This topic gives an overview of;
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.
Consider the following problems:
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.
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.
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.
The simplest expressions are monomials. They consist of only one term. To begin with we shall learn how to add or subtract like terms.
3x + 4x = (3 × x) + (4 × x)
= (3 + 4) × x (using distributive law)
= 7 × x = 7x or 3x + 4x = 7x
8xy + 4xy + 2xy = (8 × xy) + (4 × xy) + (2 × xy)
= (8 + 4 + 2) × xy
= 14 × xy = 14xy or 8xy + 4xy + 2xy = 14 xy
7n – 4n = (7 × n) – (4 × n)
= (7 – 4) × n = 3 × n = 3n or 7n – 4n = 3n
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.
Let us take some examples:
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
Note we have put like terms together; the single unlike term 8z will remain as it is. Therefore, the sum = 10x + 6 + 8z
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
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