Property of Equality

In Mathematics, two things are equal if and only if they are exactly similar in every way i.e. they have the same Mathematical value and Mathematical properties. Mathematics uses the equality sign (=) to represent equality. For example, the statement A = B implies that A and B are equal.

What is the Property of Equality?

Property of equality is the fact or truth that is applied to two or more quantities related by an equal sign. Let us learn different properties of equality with examples.

Addition Property of Equality

The Addition property of equality states that when the same number is added from both sides of the equation, the equation still holds true. In other words, it states that if,

X = Y, 

then X + Z = Y + Z

Example: 

We know that, 4 + 3 = 7

Here, if we add 5 to both sides of the equation, the equation still holds true.

i.e. 4 + 3 + 5 = 7 + 5

12 = 12

LHS = RHS

Subtraction Property of Equality

The subtraction property of equality states that when the same number is subtracted from both sides of the equation, the equation still holds. In other words it states that if,

X = Y, then X -  Z = Y -  Z

Example: 

We know that, 9 - 3 = 6

Here, if we subtract 2 to both sides of the equation, the equation still holds

9 - 3 - 2 = 6 - 2

4  = 4

LHS = RHS

Multiplication Property of Equality

The multiplication property of equality states that when both sides of an equation are multiplied by the same number, the two sides remain equal. 

In other words, it states that if x, y, and z are real numbers such that x = y, then 

a x c  = b x c .

Example, If a = 10, b = 10, and c = 12, then

a x c = b x c 

10 x 12 = 10 x 12

120 = 120

Therefore, LHS = RHS

Division Property of Equality

The division property of equality states that when both sides of an equation are divided by the same non-zero number, two sides remain equal. In other words, it states that  if a = b, and c ≠ 0, then a ÷ c = b ÷ c

Example:

If a = 10, b = 10, and c = 5, then 

a ÷ c = b ÷ c

10 ÷ 5 = 10 ÷ 5

2  = 2 

Therefore, LHS = RHS

Symmetric Property of Equality

The symmetric property of equality states that if we interchange the sides of an equation, the equation still holds. In other words, it states that if X = Y, then Y = X.

Example: If 3 + 5 = 8, then 8 =?

3 + 5 = 8  or 

8 = 5 + 3

Reflexive Property of Equality

The reflexive property of equality states that a number is always equal to itself. In other words, it states that if x is a number, then x = x.

Example: 

2 = 2

Transitive Property of Equality

The transitive property of equality states that when given 3 numbers a, b, and c then,

 a = b, b= c, then a = c. 

Example:

If a = b, and b = 3, then c is also equal to 3 according to the transitive property of equality.

Substitution Property of Equality

The substitution property of equality states that if two quantities are equal, then one can replace the other in any Mathematical equation or expression.

In other words, it states that if a = b, then b can be substituted for a, in any Mathematical expression.

Example:

If a = 2 and a + 3 = 5, then 2 can be substituted in a + 3 = 5 to get 2 + 3 = 5.

Conclusion

In short, the property of equality is the truth about any quantity that is related by an equal sign. By understanding the property of equality thoroughly, we are able to simplify, manipulate, balance, and solve Mathematical equations easily. Also, we are able to draw conclusions supported by valid reasons.