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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