IDENTITY AND EQUALITY PROPERTIES

Here we are going to see the properties of identity and equality.

1. Additive identity

2. Multiplicative identity

3. Multiplicative property of zero

4. Multiplicative inverse

5. Reflexive 

6. Symmetric

7. Transitive 

8. Substitution 

Additive Identity Property

The sum of any number and 0 is equal to the number. Thus, 0 is called the additive identity.

For any number a, the sum of a and 0 is a.

a + 0 = 0 + a  =  a

Example :

5 + 0 = 5

0 + 5 = 5

5 + 0 = 0 + 5 = 5

Multiplicative Identity Property

The product of any number and 1 is equal to the number, 1 is called the multiplicative identity.

a ⋅ 1 = 1 ⋅ a = a

Example :

5 ⋅ 1 = 5

1 ⋅ 5 = 5

5 ⋅ 1 = 1 ⋅ 5 = 5

Multiplicative Property of Zero

The product of any number and 0 is equal to 0. This is called the Multiplicative Property of Zero. 

a ⋅ 0 = 0 ⋅ a = 0

Example :

5 ⋅ 0 = 0

0 ⋅ 5 = 0

5 ⋅ 0 = 0 ⋅ 5 = 0

Multiplicative Inverse

For every number a/b, where a, b ≠ 0 , there is exactly one number b/a such that the product of a/b and b/a is 1.

(a/b) ⋅ (b/a) = (b/a) ⋅ (a/b) = 1

Example :

(2/3) ⋅ (3/2)  = 6/6  =  1

(3/2) ⋅ (2/3)  = 6/6  =  1

(2/3) ⋅ (3/2) = (3/2) ⋅ (2/3) = 1

Reflexive 

Any quantity is equal to itself.

For any number a, a  =  a.

Example :

7 = 7

2 + 3 = 3 + 2

Symmetric

If one quantity equals a second quantity, then the second quantity equals the first.

For any numbers 'a' and 'b', if a = b then b = a. 

Example :

If 9 = 6 + 3, then 6 + 3 = 9.

Transitive

If one quantity equals a second quantity the second quantity equals a third quantity, then then the first quantity equals the third quantity.

For any numbers a, b and c. if a = b and b = c, then a = c. 

Example :

If 5 + 7 = 8 + 4 and 8 + 4 = 12, then 5 + 7 = 12.

Substitution 

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

If a = b, then a may be replaced by b is any expression.

Example :

If n = 15, then 3n = 3(15).

Solve the equation. Tell which algebraic property of equality you used. 

Problem 1 :

h - 6 = 2

Solution :

h - 6 = 2

Using addition property of equality adding 6 on both sides.

h - 6 + 6 = 2 + 6

h = 8

So, the value of h is 8.

Problem 2 :

x/3 = 9

Solution :

x/3 = 9

Multiplication property of equality, we get

3 ⋅ (x/3) = 9 ⋅ 3

x = 9

So, the value of x is 9.

Problem 3 :

4m = 12

Solution :

4m = 12

Using division property, we get

4m/4 = 12/4

m = 3

So, the value of m is 3.

Problem 4 :

k + 8 = -9

Solution :

k + 8 = -9

Using subtraction property of equality, we get

k + 8 - 8 = -9 - 8

k = -17

So, the value of k is -17.

Problem 5 :

n + 2 = 6

Solution :

n + 2 = 6

Using subtraction property of equality, we get

n + 2 - 2 = 6 - 2

n = 4

So, the value of n is 4.

Problem 6 :

p/6 = -2

Solution :

p/6 = -2

Using multiplication property of equality, we get

6 ⋅ (p/6) = (-2) ⋅ 6

p = -12

So, the value of p is -12.

Problem 7 :

q - 3 = -8

Solution :

q - 3 = -8

Using additive property of equality, we get

q - 3 + 3 = -8 + 3

q = -5

So, the value of q is -5.

Problem 8 :

q - 3 = -8

Solution :

q - 3 = -8

Using additive property of equality, we get

q - 3 + 3 = -8 + 3

q = -5

So, the value of q is -5.

Identify the IDENTITY OR EQUALITY PROPERTY that is shown in each statement. 

Problem 9 :

1) 8 + 0 = 8

2) If 5 = 3 + 2, then 3 + 2 = 5

3) 0 ⋅ 5 = 0 

4) -x = (-1)x

5) If 11 = 6 + 5, then 6 + 5 = 11

6) 38 ⋅ 0 = 0

7) 5 + x – 5 = x

8) 1 ⋅ 38 = 38

9) a + b + 0 = a + b

10) (1/3) ⋅ (3 x) = x

Solution :

1) 8 + 0 = 8

Additive identity property.

2) If 5 = 3 + 2, then 3 + 2 = 5

Symmetric property

3) 0 ⋅ 5 = 0 

Multiplicative property of zero.

4) -x = (-1)x

Property of negative one.

5) If 11 = 6 + 5, then 6 + 5 = 11

Symmetric property

6) 38 ⋅ 0 = 0

Multiplicative property of zero.

7) 5 + x – 5 = x

Additive inverse property.

8) 1 ⋅ 38 = 38

Multiplicative inverse property.

9) a + b + 0 = a + b

Additive identity.

10) (1/3) ⋅ (3 x) = x

Multiplicative inverse property.

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