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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 x z = y z .

Multiplication Property

For all real numbers x , y , and z ,

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

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 ) = x y + x z

For more, see the section on the distributive property