1.1 SKM & PP 1 Getting Started with Algebra What isAlgebra? is a branch of mathematics in which symbols, usually letters, are used to represent quantities

1.1 S KM & PP 1

Getting Started with Algebra

What isAlgebra ?

is a branch of mathematics in which symbols, usually letters,are used to represent quantities that can be replaced by a number oran expression.

Algebra

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Who invented Algebra ?

is a reasoning skill and language that developed and evolved along with civilization.

No one person invented

Algebra

Algebra

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Where did the word

Algebraoriginate ?

The word is from Kitab al-Jabr wa-l-Muqabalawhich was a book written in approximately 820 A.D. by a Persian mathematician.

Algebra

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Variables

A is

variable

a letter used to represent various

numbers.

“x” is frequently used as the variable, but many

other letters can be used.

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Variables

For example, jean sizes are often given by waist

and leg inseam measurements.waist

measurement

leg inseammeasurement

1.1 S KM & PP 6

Define each Variable

We must always define what quantity or measurement

the letter represents.Here are three examples:

= waist measurement

= leg inseammeasurement

= unknown number

1.1 S KM & PP 7

Constants

A is

constant

a letter used to represent a number that doesn’t change its value

in the problem. For example:

= the speed of light in Einstein’s equation E = mc2

= “pi” = 3.1416….

1.1 S KM & PP 8

AlgebraicExpressions

An algebraic expressionis a mathematical phrase

using variables, constants, numerals, & operation signs. An algebraic

expressionwill NOT have any of the

following symbols:

= > < > <

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Algebraic Expressions: Examples

5x x is the variable.

+ is the operation

5 is a numeral and a constant.

5x algebraic expressionis the

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p3p is the variable.

. is the indicated operation


p3 algebraic expressionis the

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z9 z is the variable.

- is the operation


z9 algebraic expressionis the

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y5

y is the variable.

÷ is the operation


y5

algebraic expressionis the

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y5x2

x and y are variables.

and + are operations

2 and 5 are numerals and constants.y5x2 algebraic expressionis the

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2y5x

x and y are variables.

+ ÷ are operations

5 and 2 are numerals and constants.

2x5x

algebraic expressionis the

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Substitution

When a variable is replaced with a

numerical value, that is called substitution.

Sometimes, in higher mathematics, a variable is

replaced with an expression. That is also

called substitution.

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Evaluate an Algebraic Expression

When a numerical value is substituted into an algebraic expression

and then simplified, that is called

evaluatingthe expression.

Evaluating means you will compute a numerical

value.

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Evaluate an Expression:Example 1a

5x Evaluate

4x

5)4(

9

when

5x

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Evaluate an Expression:Example 1b

5x Evaluate

0x

5)0(

5

when

5x

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Evaluate an Expression:Example 1c

5x Evaluate

3x

5)3( 8

when

5x

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Evaluate an Expression:Example 1d

5x Evaluate

5x

5)5(

0

when

5x

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Evaluate an Expression: Example 2

p3Evaluate

5pa) when

p3 )5(3 152p b) when

p3 )2(3 6

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Evaluate an Expression: Example 3

z9 5zwhen

z9

459

Evaluate

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Evaluate an Expression:

Example 4a

y5

5ywhen

y5

55

1

Evaluate

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Evaluate an Expression:

Example 4b

y5

0y when

y5

05

undefined

Evaluate

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Evaluate an Expression:Example 5

y5x2 4x When and 1y

y5x2 )1(5)4(2

)5(813

Evaluate

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Evaluate an Expression: Example 6a

25

yxEvaluat

e

4x When and 1y

25

yx

2)1(5)4(

39

3

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Evaluate an Expression: Example 6b

2y5x

Evaluat

e

8y

2y5x

2)8(5)7(

612

2

andWhenx 7

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Evaluate an Expression: Example 6c

2y5x

Evaluat

e

4x When and 2y

2y5x

2)2(5)4(

09

undefined

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Application: Area of a Rectangle

The AREA of a Rectangle

Area = length x width

A=l.w

cm6w

cm16l

cm6cm16A 2cm96A

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Translating: English into Algebra

In order to solve problems, English phrases must be

translated into the language of algebra.

The following slides list keywords which can

help us translate.

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English & Algebra ADDITION

The following words translate as ADDITION:

•Plus•Sum•Add•Added to•Total•More than•Increased by

1.1 S KM & PP 32

X + 7

The following phrases would translate to :

•A number plus seven

•The sum of a number and seven

•Add a number and seven

•Seven added to a number

•The total of seven and a number

•Seven more than a number

• A number increased by seven

7x

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English & Algebra SUBTRACTION

The following words translate as SUBTRACTION:

•Minus•Difference•Subtract•Subtracted From•Take away•Less Than•Decreased by

1.1 S KM & PP 34

X - 7


•A number minus seven

•The difference of a number and seven

•Subtract a number and seven

•Seven subtracted from a number

•Seven take away a number

•Seven less than a number

• A number decreased by seven

7x

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English & Algebra MULTIPLICATION

The following words translate as MULTIPLICATION:

•Multiplied by•Multiply•Product•Times•Of

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7x

The following phrases would translate to : x7

•A number multiplied by seven

•Multiply seven and a number

•The product of a number and seven

•The product of seven and a number

•Seven times a number

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English & Algebra DIVISION

The following words translate as DIVISION:

•Divided by•Divide•Quotient

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x/7


7x

•A number divided by seven

• Divide a number by seven

•The quotient of a number and seven

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7/x


x7

•Seven divided by a number

• Divide a seven by a number

•The quotient of seven and a number

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“OF”

x21

•“Half of a number” would be

2x

x5.0or

or

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“OF”

x10030

•“Thirty percent of a number” is:

x3.0or

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“Twice” or “Double”

x2

•“Twice a number” is:

x2

•“Double a number” is:

1.1 S KM & PP 43

Translate a Phrase

x2

“Seven more than twice a number”

7

Seven more than

twice a number

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Translate a Phrase (watch for the

comma)

2x

“the quotient of seven anda number increased by two”

7

2x7

“the quotient of seven and a number,

increased by two”

1.1 S KM & PP 45

Salary Increase?

Suppose you will get a salary increase of 3%.

Let s represent your old salary.

The increase is 3% of your current salary, so 0.03s is the

increase.Your new salary will be the sum

of the old salary and the increase.

So, s + 0.03s is your new salary.

1.1 S KM & PP 46

Discount?

Suppose the bookstore has all merchandise on sale for 15%

off. Let p represent the regular

price.The discount is 15% of the

regular price, so 0.15p is the discount.

The sale price will be the difference of the regular price

and the discount

So, p - 0.15p is the sale price.

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1.1 SKM & PP 1 Getting Started with Algebra What isAlgebra? is a branch of mathematics in which symbols, usually letters, are used to represent quantities