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Reference

# Properties of Equality

Rule

Adding the same number to both sides of an equation results in an equivalent equation. Let and be real numbers.

If then

The Addition Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider an example.
By adding to both sides of the equation, the variable can be isolated and the solution to the equation can be found.

Simplify left-hand side

The Addition Property of Equality also holds true if and are complex numbers.
Rule

## Subtraction Property of Equality

Subtracting the same number from both sides of an equation results in an equivalent equation. Let and be real numbers.

If then

The Subtraction Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider an example.
By subtracting from both sides of the equation, the variable can be isolated and the solution to the equation can be found.

Simplify left-hand side

Note that the Subtraction Property of Equality also holds true if and are complex numbers.
Rule

## Multiplication Property of Equality

Given an equation, multiplying each side of the equation by the same number yields an equivalent equation. Let and be real numbers.

If then

The Multiplication Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example.
Here, by multiplying both sides of the equation by the variable was isolated and the solution of the equation was found. Note that the Multiplication Property of Equality also holds true if and are complex numbers.
Rule

## Division Property of Equality

Dividing each side of an equation by the same nonzero number yields an equivalent equation. Let and be real numbers.

If and then

The Division Property of Equality is an axiom, so it does not need a proof to be accepted as true. This property is one of the Properties of Equality that can be used when solving equations.
As can be observed, by dividing both sides of the equation by the variable was isolated and the solution of the equation was found. Note that the Division Property of Equality also holds true if and are complex numbers.
Rule

## Reflexive Property of Equality

For any real number, the number is equal to itself.

This property is an axiom, so it does not need a proof. This property is used to solve equations. For example, consider the equation below.
The sum on the left-hand side can be interpreted as a number, so this sum has to equal This implies that must be equal to
Rule

## Symmetric Property of Equality

For all real numbers, the order of an equality does not matter. Let and be real numbers.

If then

This property can be used together with other Properties of Equality to solve equations or to isolate a variable and express it in terms of another variable. In the example below, the Symmetric Property of Equality is used to express in terms of
This property is an axiom, so it does not need a proof to be accepted as true. The Symmetric Property of Equality also holds true if and are complex numbers.
Rule

## Transitive Property of Equality

For all real numbers, if two numbers are equal to the same number, then they are equal to each other. Let and be real numbers.

If and then

This property can be used together with other Properties of Equality to solve equations.
Since this property is an axiom, it does not need a proof to be accepted as true. The Transitive Property of Equality also holds true if and are complex numbers.
Rule

## Substitution Property of Equality

If two real numbers are equal, then one can be substituted for another in any expression.

If then can be substituted for in any expression.

Since the Substitution Property of Equality is an axiom, it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example.
Solve by substitution
By substituting with in Equation (II), the value of was obtained. Note that the Substitution Property of Equality also holds true if and are complex numbers.