Algebraic Expressions Class 7 Notes: Chapter 12

Introduction to Algebraic Expressions

Constant

Constant is a quantity which has a fixed value.

Terms of Expression

Parts of an expression which are formed separately first and then added are known as terms. They are added to form expressions.

Example: Terms 4x and 5 are added to form the expression (4x +5).

Coefficient of a term

The numerical factor of a term is called coefficient of the term.
Example: 10 is the coefficient of the term 10xy in the expression 10xy+4y.

Algebra as Patterns

Writing Number patterns and rules related to them

  • If a natural number is denoted by n, its successor is (n + 1).
    Example: Successor of n=10 is n+1 =11.
  • If a natural number is denoted by n, 2n is an even number and (2n+1) an odd number.
    Example: If n=10, then 2n = 20 is an even number and 2n+1 = 21 is an odd number.

Writing Patterns in Geometry

  • Algebraic expressions are used in writing patterns followed by geometrical figures.
    Example: Number of diagonals we can draw from one vertex of a polygon of n sides is (n – 3).

Writing Patterns in Geometry

Definition of Variables

  • Any algebraic expression can have any number of variables and constants.
    • Variable

      • A variable is a quantity that is prone to change with the context of the situation.
      • a,x,p,… are used to denote variables.
    • Constant

  • It is a quantity which has a fixed value.
  • In the expression 5x+4, the variable here is x and the constant is 4.
  • The value 5x and 4 are also called terms of expression.
  • In the term 5x, 5 is called the coefficient of x. Coefficients are any numerical factor of a term.

Factors of a term

Factors of a term are quantities which can not be further factorised. A term is a product of its factors.
Example: The term –3xy is a product of the factors –3, x and y.

Formation of Algebraic Expressions

  • Variables and numbers are used to construct terms.
  • These terms along with a combination of operators constitute an algebraic expression.
  • The algebraic expression has a value that depends on the values of the variables.
  • For example, let 6p2−3p+5 be an algebraic expression with variable p
    The value of the expression when p=2 is,
    6(2)− 3(2) + 5
    ⇒ 6(4) − 6 + 5 = 23
    The value of the expression when p=1 is,
    6(1)− 3(1) + 5
    ⇒ 6 − 3 + 5 = 8

Like and Unlike Terms

Like terms

  • Terms having same algebraic factors are like terms.
    Example: 8xy and 3xy are like terms.

Unlike terms

  • Terms having different algebraic factors are unlike terms.
    Example: 7xy and −3x are unlike terms.

Monomial, Binomial, Trinomial and Polynomial Terms

Types of expressions based on the number of terms

Based on the number of terms present, algebraic expressions are classified as:

  • Monomial: An expression with only one term.
    Example: 7xy, −5m, etc.
  • Binomial: An expression which contains two, unlike terms.
    Example: 5mn+4, x+y, etc
  • Trinomial: An expression which contains three terms.
    Example: x+y+5, a+b+ab, etc.

Polynomials

  • An expression with one or more terms.
    Example: x+y, 3xy+6+y, etc.

Addition and Subtraction of Algebraic Equations

  • Mathematical operations like addition and subtraction can be applied to algebraic terms.
  • For adding or subtracting two or more algebraic expression, like terms of both the expressions are grouped together and unlike terms are retained as it is.
  • 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 like terms.
  • 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.
  • For example, 2y + 3x − 2x + 4y
    ⇒ x(3−2) + y(2+4)
    ⇒ x+6y
  • Summation of algebraic expressions can be done in two ways:
    Consider the summation of the algebraic expressions 5a2+7a+2ab and 7a2+9a+11b
  • Horizontal method
    5a2+7a+2ab+7a2+9a+11b
    = (5+7)a2+(7+9)a+2ab+11b
    = 12a2+16a+2ab+11b
  • Vertical method
    5a2+7a+2ab
    7a2+9a+11b
    ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
    12a2+16a+2ab+11b
    ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Solving an Equation

Use of algebraic expressions in the formula of the perimeter of figures

Algebraic expressions can be used in formulating perimeter of figures.
Example: Let l be the length of one side then the perimeter of:
Equilateral triangle = 3l
Square = 4l
Regular pentagon = 5l

Use of algebraic expressions in formula of area of figures

Algebraic expressions can be used in formulation area of figures.
Example: Let l be the length and b be the breadth then the area of:
Square = l2
Rectangle = l×b = lb
Triangle = \(\frac{b\times h}{2}\), where b and h are base and height, respectively.

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