Addition of Algebraic Expressions

In mathematics, just like we add many numbers as we can and find the sum, we add two or many algebraic expressions too. However, for the addition of algebraic expressions, we combine all the like terms and then add them.

What Is Addition of Algebraic Expressions?

Addition of algebraic expressions is quite similar to the addition of numbers. However, the addition of algebraic expressions requires categorizing the terms in an algebraic expression into two types – like and unlike terms. Then, taking up the like terms and adding them.

Like terms are the terms that have the same power for the same variables. In like terms, one can only change the numerical coefficient. In the below examples, only the numerical coefficient differs and we have the same variable in each of the like terms raised to the same power:

• 5x and 13x
• 7y³ and 3y³

Terms that have different variables or the same variables raised to different powers are known as, unlike terms. Examples of unlike terms are,

• 5x and 5y
• 2m⁵ and 8m³

How to do Addition of Algebraic Expressions?

For adding algebraic expressions, we first need to collect the like terms and then add them. The sum of the like terms is a like term only whose coefficient is the sum of the coefficients of the like terms.     

Can we add 3 pencils and 3 erasers? The answer is NO. We cannot add 3 pencils and 3 erasers, as they are two different objects. Similar is the case of terms in an algebraic expression. We cannot add two or more unlike terms. An important point to remember is that to add polynomials (algebraic expressions), we can only add like terms. There are two methods to do the addition of algebraic expressions:

• Horizontal method of Addition of Algebraic Expressions
• Column method for Addition of Algebraic Expressions

Horizontal Method of Addition of Algebraic Expressions

Steps to be followed to do the addition of algebraic expressions by the horizontal method is written below:

• Step 1: Write all the expressions in a horizontal line by putting them into brackets and put an addition sign in between.
• Step 2: Group all the like terms together from all the expressions and rewrite the expression so formed.
• Step 3:  Add numerical coefficients of all the like terms followed by the common variable.
• Step 4:  Rewrite the simplified expression, and make sure all the terms in the final answer should be unlike terms.

Column Method for Addition of Algebraic Expressions

• Step 1: Write all the expressions one below the other. Make sure to like terms in one column. 

2x² + 3x − 4y + 7

         5x + 4y − 3 

If there a term whose like term is not there in the second expression, for example, then either write below it or leave that column blank.

• Step 2: Add the numerical coefficient of each column (like terms) and write below it in the same column followed by the common variable. 

• Step 3: Rewrite the final answer, 2x²+8x+4.

Let us take an example to understand it in a better way.

Add 2m² + mn -7n, 3m² +7mn + 5n and (-5mn + n).

Horizontal Method

• Step 1- (2m²+mn-7n)+(3m²+7mn+5n)+(-5mn+n)
• Step 2- (2m²+3m²)+(mn+7mn-5mn)+(-7n+5n+n)
• Step 3-  (5m²+3mn-n)

∴∴ The final answer is (5m²+3mn-n).

Column Method

Tips and Tricks

• We can ignore the order of variables in like terms in an algebraic expression. For example,  3a + 5b, and, 7b + 4a both are like terms.
• We can ignore writing 1 as the numerical coefficient of any term. For example, we write mn and not 1mn. 1mn looks odd, isn’t it?

Challenging Question

Add the given expressions:  1.52xy+57y21.52xy+57y2, 5xy+32y25xy+32y2 and 56xy+113y2−456xy+113y2−4