# Symmetric property of equality

The symmetric property of equality is one of the basic properties of equality in mathematics. There are 9 basic properties of equality, discussed further below.

## What is the symmetric property of equality

The symmetric property of equality states that for two variables, a and b:

if a = b, then b = a

In other words, the symmetric property of equality states that regardless which side of an equal sign any given variables are on, the two variables (or expressions) are equal. Thus, it does not matter if we completely switch the positions of the variables across the equal sign, as long as we keep them the same and simply trade their positions, the equation will still be true. Below are some symmetric property of equality examples:

Examples

- Since 3 + 4 = 7, 7 + 4 = 3
- Since 3 × 7 = 21, 21 = 3 × 7
- Since 3x + 7 = 10, 10 = 3x + 7
- Since 3x + 7x = 10x, 10x = 3x + 7x

In all of these examples, we wrote each side of the equation in the same order, only switching their positions. However, using the first example, we also could write since 3 + 4 = 7, 7 = 4 + 3. This includes other symmetric properties (discussed below) however, not just that of equality.

The symmetric property of equality is used widely throughout mathematics in areas such as arithmetic, algebra, computer science, and more. In algebra, the property is used to manipulate algebraic expressions in order to solve them. For example, since the two expressions on either side of an equation are the same, we can perform arithmetic operations on either side of the equation as long as we perform the corresponding arithmetic operation on the other side, and the expressions on either side will remain equal.

Example

Since 3x + 7 = 10, we can subtract 7 from both sides of the equation, and the equation will still be the same:

3x + 7 - 7 = 10 - 7

3x = 3

From here, we can solve for x by dividing by 3 to find that:

x = 1

This is the basis of solving simple algebraic equations.

## Property of equality

A property of equality describes a characteristic that applies to all quantities that are related by an equal sign. The symmetric property of equality is just 1 of 9 of these properties. Below are the 9 basic properties of equality.

### Addition property

The addition property of equality states that if a common value is added to two equal quantities, equality is retained. Given variables a, b, and c, such that a = b, the addition property of equality states:

a + c = b + c

### Transitive property

The transitive property of equality states that if two variables or expressions are equal to a third common variable, then the two variables are equal to each other. Given variables a, b, and c, the transitive property of equality states that if a = b and b = c, then:

a = c

### Subtraction property

The subtraction property of equality states that when a common term is subtracted from two equal terms, equality is retained. Given variables a, b, and c such that a = b, the subtraction property of equality states:

a - c = b - c

### Multiplication property

The multiplication property of equality states that when two equal quantities are multiplied by a common term, equality is retained. Given variables a, b, and c such that a = b, the multiplication property of equality states:

ac = bc

### Division property

The division property of equality states that when two equal quantities are divided by a common term, equality is retained. Given variables a, b, and c such that a = b, the division property of equality states:

a/c = b/c

### Symmetric property

The symmetric property of equality states that it does not matter whether a term is written on the left- or right-hand side of the equal sign; equality is retained in either case. Given variables a and b such that a = b, the symmetric property of equality states:

a = b is the same as b = a

### Reflexive property

The reflexive property of equality states that all variables are equal to themselves. Given variable a, the reflexive property of equality states:

a = a

### Substitution property

The substitution property of equality states that equal quantities can replace each other at any time in any expression. There is no concise way to express the substitution property of equality. As an example, given variables a, b, and c such that a = b, we can substitute a or b for each other in any expression. Below is one substitution property of equality example:

a - 4 = c can be replaced with b - 4 = c

### Distributive property

The distributive property of equality states that equality is retained after distributing with multiplication. This is true for any number of terms, but the example below uses two terms. Given variables a, b, and c, the distributive property of equality states:

a(b + c) = ab + ac

## Property of equality table

The following property of equality table summarizes the properties of equality.

Addition property | If a = b, then a + c = b + c |

Transitive property | If a = b and b = c, then a = c |

Subtraction property | If a = b, then a - c = b - c |

Multiplication property | If a = b, then ac = bc |

Division property | If a = b, then a/c = b/c |

Symmetric property | If a = b, then b = a |

Reflexive property | a = a |

Substitution property | If a = b and a = c, then b = c |

Distributive property | a(b + c) = ab + ac |