Addition of Algebraic Expressions

Addition of positive-like terms: 

To add the positive like terms, we follow the below steps:
1. Obtain all like terms.
2. Find the sum of the numerical coefficients of all terms.
3. Write the required sum as the like term whose numerical coefficient is the number obtained in the second step, and the variable factor is the same as the variable factors of the given like terms.

Example: Add 4xy, 5xy and 9xy

Solution: The sum of the numerical coefficients of the given like terms is $4+5+9=18.$

Thus, the sum of the given like terms is another like term whose numerical coefficient is 18.

Therefore, 4xy + 5xy + 9xy = 18xy$\mathrm{}$

Addition of Negative Like Terms: 

To add the negative like terms, we follow the below steps:

1. Obtain all like terms.
2. Obtain the sum of the numerical coefficients without negative signs of all like terms.
3. Write an expression as a product of the number obtained in the second step, with all the variable coefficients preceded by a minus sign.

Example: Add –7xy, –3xy and – 8xy

Solution: The numerical coefficients (without the negative sign) of the given like terms are 7$,3,8.$

Therefore, the sum of the numerical coefficients$=7+3+8=18$
So, the sum of the given like terms is another like term whose numerical coefficient is $18.$

Hence, –7xy –3xy –8xy = –18xy

If positive and negative like terms are involved, add the coefficients according to the general rule of addition of integers or rational numbers and continue like the above two methods.

Horizontal Method of Addition 

In this method, all expressions are written in a horizontal line and then the terms are arranged to collect all the groups of like terms and then added or subtracted as required.

Example:  Add 3x + 2y and x + y$$

Solution: Adding 3x + 2y and x + y$$ using the horizontal method shown below.
(3x + 2y) + (x + y)
Grouping the like terms, we get
(3x + x) + (2y + y)
⇒ (3 + 1)x + (2 + 1)y
= 4x + 3y
So, (3x + 2y) + (x + y) = 4x + 3y

Column Method of Addition

In this method, we write the terms of the given expressions in the same order in the form of rows with like terms below each other and add column-wise.

Add: 6a + 8b – 7c, 2b + c – 4a and a – 3b – 2c

Solution: 

 6a + 8b – 7c

 – 4a + 2b +  c

     a – 3b – 2c

   3a + 7b – 8c

  = 3a + 7b – 8c

Subtraction of Algebraic Expressions

To subtract an algebraic expression from another, we should change the signs (from $+$ to $-$ or from $-$ to $+$) of all the terms of the expression to be subtracted and then the two expressions are added.

Example: Subtract −8a from −3a

Solution: −3a − (–8a) = −3a + 8a = 5a

Horizontal Method of Subtraction 

Example: Subtract: a${}^2$–3ab${}^{\mathrm{}}$ from 2a${}^2$ – 7ab${}^{\mathrm{}}$

Solution: Subtracting a${}^2$–3ab${}^{}$${}^{\mathrm{}}$ from 2a${}^2$ – 7ab${}^{}$ using the horizontal method is shown below.

2a${}^2$ – 7ab${}^{}$ – (a${}^2$–3ab)

${}^{}$${}^{}$Grouping the like terms, we get

(2a${}^2$ – a${}^2$) –  7ab${}^{}$ + 3ab = a${}^2$ –  4ab${}^{}$  
Therefore, (2a${}^2$ – 7ab)${}^{}$ – (a${}^2$–3ab) = a${}^2$ –  4ab${}^{}$  

Column Method of Subtraction 

Example: Subtract 4a + 5b – 3c from 6a – 3b + c

Solution: 

   6a  – 3b +   c 

+ 4a + 5b – 3c 

(-)   (-)   (+) 
_____________
  2a  – 8b  + 4c 
_____________

Example: Subtract 3x² – 6x – 4 from 5 + x – 2x².