Objective:

This topic gives an overview of; 

• Adding and subtracting like terms
• Adding and subtracting general algebraic expressions

Operations on Algebraic Expressions

We can perform arithmetic operations on algebraic expressions. To add the like terms in an algebraic expression, multiply the sum of their coefficients with their common algebraic factors. For example, let us add.

Addition and Subtraction of Algebraic Expressions

Consider the following problems:

1. Sarita has some marbles. Ameena has 10 more. Appu says that he has 3 more marbles than the number of marbles Sarita and Ameena together have. How do you get the number of marbles that Appu has?

Since it is not given how many marbles Sarita has, we shall take it to be x. Ameena then has 10 more, i.e., x + 10. Appu says that he has 3 more marbles than what Sarita and Ameena have together. So we take the sum of the numbers of Sarita’smarbles and Ameena’s marbles, and to this sum add 3, that is, we take the sum of x, x + 10 and 3.

2. Ramu’s father’s present age is 3 times Ramu’s age. Ramu’s grandfather’s age is 13 years more than the sum of Ramu’s age and Ramu’s father’s age. How do you find Ramu’s grandfather’s age?

Since Ramu’s age is not given, let us take it to be y years. Then his father’s age is 3y years. To find Ramu’s grandfather’s age we have to take the sum of Ramu’s age (y) and his father’s age (3y) and to the sum add 13, that is, we have to take the sum of y, 3y and 13.

3. In a garden, roses and marigolds are planted in square plots. The length of the square plot in which marigolds are planted is 3 metres greater than the length of the square plot in which roses are planted. How much bigger in area is the marigold plot than the rose plot? 

Let us take l metres to be length of the side of the rose plot. The length of the side of the marigold plot will be (l + 3) metres. Their respective areas will be l² and (l + 3)². The difference between (l² + 3)² and l² will decide how much bigger in area the marigold plot is.

In all the three situations, we had to carry out addition or subtraction of algebraic expressions. There are a number of real life problems in which we need to use expressions and do arithmetic operations on them. In this section, we shall see how algebraic expressions are added and subtracted.

Adding and subtracting like terms

The simplest expressions are monomials. They consist of only one term. To begin with we shall learn how to add or subtract like terms.

• Let us add 3x and 4x. We know x is a number and so also are 3x and 4x. Now,

3x + 4x = (3 × x) + (4 × x)

= (3 + 4) × x (using distributive law)

= 7 × x = 7x  or 3x + 4x = 7x

• Let us next add 8xy, 4xy and 2xy

8xy + 4xy + 2xy = (8 × xy) + (4 × xy) + (2 × xy)

= (8 + 4 + 2) × xy                                      

= 14 × xy = 14xy  or 8xy + 4xy + 2xy = 14 xy          

• Let us subtract 4n from 7n.

7n – 4n = (7 × n) – (4 × n)

= (7 – 4) × n = 3 × n = 3n or  7n – 4n = 3n

• In the same way, subtract 5ab from 11ab.

11ab – 5ab = (11 – 5) ab = 6ab

Thus, the sum of two or more like terms is a like term with a numerical coefficient equal to the sum of the numerical coefficients of all the like terms. Similarly, the difference between two like terms is a like term with a numerical coefficient equal to the difference between the numerical coefficients of the two like terms.

Note, unlike terms cannot be added or subtracted the way like terms are added or subtracted. We have already seen examples of this, when 5 is added to x, we write the result as (x + 5). Observe that in (x + 5) both the terms 5 and x are retained. Similarly, if we add the unlike terms 3xy and 7, the sum is 3xy + 7. If we subtract 7 from 3xy, the result is 3xy – 7.

Adding and Subtracting General Algebraic Expressions

Let us take some examples:

• Add 3x + 11 and 7x – 5 . 

The sum = 3x + 11 + 7x – 5 .

Now, we know that the terms 3x and 7x are like terms and so also are 11 and – 5.

Further 3x + 7x = 10 x and 11 + (– 5) = 6. We can, therefore, simplify the sum as: The sum = 3x + 11 + 7x – 5

 = 3x + 7x + 11 – 5(rearranging terms)

 = 10x + 6

Hence, 3x + 11 + 7x – 5 = 10x + 6

• Add 3x + 11 + 8z and 7x – 5. 
  
  The sum = 3x + 11 + 8z + 7x – 5.
  
  = 3x + 7x + 11 – 5 + 8z (rearranging terms)

Note we have put like terms together; the single unlike term 8z will remain as it is. Therefore, the sum = 10x + 6 + 8z

•  Subtract a – b from 3a – b + 4

The difference = 3a – b + 4 – (a – b)

= 3a – b + 4 – a + b

Observe how we took (a – b) in brackets and took care of signs in opening the bracket. Rearranging the terms to put like terms together,

The difference = 3a – a + b – b + 4

 = (3 – 1) a + (1 – 1) b + 4

The difference = 2a + (0) b + 4 = 2a + 4   or  3a – b + 4 – (a – b) = 2a + 4

Summary:

•  Algebraic expressions are formed from variables and constants. We use the operations of addition, subtraction, multiplication and division on the variables and constants to form expressions. For example, the expression 4xy + 7 is formed from the variables x and y and constants 4 and 7. The constant 4 and the variables x and y are multiplied to give the product 4xy and the constant 7 is added to this product to give the expression.

• The sum (or difference) of two like terms is a like term with coefficient equal to the sum (or difference) of the coefficients of the two like terms. Thus, 8xy – 3xy = (8 – 3 )xy, i.e., 5xy.

• When we add two algebraic expressions, the like terms are added as given above; the unlike terms are left as they are. Thus, the sum of 4x2 + 5x and 2x + 3 is 4x2 + 7x + 3; the like terms 5x and 2x add to 7x; the unlike terms 4x2 and 3 are left as they are..