1. Identity

An equation
is classified as an identity when it is true for ALL real numbers for which both sides of the equation are defined An identity is an equality that states a fact, such
as the following examples:

- 9 + 5 = 14
- 2n + 5n = 7n
- 6(x-3) = 6x - 18

Notice that equation 3 merely shows the factored form of 6x - 18
and holds true when any value of x is substituted. For example, if x = 5, it becomes

6(5-3)
= 6(5) – 18

6(2)
= 30 – 18

12
= 12

If x assumes the negative value - 10, this identity becomes

6(-10-3) = 6(-10) – 18

6 (-13) = -60 – 18

-78 = -78

The
expressions on the two sides of the equality are identical.

2. Conditional Equation

A conditional
equation is an equation that is not an identity, but has at least one real number solution

A statement such as 2x - 1 = 0 is an equality only when x has one particular value. Such a statement is called a CONDITIONAL
EQUATION, since it is true only under the condition that x = 1/2. Likewise, the equation y - 7 = 8 holds true only if y =
15.

The value of the variable for which an equation in
one variable holds true is a ROOT, or SOLUTION, of the equation. When we speak of solving equations in algebra, we refer to
conditional equations. The solution of a conditional equation can be verified by substituting for the variable its value,
as determined by the solution.

The solution. is correct if the equality reduces to an identity.
For example, if 1/2 is substituted for x in 2x - 1 = 0, the result is

2(1/2) – 1 = 0

1 – 1 = 0

0 = 0 (an identity)

The identity is established for x = 1/2, since the value of each
side of the equality reduces to zero.