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EXIT

Created by Professor James A. Sinclair, Ph.D. MMXI

Dr J’s easy

ALGEBRA

Part 1


EXIT


Common Algebra terminology

Variables

Variables are letters that are used instead of numbers in a variety of formulas and equations. The most typical variable letter used is “x” representing the unknown number or value.For example: 2x, 2x2+5(a+b) ÷ (2b+3a) = 0

All of the letters used in the above examples are variables.

Coefficients

Coefficients are numbers typically used in front of the variables For example: 5x – 5 is a coefficient

Equations

Equations are mathematical statements that have an Equal sign (=). It typically represent a situation requiring a solution of a some numerical value.For example:

10x - 5 = 012y+10 =242x2+5 = 4x+5


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Expression

Equations are mathematical statements that does have an Equal sign (=). It typically An expression can contain numbers, variables, and operators such as (plus, minus, divide, or multiply)For example:

10x · 512y+102x2+5

Factors

Factors are numbers of terms that when multiply together arrive at the specific number:For instance:

3 and are factors of 125 an 2 are factors of 103 and 5 are factors of 152x and 4x are factors of 8x


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Integers

Integers are whole numbers that can have a positive (+) or a negative (-) signsFor example:10532-7-64

Prime numbers

Prime numbers are positive numbers that can only divide by itself or by 1.

2,3,5,7, 11, and 13 are examples or prime numbers

TermsAre numbers, variables, or a combination of numbers and variables, typically separated by operator signs. For example:

2x+y5x3z+4y25+3x


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Number line

0 1 2 3 4 5-1-2-3-4

Number line above represents an endless series of numbers that go in both directions from the zero (0).Numbers to the right of zero represent positive numbers, and to the left of zero, represent the negative numbers.

For example number 3 is represented below:

0 1 2 3 4 5-1-2-3-4

0 1 2 3 4 5-1-2-3-4

Number 1

0 1 2 3 4 5-1-2-3-4

Number -2

0 1 2 3 4 5-1-2-3-4

Number -3.5

0 1 2 3 4 5-1-2-3-4

Number 2.5


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Sometimes it is hard to understand the negative numbers on the number line. To make it a little easier to visualize, you can imagine the following example:

If this year the company made a profit of $20,000 it would be represented by +20,000. And if some company lost $10,000 it would be represented by a number -10,000. This would be a clear indication of a loss instead of a gain (profit). When companies see these negative numbers in their report, they become instantly aware that something needs to be done to illuminate the cause of this “negative” activity.

Number 0 on the number line represents a starting point. It represents no value at all (no positive or negative.

However, in mathematics, zero does represent a result of expressions and many equations. For example:

10-10 =05x+2 =0


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Adding positive numbers and variables:

When adding positive numbers and variables, no special considerations are needed, simply add the terms as usualFor example:

10+5 =155x+3x = 8x32 +32 = 645+5+10=204y+6y+10y=20y

Adding negative numbers and variables:

When adding two negative numbers and variables, special considerations are needed. simply add the terms as they were both positive, and then put a negative sign in front of the result. For example:

(-2)+(-5) = -7(-10)+(-4) = -14(-4x)+(-5x) = -9x(-6)+(-4)+(-5) = -15

The result of addition is called the SUM


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Subtracting positive numbers and variables:

When subtracting positive numbers and variables, no special considerations are needed, simply subtract the terms as usual. For example:

10-5 = 55x-3x = 2x32 -32 = 015-5-2= 820y-10y-2y=8y

Subtracting negative numbers and variables:

When subtracting negative numbers and variables, special considerations are needed. Simply change the subtraction problem into addition problem. For example:

10-(-5) = 10+(+5) = 1520-(-10) = 20+(+10) = 30-12-(-7) = -12+(+7) = -5

NOTE:In the last example, because -12 is a

greater value number, we subtracted 7 from 12 and the number became -5

because 12 had a negative value.

The result of Subtraction is called the DIFFERENCE


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Multiplying numbers and variables:

When multiplying numbers and variables in mathematics, one of several ways may be used to represent the multiplication procedure. For example:

10x5 = 25 (x is used to show multiplication sign)

5x·3x = 15x (a dot is used to show multiplication sign)

2y or 10x (when no operator is shown between the numbers and letters, it also mean that a multiplication sign exist between them). Therefore the expression 2y means 2·y or 10x means 10·y

10 * 5 = 50 (sometimes an asterisk (*) is used to show multiplication sign)

Dividing numbers and variables:

When dividing numbers and variables also one of several ways may be used to represent the division procedure. For example:

10÷5 = 2 (divide symbol used to show division)

15x / 3x = 5 (a slash is used to show division sign)

Sometimes a single line is used to demonstrate division:

10+5x

3x+238

or

The result of multiplication is called the PRODUCT

The result of division is called the QUOTIENT


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Multiplying numbers with the same sign:

When multiplying or dividing numbers or variables, where both terms have the same sign (plus or minus), the result would always be positive:

10·5 = 50

(-5x) · (-3x) = 15x

-7 x -3 = 21

25÷5 = 5

(-10)÷(-2) = 5

(-a) ÷ (-b) = a ÷ b

Multiplying numbers with the different signs:

When multiplying or dividing numbers or variables, where terms have different signs (plus or minus), the result would always be negative:

-5·5 = -25

5x · (-3x) = -15x

-7 x 3 = -21

-10÷5 = -2

-10y÷2y = -5

a · -b = -ab


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Absolute values

Absolute value, regardless of the sign simply show how far a given value is from the 0 on the number line. Absolute values are shown between two thin lines and are always positive. For example:

-8 = 8

-32 = 32

In absolute values, the sign inside the lines is simply disregarded

However, if the minus sign appears before the lines, the result is quite different. For example:

8 = 8

-8 = -8-

8= -8-

10-4 = -6-

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