Variables, Constants and Coefficients
Variable A quantity which can take various numerical values is known as a variable (or a literal).
Variables can be denoted by using the letters a, b, c, x, y, z, etc.
Constant A quantity which has a fixed numerical value is called a constant.
For example, 3, 25, and 8, 9 ,13- 12 are constants.
A number or a combination of numbers formed by using the arithmetic operations is called a numerical expression or an arithmetic expression.
For example, 3 + (4 × 5), 5 – (4 × 2), (7 × 9) ÷ 5 and (3 × 4) – (4 × 5 – 7) are numerical expressions.
An algebraic expression is a combination of variables and constants connected by arithmetic operations
(i) 5 added to y y + 5
(ii) 8 subtracted from n n – 8
(iii) 12 multiplied by x 12 x
(iv) p divided by 3 p/3
A term is a constant or a variable or a product of a constant and one or more variables.
3x2, 6x and – 5 are called the terms of the expression 3x2 + 6x + 5
A term could be
(i) a constant
(ii) a variable
(iii) a product of constant and a variable (or variables)
(iv) a product of two or more variables
In the expression a2 + 7a + 3, 2+ + the terms are a2 , 7a and 3. The number of terms is 3.
The coefficient of a given variable or factor in a term is another factor whose product with the given variable or factor is the term itself. If the coefficient is a constant, it is called a constant coefficient or a numerical coefficient.
In the term 5xy,
Coefficient of xy is 5 (numerical coefficient),
Coefficient of 5x is y,
Coefficient of 5y is x.
Like terms and Unlike terms
Terms having the same variable or product of variables with same powers are called Like terms.
Terms having different variable or product of variables with different powers are called Unlike terms.
Example 1. (i) x, -5x, 9x are like terms as they have the same variable x
(ii) 4x y, 7yx 2 2 - are like terms as they have the same variable x y 2
Example 2 (i) 6x, 6y are unlike terms
(ii) 3xy , 5xy, 8x, 10y 2 - are unlike terms
Degree of an Algebraic expression
Consider the expression 8x2 + 6x + 7 It has 3 terms 8x2, -6x and 7.
In the term 8x2 the power of the variable x is 2.
In the term 6 x, the power of the variable x is 1.
The term 7 is called as a constant term or an independent term.
The term 7 is 7 x 1= 7x0 in which the power of the variable x is 0.
In the above expression the term 8x2 has the highest power 2. So the degree of the expression 8x2 – 6x + 7 is 2.
“ The degree of an expression of one variable is the highest value of the exponent of the variable. The degree of an expression of more than one variable is the highest value of the sum of the exponents of the variables in different terms.”
Note: The degree of a constant is 0.
Addition and subtraction of expressions is same as Adding and subtracting like terms
To fi nd the sum of two or more like terms, we add the numerical coefficient of the like terms. Similarly, to fi nd the difference between two like terms, we find the difference between the numerical coefficients of the like terms. There are two methods in finding the sum or difference between the like terms namely,
(i) Horizontal method
(ii) Vertical method
(i) Horizontal method: In this method, we arrange all the terms in a horizontal line and then add or subtract by combining the like terms.
Add 3x and 5x.
3x + 5x =( 3 + 5 )´ x = 7´ x= 7x
(ii) Vertical method: In this method, we should write the like terms vertically and then add or subtract.
+ 7 a
Subtracting a term is the same as adding its inverse. For example subtracting + 3a is the same as adding – 3a.
Subtract -2xy from 9xy.
– 2 xy
(+) (change of sign)
Notes Unlike terms cannot be added or subtracted the way like terms are added or subtracted.
For example when 7 is added to x we write it as x + 7 in which both the terms 7 and x are retained.
Similarly, if we add the unlike terms 4xy and 5, the sum is 4xy + 5. If we subtract 6 from 5pq the result is 5pq-6.
1) What should be subtracted from 4p + 6q + 14 to get -5p + 8q + 20?
2. Three sides of a triangle are 3a + 4b - 2, a - 7 and 2a - 4b + 3. What is its perimeter?
3. The sides of a rectangle are 3x + 2 and 5x + 4. Find its perimeter.
4. Ram spends 4a+3 rupees for a shirt and 8a - 5 rupees for a book. How much does he spend in all?
5. A wire is 10x - 3 metres long. A length of 3x + 5 metres is cut out of it for use. How much wire is left out?
6. (iii) If A = 8x - 3y + 9, B =- y - 9 and C = 4x - y - 9 fi nd A + B - C.