Algebraic Laws:

Several operations on real numbers are based on the commutative, associative & distributive laws.  The effectual use of these laws is significant. These laws will be beginning in written form as well as algebraic form, where letters/symbols are used to represent an unknown number.

The commutative laws imply that numbers can be added or multiplied in any order.

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

Commutative Law of Multiplication:  a(b) = b(a)

An associative laws state that in addition or multiplication there are numbers that can be grouped in any order.

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

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

The distributive laws involve both addition and multiplication and state the following.

Distributive law:  a(b + c) = ab + ac

Distributive law:  (a + b)c = ac + bc

The below given list of axioms pertains to the real number system where a, b, and c represent any real numbers. These properties must be true for the algebraic laws to apply.

Closure Properties    1.         a + b is a real number

                                    2.         ab is a real number

Identity Properties    3.a + 0 = a

                                    4. a (l) = a

Inverse Properties     5. For each real number, a, there exists a real number, -a, like that a + (-a) = 0

                                    6.  For each real number, a0, there exists a real number, l/a, such that a (1/a) = 1

An equation is a statement of equality.  For instance, 4 + 3 = 7 an equation can also be written with one or more unknowns (or variables).  The equation x + 7 = 9 is an equality only while the unknown x = 2.  The number 2 is known as the root or solution of this equation.

The end product of algebra is solving a mathematical equation(s).  The operator generally will be included in the solution of equations which are linear, quadratic, or simultaneous in nature.