Addition of Algebraic Expressions



In addition of algebraic expressions while adding algebraic expressions we collect the like terms and add them. The sum of several like terms is the like term whose coefficient is the sum of the coefficients of these like terms.


Two ways to solve addition of algebraic expressions.

Horizontal Method: 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.

Column Method: In this method each expression is written in a separate row such that there like terms are arranged one below the other in a column. Then the addition of terms is done column wise. 

Following illustrations will illustrate these methods. 

Examples on addition of algebraic expressions:

1. Add: 6a + 8b - 7c, 2b + c - 4a and a - 3b - 2c

Solution: 

Horizontal Method:

(6a + 8b - 7c) + (2b + c - 4a) + (a - 3b - 2c) 

= 6a + 8b - 7c + 2b + c - 4a + a - 3b - 2c

Arrange the like terms together, then add. 

Thus, the required addition

= 6a - 4a + a + 8b + 2b - 3b - 7c + c - 2c

= 3a + 7b - 8c


Column Method:

Solution:

Writing the terms of the given expressions in the same order in form of rows with like terms below each other and adding column wise;

    6a + 8b - 7c

 - 4a + 2b +  c

     a - 3b - 2c

   3a + 7b - 8c

  = 3a + 7b - 8c




2. Add: 5x² + 7y - 8, 4y + 7 - 2x² and 6 – 5y + 4x².

Solution:


Writing the given expressions in descending powers of x in the form of rows with like terms below each other and adding column wise;

    5x² + 7y - 8

 - 2x² + 4y + 7

   4x² – 5y + 6
   ___________
   7x² + 6y + 5
   ___________

= 7x² + 6y + 5



3. Add: 8x² - 5xy + 3y², 2xy - 6y² + 3x² and y² + xy - 6x².

Solution:


Arranging the given expressions in descending powers of x with like terms under each other and adding column wise;

   8x² - 5xy + 3y²

   3x² - 2xy - 6y²

 -6x² +  xy +  y²
  _____________
  5x² - 2xy - 2y²
  _____________

= 5x² - 2xy - 2y²




4. Add: 11a² + 8b² - 9c², 5b² + 3c² - 4a² and 3a² - 4b² - 4c².

Solution:


Writing the terms of the given expressions in the same order in form of rows with like terms below each other and adding column wise;

  11a² + 8b² - 9c²

 - 4a² + 5b² + 3c²

   3a² - 4b² - 4c²
 ________________
 10a² + 9b² - 10c²
 ________________

= 10a² + 9b² - 10c²

5. Add the 3x + 2y and x + y.

Solution:

Horizontal Method:

(3x + 2y) + (x + y)

Arrange the like terms together, then add.

Thus, the required addition

= 3x + 2y + x + y

= 3x + x + 2y +y

= 4x + 3y

Column Method:

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in same vertical column and then add the different groups of like terms.

3x + 2y

+ x + y
_________
4 x + 3y



6. Add: x + y + 3 and 3x + 2y + 5

Solution:

Horizontal Method:

(x + y + 3) + (3x + 2y + 5)

= x + y + 3 + 3x + 2y + 5

Arrange the like terms together, then add.

Thus, the required addition

= x + 3x + y + 2y + 3 + 5

= 4x + 3y + 8

Column Method:

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in same vertical column and then add the different groups of like terms.

x + y + 3

+ 3x + 2y + 5
_________________
4x + 3y + 8



7. Add: 2x + 3y + z and 2x - y - z

Solution:

Horizontal Method:

(2x + 3y + z) + (2x - y – z)

=2x + 3y + z + 2x - y – z

Arrange the like terms together, then add.

Thus, the required addition

= 2x + 2x + 3y - y + z - z

=4x + 2y

Column Method:

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in same vertical column and then add the different groups of like terms.

2x + 3y + z

+ 2x - y - z
_____________
4x + 2y




8. Add: 5x³ – 2y³ and 7x³ – 3y³

Solution:

Horizontal Method:

(5x³ – 2y³) + (7x³ – 3y³)

=5x³ - 2y³ + 7x³ – 3y³

Arrange the like terms together, then add.

Thus, the required addition

= 5x³ + 7x³ - 2y³ - 3y³

=12x³ – 5y³

Column Method:

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in same vertical column and then add the different groups of like terms.

5x³ – 2y³

+ 7x³ - 3y³
_____________
12x³ – 5y³



9. Add: a² + b² + c² – 3abc and a² – b² + c² + abc

Solution:

Horizontal Method:

(a² + b² + c² - 3abc) + (a² – b² + c² + abc)

= a² + b² + c² - 3abc + a² – b² + c² + abc

Arrange the like terms together, then add.

Thus, the required addition

= (a² + a²) + (b² – b²) + (c² + c²) - 3abc + abc

= 2a² + 2c² -2abc

Column Method:

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in same vertical column and then add the different groups of like terms.

a² + b² + c² – 3abc

+ a² – b² + c² + abc
__________________
2a² + 0 + 2c² – 2abc



10. Add: xy² + 4x²y – 7x²y - 3xy² + 3 and x²y + xy²

We have;

xy² + 4x²y -7x²y - 3xy² + 3

= - 2xy² - 3x²y + 3

Solution:

Horizontal Method:

(xy² + 4x²y – 7x²y - 3xy² + 3) +(x²y + xy²)

= (-2xy² - 3x²y + 3) + x²y + xy²

= -2xy² - 3x²y + 3 + x²y + xy²

Arrange the like terms together, then add.

Thus, the required addition

= -2xy² + xy² - 3x²y + xy² + 3

= - xy² - 2x²y + 3

Column Method:

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in same vertical column and then add the different groups of like terms.

- 2xy² - 3x²y +3

+ xy² + x²y
________________
- xy² - 2x²y + 3



11. Add: 5x² + 7y - 6z², 4y + 3x², 9x² + 2z² - 9y and 2y - 2x².

Solution:

Horizontal Method:

(5x² + 7y - 6z²) + (4y + 3x²) + (9x² + 2z² - 9y) + (2y - 2x²).

= 5x² + 7y - 6z² + 4y + 3x² + 9x² + 2z² - 9y + 2y - 2x²

Arrange the like terms together, then add.

Thus, the required addition

= 5x² + 3x² + 9x² - 2x² + 7y + 4y - 9y + 2y - 6z² + 2z²

= 15x² + 4y - 4z²

Column Method:

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in same vertical column and then add the different groups of like terms.

5x² + 7y - 6z²

+ 3x² + 4y

+ 9x² - 9y + 2z²

- 2x² + 2y
________________
15x² + 4y - 4z².


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