Properties of Equality

The following are the properties of equality for real numbers. Some textbooks list just a few of them, others list them all. These are the logical rules which allow you to balance, manipulate, and solve equations.

PROPERTIES OF EQUALITY
Reflexive Property

For all real numbers x , x=x .

A number equals itself.

These three properties define an equivalence relation

 

Symmetric Property

For all real numbers xandy ,

if x=y , then y=x .

Order of equality does not matter.

Transitive Property

For all real numbers x,y,andz ,

if x=y and y=z , then x=z .

Two numbers equal to the same number are equal to each other.

Addition Property

For all real numbers x,y,andz ,

if x=y , then x+z=y+z .

These properties allow you to balance and solve equations involving real numbers
Subtraction Property

For all real numbers x,y,andz ,

if x=y , then xz=yz .

Multiplication Property

For all real numbers x,y,andz ,

if x=y , then xz=yz .

Division Property

For all real numbers x,y,andz ,

if x=y , and z0 ,

then x z = y z .

Substitution Property

For all real numbers xandy ,

if x=y , then y can be substituted for x in any expression.

Distributive Property

For all real numbers x,y,andz ,

x( y+z )=xy+xz

For more, see the section on the distributive property