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 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 