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 x and y,

if x = y, then y = x.

Order of equality does not matter.

Transitive Property

For all real numbers x, y, and z ,

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, and z,

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, and z,

if x = y, then xz = y – z.

Multiplication Property

For all real numbers x, y, and z,

if x = y, then xz = yz.

Division Property

For all real numbers x, y, and z,

if x = y, and z ≠ 0, then x/z = y/z.

Substitution Property

For all real numbers x and y ,

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

Distributive Property

For all real numbers x, y, and z,

x(y + z) = xy + xz.

For more, see the section on the distributive property