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
Thus, the sum of the given like terms is another like term whose numerical coefficient is 18.
Therefore, 4xy + 5xy + 9xy = 18xy
Addition of Negative Like Terms:
To add the negative like terms, we follow the below steps:
- Obtain all like terms.
- Obtain the sum of the numerical coefficients without negative signs of all like terms.
- 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
Therefore, the sum of the numerical coefficients
So, the sum of the given like terms is another like term whose numerical coefficient is
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–3ab from 2a – 7ab
Solution: Subtracting a–3ab from 2a – 7ab using the horizontal method is shown below.
2a – 7ab – (a–3ab)
Grouping the like terms, we get
(2a – a) – 7ab + 3ab = a – 4ab
Therefore, (2a – 7ab) – (a–3ab) = a – 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.
Solution:
Arranging the terms of the given expressions in descending powers of x and subtracting column-wise;
– 2x² + x + 5
+ 3x² – 6x – 4
(-) (+) (+)
_____________
– 5x² + 7x + 9
_____________
Solution:
9x – 5y + z
+ 3x + y – 3z
(-) (-) (+)
_____________
6x – 6y + 4z
_____________