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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