You have probably come across many equations and identities in mathematics. However, what do we actually mean by the term ‘equation’ or an ‘identity’? Sometimes it can be difficult to distinguish between equations and identities. In this article, we will be looking at equations, identities as well as difference between equation and identity.

First of all, let’s outline what we mean when talking about equations and identities. Let’s start with an expression in math.

## What is an Expression?

An expression is a collection of mathematical terms, related by mathematical operations (plus, minus, multiplication, or division).

A mathematical term is a single mathematical number(constant) or letter(variable), for example, $x$ or $5$. We could also have $5x^{2}$, where $5$ is known as the coefficient of $x$ or $x$ as a variable of $5$.

Some of the examples of mathematical expressions are $3x + 2$, $2x^{2} – 7x + 9$, $2a + 3b + 7c$, $ \frac {2}{3} s$.

## What is an Equation?

An equation is a statement that shows that two mathematical expressions shall be the same. An equation is expressed with an equal sign between two mathematical expressions. In simple terms, anything with an equal sign is an equation. Here are some examples of equations: $x = 7$, $2x + 5 = 9$, $a + b = c$, $2x^{2} + 5y = 9$, $2x^{2} + 7x – 9 = 0$.

## What is the Solution of an Equation?

It is important to observe here that two expressions may only be equal under specific conditions. The values (specific conditions), where the LHS (Left Hand Side) algebraic expression is equal to RHS (Right Hand Side) algebraic expression is called the solution of an equation.

The solution of an equation is the set of all values that, when substituted for the variables in the equation, make the equation true (LHS = RHS).

For example, $2x + 5 = 9$ is true only for $x = 2$ (Plugging $x = 2$ in $2x + 5 = 9$, we get $2 \times 2 + 5 = 9$, which is true). So, $x = 2$ is the solution of $2x + 5 = 9$.

Or, $2x^{2} + 7x – 9 = 0$ is true only for $x = 1$ and $x = -\frac {9}{2}$.

Plugging $x = 1$ in $2x^{2} + 7x – 9$, we get $2 \times 1^{2} + 7 \times 1 – 9 = 0$ or $x = -\frac{9}{2} in 2x^{2} + 7x – 9$, we get $2 \times \left( -\frac {9}{2} \right)^{2} + 7 \times \left( – \frac {9}{2} \right) – 9$

$= 2 \times \frac {81}{4}- \frac {63}{2} – 9 = \frac {81}{2} – \frac {63}{2} – 9 = \frac {18}{2} – 9 = 0 $

## What is an Identity?

There are some expressions that are always equal to each other, regardless of the values of the variables they contain. These expressions are called mathematical identities. A mathematical identity is where two mathematical expressions are always identical.

Examples of identities are $a + a = 2a$, $x \times x = x^{2}$, $ \left( a + b \right) ^{2} = a^{2} + 2ab + b^{2} $, $ a^{3} – b^{3} = \left( a – b \right) \left( a^{2} + ab + b^{2} \right)$.

For example, in $ \left( a + b \right) ^{2} = a^{2} + 2ab + b^{2}$, if you plug in any value, LHS is **always equal** to RHS.

Substituting $a = 2$ and $b = 3$,

LHS = $ \left( 2 + 3 \right) ^{2} = 5^{2} = 25$

and RHS = $2^{2} + 2 \times 2 \times 3 + 3^{2} = 4 + 12 + 9 = 25$

Or, substituting $a = 5$ and $b = -2$,

LHS = $ \left( 5 + \left( -2 \right) \right) ^{2} = \left( 5 – 2 \right) ^{2} = 3^{2} = 9$

and RHS = $5^ {2} + 2 \times 5 \times \left( -2 \right) + \left( -2 \right) ^{2} = 25 – 20 + 4 = 9$

## Difference Between Equation and Identity

Equation | Identity |

For all values of the variable(s) LHS and RHS are not equal | For all values of the variable(s) LHS is equal to RHS |

Values of variables for which LHS and RHS are equal are called solution(s) of an equation | There is no concept of a solution |

All equations are not identities | All identities are equation |

## Key Takeaways

- An expression is a collection of mathematical terms, related by mathematical operations.
- An equation is any mathematical relation expressed with an equal sign.
- Identity is where two mathematical expressions are always identical.
- Identities are expressed using the symbol which is like an equal sign with an extra line.
- The difference between an equation and an identity is that equations state equality under a specific condition, while identities show that two expressions are always equal.

## Practice Problems

- What is an algebraic expression?
- What is an equation?
- What is an identity?
- Write down the difference between equation and identity.

## FAQs

### Is every equation an identity?

Every identity is an equation, but not every equation is an identity. To know that an equation is an identity it is necessary to provide a convincing argument that the two expressions in the equation are always equal to each other.

### What makes an equation an identity?

If solving an equation leads to a true statement such as 0 = 0, the equation is an identity.