)(r2h)

rh

73

p ereV=(4/3)(r3)

r


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Formula for finding number of

diagonals of a polygon

To find the number of diagonals d in a polygon of n sides,the following formula may be usedd=(1/2)n2 (3/2)n.

1 2

Example : How many diagonals does an octagon(8 sides) have?Solu t ion : n = 8

therefore, d=(1/2)82

(3/2)8=(1/2)64 12= 32 12

74

= 20


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Exercise 3.1, p#151

Use the formula to find the value of the variableindicated. Use a calculator to save time. Roundanswers off to hundredths.

1. A=s2; find A when s=52. P=a + b + c; find P when a = 4, b = 3, and c =

3. P=2l + 2w; find P when l = 6 and w = 54. A=(1/2)bh; find A when b = 12 and h = 88.

p=i2

r; find r when p = 4000 and i = 211. V=lwh; find l when V = 18, w = 1, and h = 3= = =

75

. , ,P=1000(try the rest of the questions up to 24)


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Continue.

Solve each equation for y; then find the value of y forthe given value of x.

25. 2x + y = 8, when x = 2

28. -3x 5y = -10, when x = 0

30. 15 = 3y x, when x = 3

34. -12 = -2x 3 when x = -4

36. 2x + 5y = 20, when x = -5

Solve for the variable indicated.

38. d = rt, for r

42. V = lwh for w47. 4n + 3 = m, for n

50. Y = mx + b for x

76

57. ax + by = c, for y


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Continue.

Use the formula d = (1/2)n2 (3/2)n, to find

the number of diagonals in a figure with thegiven number of sides.

61. 10 sides

62. 6 sides

Use the formula C = (5/9)(F - 32) to find theCelsius temperature (C) equivalent the given

Fahrenheit temperature (F).63. F = 500

77

64. F = 860


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Continue.

In chemistry the ideal gas law is P = KT/V, whereP is the pressure, T is the temperature, V is thevolume, and K is a constant. Find the missingquantity.

67. T = 10, K = 1, V = 1. , , .

69. P = 80, T = 100, V = 570. P = 30, K = 2, V = 6

The sum of the first n even numbers can be found

by the formula S = n2 + n. Find the sum of thenumbers indicated.

78

. .72. First 10 even numbers.


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Continue.

Use the formulas given in the tables to workexercises 77-90.

GROUP ACTIVITY1. (a) Use the formulas presented in this

section, write an equation in d that can beused to find the shaded area in the figureshown.

(b) Find the shaded area when d = 4 feet.

=

79

d


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Continue.

2. A cereal box is to be made by folding the cardboard along thedashed lines as shown in the figure.

(a) Using the formula

volume = length . width . Height

Write an equation for the volume of the box.

(b) Find the volume of the box when x = 7 cm.

(c) Write an equation for the surface area of the box.

(d) Find the surface area when x = 7 cm.

x3x cm

cm

80

Back

Front

(6x-1)

x

(6x-1)cm


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Continue.

3. Earths diameter is 3 9 6 3 miles and themoons diameter is 2 1 6 0 miles. Themoon travels in an elliptical orbit aroundthe Earth. From the center of Earth tothe center of the moon the minimum

s ance s 2 2 1 ,4 6 3 m es an emaximum distance is 2 5 2 ,7 1 0 miles.Assuming that Earth and moon are

spheres, find (a) the nearest approachof their surfaces, (b) the farthesta roach to their surfaces and c the

81

circumference of the moon.


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Continue.

4. The Pantheon is an ancient building in Romeconstructed about A.D. 126. It is shaped like acircular cylinder with a dome on top. Theoutside circumference of the cylinder is about446 feet.

a Find the radius and diameter of the c lindricalpart of the Pantheon.

(b) If the walls of the Pantheon are 4 feet thick, findthe inside diameter of the floor of the pantheon.

(c) Find the surface area of the marble floor insidethe Pantheon.d If the hei ht of the c lindrical art of the

82

Pantheon (excluding the domed portion) is 120feet, find its inside volume.


