Unit 1 - Objective 2 - Fundamental Laws of Algebra


Fundamental Laws of Algebra

Commutative Law of Addition: a + b = b + a

Example:

2 + 3 = 3 + 2
5 = 5

Associative Law of Addition: a + (b + c) = (a + b) + c

Example:

2 + (3 + 4) = (2 + 3)+ 4
2 + 7 = 5 + 4
9 = 9

Commutative Law of Multiplication: ab = ba

Example:

(3) (4) = (4) (3)
12 = 12

Associative Law of Multiplication: a (bc) = (ab) c

Example:

2 x (3 x 4) = (2 x 3) x 4
2 x 12 = 6 x 4 
24 = 24

Distributive Law: a (b + c) = ab + ac

Example:

2 (3 + 4) = 2 (3) + 2 (4)
2 (7) = 6 + 8
14 = 14


Basic Operations

Addition of Integers

To add two real numbers with the same sign, add the numbers and give to the sum the sign of the original number.

Example:
1.  +8 + (+7) = 15
2.  -9 + (-24) = -33

To add two real numbers with different signs, take the absolute value of both numbers, subtract the smaller absolute value from the larger, and give to the answer the sign of the number with the larger absolute value.

Example:
1. -8 + (+5) |-8| = 8
|+5| = 5
8 > 5 so the answer is negative
8 - 5 = 3 so -8 + (+5) = -3
2. (-9) + 14 |+14| = 14
|-9| = 9
14 > 9 so the answer is positive
14 - 9 = 5 so 14 + (-9) = +5

Subtraction of Integers

To subtract one real number from another, change the sign of the number being subtracted and then add.

Example:
1. -6 - (-9) = -6 + 9
= 3
2. -13 - (+7) = -13 + (-7)
= -20

Multiplication and Division of Integers

The product or quotient of two real numbers with the same sign is the product or quotient of their absolute values.

Example:
1. (-8) (-9) = |-8| x |-9|
= 8 x 9
= 72
2. -27 / (-3) = |-27| / |-3|
= 27 / 3
= 9

The product or quotient of two real numbers with different signs is the additive inverse of the product or quotient of their absolute values.

Example:
1. (-8) (9) = -72
2. 81 / (-3) = -27

Order of Operations


  1. Operations inside parenthesis are done first.
  2. Multiplication and division in the order in which they appear from left to right.
  3. Addition and subtraction in the order in which they appear from left to right.
Example:
1. 6 + 10 / 2 (do the division first)
= 6 + 5
= 11
2. 8 + 9 x 2 + 16 - 12 / 4 (multiply and divide from left to right)
= 8 + 18 + 16 - 3 (add and subtract from left to right)
= 26 + 16 - 3
= 42 - 3
= 39
3. 28 - (26 - (3 - (4 - 3))) (start with the innermost parenthesis)
= 28 - (26 - (3 - 1))
= 28 - (26 - 2)
= 28 - 24
= 4

Identity Elements


The identity element for addition is 0: a + 0 = 0 + a = a

Example:

3 + 0 = 3 or 0 + 3 = 3

The identity element for multiplication is 1: a x 1 = 1 x a = a

Example:

-7 x 1 = -7 or 1 x (-7) = -7


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