**Algebraic Expressions**

**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.

**Numerical expression**

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.

**Algebraic Expression**

An algebraic expression is a combination of variables and constants connected by arithmetic operations

**Statement Expressions**

(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

**Term**

A term is a constant or a variable or a product of a constant and one or more variables.

3x

^{2}, 6x and – 5 are called the terms of the expression 3x^{2}+ 6x + 5A 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 a

^{2}+ 7a + 3, 2+ + the terms are a^{2}, 7a and 3. The number of terms is 3.**Coefficient**

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 5

*xy*,Coefficient of

*xy*is 5 (numerical coefficient),Coefficient of 5

*x*is*y*,Coefficient of 5

*y*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**.**(i)**

*Example 1.**x*, -5

*x*, 9

*x*are like terms as they have the same variable

*x*

(ii) 4

*x y*, 7*yx*2 2 - are like terms as they have the same variable*x y*2**(i) 6**

*Example 2**x*, 6

*y*are unlike terms

(ii) 3

*xy*, 5*xy*, 8*x*, 10*y*2 - are unlike terms**Degree of an Algebraic expression**

Consider the expression 8

*x*6^{2}+*x +*7 It has 3 terms 8*x*^{2}, -6*x*and 7.In the term 8

*x*^{2}*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= 7

*x*^{0}in which the power of the variable*x*is 0.In the above expression the term 8

*x*^{2}has the highest power 2. So the degree of the expression 8*x*^{2}– 6*x*+ 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 3

*x*and 5*x*.3

*x*+ 5*x*=( 3 + 5 )´ x = 7´*x= 7x***(ii) Vertical method:**In this method, we should write the like terms vertically and then add or subtract.

4

*a* + 7

*a**----------------------*

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*a*Subtracting a term is the same as adding its inverse. For example subtracting + 3

*a*is the same as adding – 3*a*.Subtract -2

*xy*from 9*xy*.**9**

*xy*

– 2

*xy* (+) (change of sign)

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*xy**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 4

*xy*and 5, the sum is 4*xy*+ 5. If we subtract 6 from 5*pq*the result is 5*pq*-6.**Try these**

1) What should be subtracted from 4

*p*+ 6*q*+ 14 to get -5*p*+ 8*q*+ 20?2. Three sides of a triangle are 3

*a*+ 4*b*- 2,*a*- 7 and 2*a*- 4*b*+ 3. What is its perimeter?3. The sides of a rectangle are 3

*x*+ 2 and 5*x*+ 4. Find its perimeter.4. Ram spends 4a+3 rupees for a shirt and 8

*a*- 5 rupees for a book. How much does he spend in all?5. A wire is 10

*x*- 3 metres long. A length of 3*x*+ 5 metres is cut out of it for use. How much wire is left out?6. (iii) If A = 8

*x*- 3*y*+ 9, B =-*y*- 9 and C = 4*x*-*y*- 9 fi nd A + B - C.
an algebraic expression consists of variables constants, operator signs (plus, minus, multiplication, division or exponentiation)

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