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Changing Application Problems into

Equations

Ver b a l Alg eb r a ic 5 more than a number x+5A number increased by 3 x+37 less than a number x-7A number decreased by 12 x-12

The product of 6 and a number 6xOne-eighth of a number (1/8)x or x/8A number divided by 3 (1/3)x or x/3

4 more than twice a number 2x+45 less than three times a number 3x-5

83

Twice the difference of a number and 4 2(x-4)


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Express Relationships between

Two Related Quantities

Two Numbers differ by 3 x x + 3

Johns age now andjohns age in 6 years

x x + 6

One number is six times x 6xthe other

One number is 12% lessthan the other

x x 0.12x

A 25 foot length of wood x 25 - x

84

cut in two pieces

The sum of twonumbers is 10

x 10 - x


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Exercise Set 3.2, p#162/163

Write as an algebraic expression.1. Five more than a number.2. Seven less than a number.3. Four times a number.4. The product of a number and eight.5. 70% of a number x.6. 8% of a number y.8. A 7% sales tax on a car costing p dollars.9. The 16% of the U.S population, p, who do not receive

adequate nourishment.

10. Only 7% of all U.S tires, t, are recycled.11. Three less than six times a number.12. Six times the difference of a number and 3.

85

13. Seven plus three-fourths of a number.14. Four times a number decreased by two.15. Twice the sum of a number and 8.16. Seventeen decreased by x.


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Continue.17. The cost of purchasing x rolls of electrical tape at $4 each.18. The rental fee for subscribing to home box office for x

months at $12 per month.19. The cost in dollars of travelin x miles at 23 cents er mile.20. The distance covered in t hours when traveling 30 miles per

hour.22. The cost of waste disposal for y months at $16 per month.23. The population growth of a city in n years if the city is

growing at a rate of 300 persons per year.24. The number of calories in x ram of carboh drates if each

gram of carbohydrates contains 4 calories.25. The number of cents in x quarters. (Quarter=a fourth of a

dollar)26. The number of cents in x quarters and y dimes.27. The number of inches in x feet.28. The number of inches in x feet and inches.

86

29. The number of ounces in e pounds.31. An average chicken egg contains 275 milligram of

cholesterol and an ounce of chicken contains about 25 mg ofcholesterol. Write the amount of cholesterol in x chickeneggs and y ounces of chicken.


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Continue.

Express as a verbal statement. (There are many acceptableanswers.)

33. x 6 six less than a number34. x + 3 (three more than a number)35. 4x + 1 (one more than four times a number)36. 3x 4 (four less than three times a number)37. 5x 7 (seven less than five times a

number).

39. 4x 2 (two less than four times a number)40. 5 x (a number subtracted from five)41. 2 3x (three times a number subtracted

from two)42. 4 + 6x (four more than six times a number)

87

. -one)

44. 3(x + 2) (three times the sum of a numberand two)


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Continue.Select a variable to represent one quantity and state whatthat variable represents. Then express the second quantityin terms of the variable.

45. Eileens salary is $45 more than Martins salary.46. A boy is 12 years older than his brother.47. A number is one-third of another.48. Two consecutive integers.49. Two consecutive even integers.50. One hundred dollars divided between two people.51. Two numbers differ b 12.52. A number is 5 less than the four times another number.53. A number is 3 more than one-half of another number.54. A Cadilac costs 1.7 times as much as a Ford.

55. A number is 4 less than three times another number.56. An 80-foot tree cut into two pieces.57. Two consecutive odd inte ers.

88

58. A number and the number increased by 12%.59. A number and the number decreased by 15%.60. The cost of an item and the cost increased by a 7% sales

tax.


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HOME WORK

romQuestion 61

to

uestion 68

89

On page # 163


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Continue.Express as an equation.

69. One number is five times another. The sum of two numbers is18.

70. Marie is 6 years older than Denise. The sum of their ages is48.

71. The sum of two consecutive integers is 47.72. The product of two consecutive even integers is 48.73. Twice a number decreased by 8 is 12.74. For two consecutive integers, the sum of the smaller and

twice the larger is 29.75. One-fifth of the sum of a number and 10 is 150.76. One train travels six times as far as another. The total

distance traveled by both trains is 700 miles.77. One train travels 8 miles less than twice the other. The total

distance traveled by both the trains is 1000 miles.79. A number increased by 8% is 92.80. The cost of a car lus a 7% tax is $13,600.

