How to add/subtract terms which contain variable/variables


Only Like-terms in an expression can be added or subtracted. 

Constant like-terms are added/subtracted same as we do addition/subtraction of integers. 

To add/ subtract like-terms containing variable/variables, we add/subtract their coefficients and write the variable/variables beside it. 

Let us understand this with following examples.


Example 1:

Simplify: 2p + 3p

Solution: 

This expression has two terms, 2p and 3p.


In 2p, 2 is coefficient and p is variable.

In 3p, 3 is coefficient and p is variable. 

(Both have same variable. So, we say 2p and 3p are like-terms.)


To add 2p and 3p, we add the coefficients 2 and 3 and write p beside it. That is:

2p + 3p = (2 + 3) p = 5p

Thus, the expression 2p + 3p simplified to 5p.


Example 2:

Simplify: k - 3k

Solution: 

This expression also has two terms, k and -3k.


In k, 1 is coefficient and k is variable.

In -3k, -3 is coefficient and k is variable. 

(So, k and -3k are like-terms.)


Thus:

k - 3k = (1 - 3) k = - 2k

Expression k - 3k simplified to - 2k.


Example 3:

Simplify: t + 5 - 8t - 4 - 2t

Solution: 

This expression has five terms, t , 5 , - 8t , - 4 and - 2t


Among these,

 t, - 8t and - 2t are like-terms.

5 and - 4 are also like-terms.


t + 5 - 8t - 4 - 2t 

= t - 8t - 2t + 5 - 4      (Gathered like-terms)           

= (1 - 8 - 2) t + 1

= - 9t + 1

Thus, t + 5 - 8t - 4 - 2t simplified to - 9t +1.


Example 4:

Simplify: y² + 3y - 4x - y² - 8x + 5

Solution: 

This expression has six terms, y² , 3y , - 4x , - y² , - 8x and 5


Among these,

y² and - y² are like-terms.

- 4x and - 8x are like-terms

3y and 5 do not have any like-term.


y² + 3y - 4x - y² - 8x + 5 


= y² - y² - 4x - 8x + 3y + 5    (Gathered like-terms)

                                        

= (1 - 1) y² + (- 4 - 8)x + 3y + 5


= - 12x + 3y + 5                   (As 0y² = 0 × y² = 0)


 Thus, y² + 3y - 4x - y² - 8x + 5 simplified to - 12x + 3y + 5


Example 5:

Can the following expression be simplified?

xy² + 4y - 2x - x²y + 5

Solution: 

This expression has five terms, xy² , 4y , - 2x , - x²y and 5

None of these terms are like-terms. So, this expression cannot be simplified.


Remember: We cannot add/subtract the terms which are not like-terms. For example 

2xy + 2x
are not like-terms, so they cannot be added.

Similarly,

3t² - 4t
are not like-terms, so they cannot be subtracted.