Laws of Algebra

Let a,b,c be three variables. Then the followings are some basic rules of algebra applicable to these variables.


1. Commutative Law for Addition

`a + b = b + a`


2. Commutative Law for Multiplication

` a.b = b.a `


3. Associative Law for Addition

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


4. Associative Law for Multiplication

` a.(b.c) = (a.b).c `


5. Distributive Law

` a.(b + c) = (a.b) + (a.c) `


6. Cancellation Law for Addition

` a + c = b + c iff a = b `


7. Cancellation Law for Multiplication

` a.c = b.ciffa = b` where c != 0


In the next section, we make use of these simple formulas to find solutions to given problems.

Example 1:

Find the value of x such that

3 + 5 = x + 3

Solution:

Using the Commutative Law for Addition stated above

`a + b = b + a`

=> For the given expression, x = 5


Example 2:

Find the value of r such that

2 . 4 = 4 . r

Solution:

Using of the Commutative Law For Multiplication stated above

` a . b = b . a`

=> For the given expression, r = 2


Example 3:

Find the value of m such that

2 . ( 3 + 1 ) = ( 2 . 3 ) + ( m . 1 )

Solution:

According to the Distributive Law, we have

`a . ( b + c ) = ( a . b ) + ( a . c )`

Through comparison of the given expression with the above formula, we conclude that

m = a = 2

We can also prove that m = 2 by method of equating the two sides of the given expression as shown below.

Given that

2 . ( 3 + 1 ) = ( 2 . 3 ) + ( m . 1 )

L.H.S (Left Hand Side) = 2 . ( 3 + 1 )

= 2 . 4 = 8

R.H.S (Right Hand Side) = ( 2 . 3 ) + ( m . 1 )

= 6 + m

By equating the two sides,

8 = 6 + m

=> m = 2 (Subtract 6 from both sides of the equation)

Hence we get the same answer as we got using the Distributive Formula.


Example 4:

Find the values of d such that

4 . ( 2 . d ) = ( 4 . 2 ) . 5

Solution:

Here we make use of the Associative Law For Multiplication which states that for any variables a, b and c

`a . ( b . c ) = ( a . b ) . c`

Through comparison we get

d = c = 5


Example 5:

For what value of x does the following identity hold?

x + 2 = 5 + 2

Solution:

Using the Cancellation law For Addition

`a + c = b + c iff a = b `

Here a = x, b = 5, c = 2

And the formula states that the identity will hold if and only if a = b i.e., if x = 5


Example 6:

Find n such that the following equation holds:

n . 3 = 5 . 3

Solution:

Using the Cancellation law For Multiplication

`a . c = b . c iff a = b ` where c != 0

Here

a = n

b = 5

c = 3 != 0

Hence the equation will hold if and only if a = b, that is if n = 5


Example 7:

What is 4 x 14 equal to? Make use of the properties stated above to simplify your solution?

Solution:

We have studied Distributive law which states that

`a . ( b + c ) = ( a . b ) + ( a . c )`

Here we can split up 14 = 10 + 4

=> 4 . ( 10 + 4 ) = ( 4 . 10 ) + ( 4 . 4 )

=> 4 x 14 = 40 + 16 = 56

Hence we can split our problem using this property and find the solution with greater ease.


Example 8:

Use distributive property and simplify the expression 3x + 9

Solution:

We have studied that in the distributive property, we distribute the operation of multiplication over addition. We can use this property to simplify the above expression.

We see that both 3, and 9 are multiples of 3. So if we take 3 out of the above expression, leaving the rest inside the parentheses, what we get is a distributed equivalent of the above expression.

=> 3x + 9 = 3 ( x + 3 )

A mentionable point here is that the property also holds for subtraction. For example

=>` 3x-9 = 3(x-3 )=3 ( x+(-3 ) )`

is also true.


Example 9:

Simplify this expression using the Associative Property for Multiplication.

2 x ( 4 x n )

Solution:

Using the Associative Property for Multiplication

` a . ( b . c ) = ( a . b ) . c`

=> 2 x ( 4 x n ) = ( 2 x 4 ) x n

=> 2 x ( 4 x n ) = 8 x n


Non-Commutative Operations

Here we need to mention that subtraction, division, matrix multiplication, vector product are all non-commutative.

Example 10:

`4-2!=2-4` => Subtraction is non-commutative

`( 4 )/( 2 )!=2/( 4 )` => Division is non-commutative

Similarly Matrix Multiplication and vector Product can be shown to be non-commutative.