Session 3: Addition and Subtraction of Algebraic Expressions – Rules, Methods, Steps, Interactives and Examples

In the lesson “Algebraic Expressions ”, we learned about algebraic expressions, their types and like and unlike terms in an algebraic expression. In this lesson, we shall study the fundamental operations of addition and subtraction of algebraic expressions. Since an algebraic expression may consist of like and unlike terms, so, the operation of addition and subtraction of algebraic expressions involves the addition or subtraction of these terms. We shall first study the addition of algebraic expressions.

Addition of Algebraic Expressions

In adding algebraic expressions, we collect different groups of like terms and find the sum of like terms in each group. The rules of addition of the numerical coefficients obey the rules of addition of integers.

Steps to Add Positive or Negative Like Monomials

To add positive or negative like monomials, we follow the following steps:

Step 1: Find the sum of the numerical coefficients of all monomials.

Step 2: The required sum is the sum of all the numerical coefficients, and the literal factor is the same as the literal factor of the given like monomials. 

1) Add 6xy and 2xy. 

2) Add −4y and −y.  

Steps to Add Positive and Negative Like Monomials 

To add positive and negative like monomials, we follow the following steps:

Step 1: Find the sum of the numerical coefficients of all positive monomials.

Step 2: Find the sum of the numerical coefficients of all negative monomials.

Step 3: The required numerical coefficient is the difference between the two numerical coefficients obtained in the above steps. The literal factor is the same as the literal factor of the given like monomials.

3) Add 7xyz and −2xyz.  

4) Add −5ab and 3ab.  

The collection of like terms can be done by any one of the following methods:

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. The following example will illustrate this method. 

5) Add 3x+2y and x+y using the horizontal method. 

Steps to Add Algebraic Expressions using the Horizontal Method

To add algebraic expressions using the horizontal method, we can use the following steps:   

Step 1: Write all addends in line with addition signs in between.

Step 2: As the sign before the bracket is +, the common multiplier is +1. So, remove the brackets without changing the sign of the terms.

Step 3: Group the like terms together. 

Step 4: Add the like terms to obtain the sum of expressions. 

Column Method:

In this method, each expression is written in a separate row such that their like terms are arranged one below the other in a column. Then the addition of like terms is done column-wise. The following example will illustrate this method. 

6) Add 7x+y and 3x+2y using the column method.

Steps to Add Algebraic Expressions Using the Column (Vertical) Method

To add algebraic expressions using the column(vertical) method, we can use the following steps:  

Step 1: The expressions are arranged vertically with the like terms in the same columns.

Step 2: Add the like terms to obtain the sum of expressions.

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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, which is to be subtracted and then the two expressions are added. 

In the subtraction of two expressions in the column method, we indicate the change of sign of every term in the expression to be subtracted below the original sign of each term. 

The following examples will illustrate the procedure.

Horizontal Method

In the case of subtraction by 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 subtract. The following example will illustrate this method.

7) Subtract 5a+2c−3b−5 from 6a+b−4c. 

Steps to Subtract Algebraic Expressions Using the Horizontal Method

To subtract algebraic expressions using the horizontal method, we can use the following steps:  

Step 1: Write the subtrahend after the minuend with a subtraction sign in between.

Step 2: As the sign before the bracket is -, the common multiplier is -1. So, replace the subtraction sign with an addition sign and write the additive inverse of the subtrahend.

Step 3: Remove the brackets and group the like terms together.

Step 4: Add the like terms to obtain the difference of expressions.

Column Method:

In the case of subtraction by this method, each expression is written in a separate row such that the like terms are arranged one below the other in a column, and then the subtraction of like terms is done column-wise. 

8) Subtract 3xy+2z² from 5xy+3z²−xz. 

Steps to Subtract Algebraic Expressions Using the Column (Vertical) Method

To subtract algebraic expressions using column(vertical) method, we can use the following steps:  

Step 1: The expressions are arranged vertically with the like terms in the same columns.

Step 2: The signs of all terms of the subtrahend are changed (from ‘+’ to ‘-’ or from ‘-’ to ‘+’).

Step 3: Add the like terms to arrive at the difference of expressions. 

Addition and Subtraction of Algebraic Expressions – Examples

Example 1 

Add .

We have, 

We can carry out the addition by both horizontal and column (vertical) method.

Horizontal Method:

Column Method:

Example 2  

Subtract .

Column Method:

Here, we arrange the expressions one below the other in columns and subtract. 

Horizontal Method: 

Example 3 

What should be added to  to obtain ?

Required expression is equal to the difference of  and . 

Required expression

Hence,  should be added to  to obtain .

Example 4

How much is  greater than ?

Required expression=

 Remember this! 

• To add two or more algebraic expressions, we collect different groups of like terms and then find the sum of like terms in each group.
• To subtract one like term from another – change the sign of the term to be subtracted and then add. 
• To subtract one algebraic expression from another – change the sign of each term of the expression to be subtracted and then add.