# Properties of Subtraction: Commutative, Associative, Inverse and other subtraction properties

In this post we will understand different properties of Subtraction.

The properties are important as they are frequently used in solving advanced algebra and calculus problems.

Also, if you are preparing for State Standardized Tests like American Regions Mathematics League (ARML), State of Texas Assessment of Academic Readiness (STAAR) etc. , these subtraction properties will help you solve questions faster.

## What are properties of Subtraction?

Here we have discussed following properties of subtraction:

(a) Commutative Property of Subtraction
(b) Associative Property of Subtraction
(c) Identity Property of Subtraction
(d) Distributive Property of Subtraction
(e) Inverse Property of Subtraction
(f) Subtraction Property of Equality

### Commutative Property of Subtraction

The subtraction of numbers is non commutative in nature.

It means that in subtraction if you change the position of numbers you will get different result.

For Instance:
If A & B are whole number such that A > B, then

A – B, will be a whole number

But when you change the location of numbers, then we get different result

B – A, will be negative integer

Let us see some examples for better understanding of concept

Example 01
Let A = 7 and B = 3

Subtraction of A – B
⟹ A – B
⟹ 7 – 3
⟹ 4

Now subtracting B – A
⟹ B – A
⟹ 3 – 7
⟹ – 4

Here A – B is not equal to B – A

Example 02
Let A = -7 and B = -3

Finding A – B
⟹ A – B
⟹ -7 – (-3)
⟹ -7 + 3
⟹ – 4

Finding B – A
⟹ B – A
⟹ – 3 – (-7)
⟹ -3 + 7
⟹ 4

Again A – B is not equal to B – A

Note that changing the location of number produced different results.

Conclusion: In subtraction, commutative property does not work

### Associative Property of Subtraction

Subtraction of two numbers are non – associative in nature.

It means that in a subtraction forming different groups will yield different results.

Let us understand this with some examples:

Example 01
Let A = 10, B = 5 & C = 2

First subtract (A – B) – C
⟹ (A – B) – C
⟹ (10 – 5) – 2
⟹ 5 – 2
⟹ 3

Now Subtract A – (B – C)
⟹ A – (B – C)
⟹ 10 – (5 – 2 )
⟹ 10 – 3
⟹ 7

You can see that different groups produce different results
So here, (A – B) – C is not equal to A – (B – C)

Example 02
Let A = -12, B = -7 & C = 1

First find (A – B) – C
⟹ (A – B) – C
⟹ [-12 -(-7)] – 1
⟹ – 12 + 7 – 1
⟹ -6

Now Find A – (B – C)
⟹ A – (B – C)
⟹ -12 – (- 7 – 1)
⟹ – 12 – (- 8)
⟹ – 12 + 8
⟹ -4

Again, (A – B) – C is not equal to A – (B – C)

Conclusion: Associative Property does not work in subtraction

### Identity Property of Subtraction

Like addition, identity property also works in subtraction of number.

If you subtract any number with 0, the result will be the same number.

Hence, subtraction with 0 does not change the identity of the number, hence the term identity property

But the property does not work when you change the location of number and subtraction 0 with any number

Subtraction 0 from any number results in negative variant of the number, the real identity of the number has been changed, hence it can’t be the identity property.

### Distributive Property of Multiplication over Subtraction

According to the property, the multiplication of subtraction of numbers is equal to subtraction of the multiplication of individual number.

Where, A, B & C can be any possible number

Distributive Property helps to simplify the complex algebraic expression.

Given below are some examples for your understanding

Example 01
Let A = 2, B = 6 and C = 3

Finding A (B – C)
⟹ 2 ( 6 – 3 )
⟹ 2 x 3
⟹ 6

Finding A.B – A.C
⟹ 2.6 – 2.3
⟹ 12 – 6
⟹ 6

Hence, A (B – C) = A (B – C) = 6

Let us move to another example….

Example 02
Let A = -3, B = 2 & C = 1

Finding A (B – C)
⟹ -3 ( 2 – 1 )
⟹ -3 x 1
⟹ -3

Finding A.B – A.C
⟹ -3.2 – (-3).1
⟹ -6 + 3
⟹ -3

Here, A (B – C) = A (B – C) = -3

Hence, distributive property works in multiplication over subtraction.
Try to remember the concept for fast problem solving

### Inverse Property of Subtraction

According to the property, subtracting any number with the same number results in number 0.

Inverse Property of subtraction is given by following expression

Where A can be any possible number

Why this property is named Inverse Property?

Because after the operation, the identity of the number is reversed to zero. Here we have completely inversed the value of number to 0.

Example
Let A = 5

Using the inverse property of subtraction
5 – 5 = 0

Using the property the value if number 5 is reversed to 0

Given below are some examples of Inverse Property

Example 01
Let A = 9

Finding A – A
⟹ 9 – 9 = 0

Example 02
Let A = -3

Finding A – A
⟹ -3 – (-3)
⟹ -3 + 3
⟹ 0

### Subtraction Property of Equality

This property is used to manipulate any given algebraic equation.

According to the property, if we subtract any number on both side of the equation, the equality of equation still holds.

For the given algebra equation;
⟹ x – 3 = 5

If we subtract same number on both side, the equation will still holds true.
Here we will subtract 7 from both sides

⟹ x – 3 – 7 = 5 – 7

x – 10 = -2

The equation x – 10 = -2 is correct, we have just manipulated it by subtraction

The subtraction property of equality can be expressed as:

Note:
We should subtract the same number on both side of equation otherwise the equation will not be right

For Example in the below equation:
x – y = 7

If we subtract different number on both sides: like subtracting 3 on the left and 2 in the right, then the resulting equation will not be correct.

x – y – 3 = 7 – 2

The above equation is not correct.

Conclusion: Subtracting same number on both sides of algebraic equation will not make the equation incorrect.

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