First, one should be introduced to the properties of equality. The Following can be said to be the properties of equality: 
a = a.
If a = b, then b = a.
If a = b and b = c, then a =c.
If a = b, then a + c = b + c.
If a = b, then a – c = b – c.
If a = b, then a x c = b x c.
If a = b and ‘c’ is not equal to 0, then a/c = b/c.
If a = b, then ‘b’ may be substituted for ‘a’ in any expression containing ‘a’.
The following can be said to be the properties of equality for real numbers. Certain textbooks list only a few of them, and others list them all. These can be said to be the logical rules that enable any individual to balance, manipulate, as well as solve equations.
PROPERTIES OF EQUALITY 

Reflexive Property 
For all the real numbers xx, x=xx=x. A number equals itself. 
These 3 properties define an equivalence relation

Symmetric Property 
For all the real numbers x and yx and y, if x=yx=y, then y=xy=x. Order of the equality does not matter. 

Transitive Property 
For all the real numbers x,y, and zx,y, and z , if x=yx=y and y=zy=z, then x=zx=z. 2 numbers equal to the same number would be equal to each other. 

Addition Property 
For all the real numbers x,y, and zx,y, and z, if x=yx=y, then x+z=y+zx+z=y+z. 
These properties enable a person to balance as well as solve equations involving the real numbers 
Subtraction Property 
For all the real numbers x,y, and zx,y, and z , if x=yx=y, then x−z=y−zx−z=y−z. 

Multiplication Property 
For all the real numbers x,y, and zx,y, and z , if x=yx=y, then xz=yzxz=yz. 

Division Property 
For all the real numbers x,y, and zx,y, and z, if x=yx=y, and z≠0z≠0, then xz=yzxz=yz. 

Substitution Property 
For all the real numbers x and yx and y, if x=yx=y, then yy can be substituted for xx in any expression. 

Distributive Property 
For all the real numbers x,y, and zx,y, and z, x(y+z) =xy+xzx(y+z) =xy+xz 
For more, one should see the section on distributive property 
The Following can be said to be the outlining of the 8 of the most rudimentary axioms of equality.
The 1^{st} axiom is known as the reflexive property or the reflexive axiom. It mentions that any quantity is actually equal to itself. This axiom governs or regulates real numbers, but can be interpreted for the geometry. Any figure with any measure of few sorts is also equal to itself.
The 2^{nd} of the rudimentary axioms can be said to be the transitive axiom, or the transitive property. It actually mentions that if 2 quantities are both equal in relation to a 3^{rd} quantity, then they can be said to be equal to each other.
The 3^{rd} major axiom can be said to be the substitution axiom. It actually mentions that if 2 quantities can be said to be equal, then 1 might be replaced by the other in any specific expression, and the result would not be changed.
The 4^{th} axiom is normally to be the partition axiom. It mentions that any quantity is actually equal to the sum of its portions. Similarly, in the geometry, the measure of any segment or any angle is actually equal to the measures of its portions.
The last 4 major axioms of equality have to do with the operations between the equal quantities.
There can be said to be some logic systems that actually do not have any notion relating to equality. This actually reflects the undecidability of the equality of 2 real numbers, which can be said to be defined by the formulas involving the specific integers, the rudimentary arithmetic operations, the logarithm as well as the exponential function. To state it in a simpler manner, there cannot be existence any algorithm in order to decide such an equality.
Example 1: Lisa and Linda actually have the same money amount. If they double their amount, they still have the same amount.
Utilizing such property to solve the equations.
Example 2: ^{x}⁄_{4 }= 5
Multiply both sides by 4.
^{x}⁄_{4 }× 4 = 5 × 4
x = 20
For checking, one can substitute the value of x in the initial equation.
^{20}⁄_{4 }= 5
5 = 5
Example 3: 1/4^{th} of the kids visiting the amusement park ‘Jump & Slide’ on holiday tried new ride ‘loopOloop’. If 75 kids tried ride, how many kids actually visited the park on that day?
‘a’ is the number of children who visited the park. 1/4^{th} of such number is 75. That is,
^{a}⁄_{4 }= 75
Solve equation for a.
Through multiplication property of equality, the equality still holds true. So, multiply two sides by 4.
^{a}⁄_{4} × 4 = 75 × 4
a = 300
Therefore, 300 kids visited the park.
1. Addition
2. Subtraction
3. Multiplication
4. Division
4
For instance, if one adds 1 to left side of equation, one should also add 1 to right side of equation.
Reflexive property.
Distributive property.
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