# Laws of Equality

Before knowing the properties of equality, let me introduce you to the properties of algebra. Below are given some properties which are applied in algebra:

1. Associative property of addition:

(a+b) +c = a + (b+c)

2. Commutative property of addition:

a + b = b + a

3. Additive property of 0:

a + 0 = 0 + a = a

4. Existence of additive inverses:

For every ‘a’ there exists (-a) so that a + (-a) = 0.

5. Associative property of multiplication:

(a x b) x c = a x (b x c)

6. Commutative property of multiplication:

a x b = b x a

7) Multiplicative identity property of 1:

a x 1 = 1 x a =a

8. Existence of multiplicative inverse:

For every ‘a’not equal to 0, there exists 1/a so that

a x 1/a = 1/a x a = 1.

9. Distributive property of multiplication over addition:

a x (b + c) = a x b + a x c

Following are the some of the solved examples based on the above given properties to make the better understanding of the concept:

1. Associative property of addition:

The way 3 numbers are grouped when adding does not change the sum.

Example: 3 + (4 + 9) = (3 + 4) + 9 = 16.

2. Commutative property of addition:

The order in which two numbers are added does not change their sum.

Example: 3 + 9 = 9 + 3 = 12.

3. Additive identity property of 0:

The sum of a number and 0 is the number itself.

Example: 16 + 0 = 0 + 16 = 16.

4. Existence of additive inverses:

The sum of a number and its compliment (opposite) is equal to 0.

Eg. 12 + (-12) = 0.

5. Associative property of multiplication:

The way 3 numbers are grouped when multiplying does not change the product.

Eg. 4 x (3 x 2) = (4 x 3) x 2 = 24.

6. Commutative property of multiplication:

The order in which two numbers are multiplied does not change their product.

Example: 4 x 8 = 8 x 4 = 32.

7. Multiplicative identity property of 1:

The product of a number and 1 is the number itself.

Example: 8 x 1 = 8

8. Existence of multiplicative inverses:

The product of a number (which is not equal to 0) and its reciprocal is equal to 1.

Example: 4 x ¼ = 1.

9. Distributive property of multiplication over addition:

When multiplying a number by a sum, the number can be multiplied by each term in the sum. Multiplication can also be distributed over subtraction.

Example: Multiplication over addition:

3 x (4 + 5) = 3 x 4 + 3 x 5 = 12 + 15 = 27.



Now, let me introduce you to the properties of equality. Following are the properties of equality:

1. Reflexive property of equality:

a = a.

2. Symmetric property of equality:

If a = b, then b = a.

3. Transitive property of equality:

If a = b and b = c, then a =c.

4. Addition property of equality;

If a = b, then a + c = b + c.

5. Subtraction property of equality:

If a = b, then a – c = b – c.

6. Multiplication property of equality:

If a = b, then a x c = b x c.

7. Division property of equality;

If a = b and ‘c’ is not equal to 0, then a/c = b/c.

8. Substitution property of equality:

If a = b, then ‘b’ may be substituted for ‘a’ in any expression containing ‘a’.

Below are given explanations and examples for the above mentioned properties of equality:

1. Reflexive property of equality:

Any number is equal to itself.

Example: 14 = 14.

2. Symmetric property of equality:

An equation may be written in the opposite order,

Example: If y = 45, then 45 = y.

3. Transitive property of equality:

Two quantities that are equal to the same thing are equal to each other.

Example: If x = 10 and 10 = y, then x = y.

4. Addition property of equality:

The same number can be added to both sides of an equation.

Example: If x = 35, then x + 4 = 35 + 4.

5. Subtraction property of equality:

The same number can be subtracted from both sides of an equation.

If x = 13, then x – 4 = 13 – 4.

6. Division property of equality:

Both sides of an equation can be divided by any non- zero number.

Example: If x = 8, then x/2 = 8/2.

7. Substitution property of equality;

A number may be substituted for its equal in any expression.

Example: If x = 80 and y = 80, then x = y.

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