# How to determine whether an equation is identity or conditional

How do you know if an equation is an identity or conditional?

Identity and conditional equations are ways in which numbers associate with each other.

**Identity equation** is when the equation is true for every value of the variable.

It is often denoted as I or E (the E is from the German Einheit, or “unity”).

The equation is satisfied by every number that is a meaningful replacement for the variable.

*For example*,

3x = 3x

3(x +1) = 3x + 3

This is an identity equation, in essence x will always be the same number.

• Zero is the identity element for addition, because any number added to 0 does not change the value of any of the other numbers in the operation ( x + 0 = x).

• The number 1 is the identity element of multiplication, as any number in an operation multiplied by 1 does not change the value of that number. Multiple identity is often written as x × 1 = x.

When an equation is false for at least one value, it is called a **conditional equation.**

For example,

6x = 12

This is conditional because it is false when x = 3 (and any number other than 2).

An equation that has no solution, such as x = x +1, is called a contradiction.

In conclusion,

1. If solving a linear equation results in a true statement such as 0 = 0, the equation is an identity. Its answer set is {all real numbers}.

2. If solving a linear equation leads to a single answer such as x = 3, the equation is conditional. Its solution set consists of a single element.

3. If solving a linear equation leads to a false statement such as −3 = 7, then the equation is a

contradiction. Its solution set is the empty set or null set.