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.

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

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)

\(\therefore\) 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:  \(\dfrac{1.5}{2}xy + \dfrac{5}{7}y^2\), \(5xy + \dfrac{3}{2}y^2\) and \(\dfrac{5}{6}xy + \dfrac{11}{3}y^2-4\) 

Topics Related to Addition of Algebraic Expressions

• Subtraction of Algebraic Expressions
• Multiplication of Algebraic Expressions
• Division of Algebraic Expressions
   

What Is the Definition of an Algebraic Expression?

An algebraic expression is a combination of terms connected by the operations such as addition, subtraction, multiplication, division, etc.

What Are the Two Basic Rules for Solving Algebraic Expressions?

Two rules for solving algebraic expressions are: 

• Only like terms can be added or subtracted.
• It is important to solve brackets first.

What Is the Rule for Adding Algebraic Terms?

The basic rule to add algebraic terms is to add only like terms.

How Do you Combine Like Terms and Simplify?

Group together all the like terms, add or subtract the numerical coefficients of the like terms and attach the common variable to it.

How Do you Multiply Algebraic Expressions?

In order to multiply algebraic expressions, we use using algebraic identities. Some of the algebraic identities are given below:

• (a + b)² =  a² + 2ab + b² 
• (a - b)²  =  a² - 2ab + b² 
• (a + b)(a − b) = a² − b² 
• (x+a) (x+b) = x²+x(a+b)+ab 

How Do you Add Unlike Terms?

Unlike terms cannot be added. We write it as it is in our final answer.

How To Add Algebraic Expressions with Exponents?

We can add terms having the same variable raised to the same exponent. For example, 5x³+8x³=13x³

How To Add Rational Algebraic Expressions?

In rational algebraic expressions, we first group together the like terms. Then, we add the numerical coefficients of the common variable by taking the LCM of the denominator. It is the same as adding fractions. For example:

• \(\dfrac{1}{x} + \dfrac(2}{x} = \dfrac{3}{x}\) (common denominators)
• \(\dfrac{1}{x} + \dfrac(1}{2x} = \dfrac{3}{2x} = \dfrac{3}{2x}\) (different denominators)