90

81. The cost of a jacket at a 25%-off sale is $65.82. The cost of a meal plus 15% tip is $18.83. The cost of a videocassette recorder reduced by 20% is $215.84. The product of a number and the number plus 5% is 120.


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CHALLENGE

93-104,(p#164)

equation as a

91

.

Bring books in next class


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3.3 Solving Application problems

To solve the word problemo solve the word problem1. Read the question carefully.2. If possible draw a sketch to help visualize the

problem.3. Determine which quantity you are being asked to

in . C oose a etter to represent t is un nownquantity. Write down exactly what this letterrepresents. If there is more than one quantity,

represent all unknown quantities in terms of thisvariable.

92

. .5. Solve the equation for the unknown quantity.6. Answer the question or questions asked.7. Check the solution in the original stated problem.


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Exercise Set 3.3 p# 174

DISCUSSION

93


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Group Activity/Challenging

Problems

1. To find the average of a set of values,you find the sum of the values anddivide the sum by the number of values.

(a) If Paul's first three test rades are 7488, and 76, write an equation that canbe used to find the grade that Paul must

get on his fourth exam to have an 80average.

94

(b) Solve the equation from part (a) anddetermine the grade Paul must receive.


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Continue.

2. At a basketball game Dukeuniversity scored 78 points. Dukemade 12 free throws (1-pointeach . Duke also made 4 times asmany 2-points field goals as 3-points field goals (field goals made

from more than 18 feet from thebasket). How many 2-point field

95

goa s an ow many -po n s egoals did Duke made?


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EXPONENTS

In the expression xn, x is calledthe base and n is called theexponent. xn is read as x to thenth power.

96

BASE


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What an exponent represents??

An exponent tells us how manytimes a number is multiplied withitself.

or examp e means

105 = 10 x 10 x 10 x 10 x 10

= 100000

97

73 = 7 x 7 x 7 = 343


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Continue.

4=

EXPANDED FORM

36EXPONENTIAL FORM STANDARD FORM

98

216

1296


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LAWS OF EXPONENTS

Certain mathematical operation can bedone whenever we have variables whichcontain exponents and equal bases. These

operations are called laws of exponent.

Pr o d u ct R u l e fo r E x p o n e n t sPr o d u ct R u l e fo r E x p o n e n t s

Qu o t ie n t R u l e fo r E x p o n e n t sQu o t ie n t R u l e fo r E x p o n e n t s

Z e r o Ex p o n e n t R u leZ e r o Ex p o n e n t R u le

99

Each one is explained as underEach one is explained as under


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Product ru le fo r ex ponent s

xn . xm = xn+m

Thus when same bases aremultiplied, the exponents are added

xn

. xm

x=n+m

100

Bases are same Exponents are Added


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Examples (Exercise Set 4.1, p# 195)

Simplify.1. x2 . x4 = x6

2. x5 . x4 = x9

3. y . y2 = y34. . = =

5. 32 . 33 = 35 = 2436. x4 . x2 = x6

7. y3 . y2 = y5

8. x3 . x4 = x7

101

9. y4 . y = y5


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Quot ient Rule for Ex ponent s

0, = xxx

x nmn

m

Bases are same

Since division by0is notallowed

102

Exponents are subtracted


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Continue.

EXAMPLES ONUOTIENT

RULE(EXERCISE SET 4.1, P#195)

103

(10 - 20)


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Pow er Rule for Ex ponent s

(xm)n = xm.n

Keeping the base & multiplying the exponents

The power rule indicates that when we raise anexponential expression to a power, we keep the

Exponential expression raise to a power

104

base and multiply the exponents.


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Expanded Power Rule for Exponents

0,0, =

yb

yb

xa

by

axmm

mmm

This rules illustrates that every factor within the

parenthesis is raised to the power outside thearenthesis.

(As division by zero is not allowed)

105


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Zero Ex ponent Rule

0,10 = xx

y e zero exponen ru e, any rea num er, excep ,raised to the zero power equals 1.

NOTE: 00 is not a real number.

106

form.


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Continue.

EXE CISE 4.1

PAGE #

107


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Negat ive Ex ponent s

2533

== xxx

We will develop this rule as follows:Using Quotient rule we have1.

x

25

3 1.. xxxx ==

Again by dividing out common factors,2.

108

.... xxxxxxx

We see that is equal to both & . Therefore

must equal

5

3

x

x 2x2

1

x

2x

2

1

x


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Negative Exponents Rule

0, = xx

xm

m

It means that when a variable or a number israised to a negative exponent, the expressionmay be rewritten as 1 divided by the variableor number to the positive exponent.

N o te : When we are asked to simplify an exponentialex ression, our final answer should contain no ne ative

109

exponent.Also when a factor is moved from the denominator to thenumerator or from numerator to the denominator, the signOf the exponent changes.


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Continue.

DISCUSSION

110

P#202/203


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System of Linear equation

of linear equations.

Sometimes in business we deal with manyvariables and unknown quantities. For example acompany consider overhead cost, cost of material,

, ,price of the item, and a host of other items whenseeking to maximize their profit. The business may

express the relationship between the variables inequation or inequalities. These equations or

111

inequalities. The solution of the system ofequations or inequalities gives the values of thevariables for which the company can maximizeprofit.


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EXAMPLES

63

1072

=+

=+

yx

yx

123

06174

=+

=+

zyx

zyx

(2 variables, 2 equations) (3variables, 2 equations)

105

54

1243

32

321

=+=

=++

xx

xx

xxx

653

074

132

21

21

21

==+

=+

xxxx

xx

112

23 21 =+ xx

(3 variables, 3 equations) (2 variables, 4 equations)

In this course we will study linear system of equations in two variables


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Solution of System of Equations

By the solution of linear system we mean theorder pair(s), which satisfy all the equations inthe system simultaneously.

e.g Consider the system

It has the solution 1 6 since this order air

42

5

+=

+=

xy

xy

113

satisfy both the equations in the system. But(2,7) is not a solution to the system as it satisfythe first equation but not the second.


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Continue

A system can be solved by the followingmethods:

1. The graphical method

2. The substitution method

3. The addition method

While solving a linear system, we faceone of the following three situations:

a) The system is consistent (sol: exists)

114

b) e sys em s ncons s en no so :

c) The system is dependent (infinite sol:)


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Continue.

Consistent, Inconsistent, Dependent systemy

y

y

.

Solution

Line 1 Line 2 Line 1

Line 2

xLine 2

x(a)

115

Line 1(b)

(c)

Inconsistent System

Dependent System


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Solution by Graphical Method

To solve a linear system of equationsgraphically, we graph each equation anddetermine the point(s) of intersection.

graphical method, because often thesolution by this method may beincorrect since we have to estimate the

116

.Thus we follow the other two methods.


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Solution of System of equations by

Substitution

To solve a system of equations bysubstitution

1. Solve for a variable in either equation. (To avoidworking with fractions, prefer to solve for thevar a e w t t e numer ca coe c ent o .

2. Substitute the expression found for the variablein step 1 into the other equation.

3. Solve the equation in step 2 to find the value ofone variable.

117

4. Substitute the value found in step 3 into theequation found in step1 to find the othervariable.


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Exercise Set 8.2 (p# 454)

1.

132

42

=

=+

yx

yx

Solve the system of equations by substitution

(1)

(2)

o : o v ng or x

42 += yx (3)

Putting this value of in (2)x

1348

13)24(2

=

=

yy

yy

118

1

178

=

=

y

y

Putting in (3), we get1=y 2=xThus the solution is (2,1)


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Continue.

DISCUSSION ON

8.2

119

(P#454/455)


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Solution by AdditionMethod

This method is also called theelimination method.

This is often an easier method to solvea linear system of equations.

In this method we add or subtract thegiven equations to get a third equation

which contains only one variable.

120

obtain one equation containing onlyone unknown.


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TheMethod

. ,variables to the left side of the equal sign andthe constants to the right side of the equal sign.

2. If necessary, multiply one or both equations bya constant(s) so that when the equations are

variable.

3. Add the equations. This gives a single equation

which contains only one variable.4. Solve the equation obtained in step 3.

121

5. Put the value obtained from step 3 into either ofthe original equation. Solve this equation for theremaining variable.


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DISCUSSION

SET 8.3,

Documents

01 - College Algebra 01 [Compatibility Mode]