Variables: A variable is denoted by Alphabets a, b, x, y, m, n, etc. The value of a variable is not fixed and can take various values. 

Constant: Constant is a fixed value, denoted by any numerical.

Algebraic expression: An algebraic expression is formed by the combination of variables and constants when arithmetic operations, addition, subtraction, multiplication and division, is applied to variables and constants. 

For example, 5x+4

This expression is formed when a variable 'x' is multiplied by a constant 5 and then adding a constant 4 to 5x. Similarly, an expression 'x²' is formed when a variable x is multiplied by itself.

Terms are added to form and expression. 

For example: 5x² + 4y 

Here, 5x² and 4y are 2 separate terms of the expression 5x² + 4y. 

1. Factors of a term: Considering the same algebraic expression 5x² + 4y, the term 5x² is formed by multiplying 5, x and x. Therefore, the variable x and constant 5 will be considered factors of the term 5x². Similarly, 4 and y are factors of the term 4y. 

2. Coefficients: The numerical value of a term is called the coefficient of the term. For example: in the expression 5x² + 4y, 5 is the coefficient of the term 5x² and the coefficient of x. Similarly, 4 is the coefficient of the term 4y and y.

3. Like and Unlike terms: They are called Like terms when the terms have the same algebraic factors (variables), and Unlike terms, if they have different algebraic factors(variables).
   For example: In the expression '2xy-3y+5xy-6', 2xy and 5xy are Like terms and -3y and -6 are Unlike terms. Similarly, -3y and 2xy are also Unlike terms.

Monomial: An expression with a single term is called a monomial.
For example: 7y, 5xy, -3 etc. 

Binomial: An expression with two, Unlike terms, is called a Binomial.
For example: 6xy+ 7m, 8x+9 etc.
Note: The terms cannot be Like terms, if added with each other to form a monomial. 

Trinomial: An expression with three, Unlike terms, is called a trinomial.
For example, 3x³+ 4y+ 6

Polynomial: An expression with one or more terms is called a polynomial. Therefore, a monomial, binomial and trinomial all are polynomials.

Q. Classify the following algebraic into monomials, binomials and trinomials.

(i) 4y – 7z

Solution:- Binomial.

An expression that contains two unlike terms is called a binomial.

(ii) y2

Solution:- Monomial.

An expression with only one term is called a monomial.

(iii) x + y – xy

Solution:- Trinomial.

An expression that contains three terms is called a trinomial.

(iv) 100

Solution:- Monomial.

An expression with only one term is called a monomial.

(v) ab – a – b

Solution:- Trinomial.

An expression that contains three terms is called a trinomial.

(vi) 5 – 3t

Solution:- Binomial.

An expression that contains two unlike terms is called a binomial.

• Addition and subtraction are applied to each algebraic term in an expression.
• Only Like terms can be added and subtracted with each other, Unlike terms are added as it is in the final answer. 
• When we add and subtract 2Like terms only, their coefficient is added or subtracted, not the variables. 
• For example: 3xy+ 4xy = 7xy, not 7*2x*2y.
• Summation and subtraction of algebraic expression can be done in 2 ways: 
  1. Horizontal method
  2. Vertical method

Consider the summation of the two algebraic expressions:
( 7a²+ 6ab +3b + 12)+ (8a²+ 5ab +6)

Horizontal method

(7+8) a² + (6+5) ab + 3b +18
15a²+11ab+3b+18

Vertical method

7a²+6ab+3b + 12
8a²+5ab+0b + 6

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
15a²+11ab+3b+18

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Finding the Values of an Expression

To find the value of an expression, we first need to find the value of a variable. We can find its value if we know the numerical value of all the variables of all the terms in an algebraic expression. 

For example, to find the area of a square, we need to know the value of the side L. We can automatically find the value of the expression for the area, i.e. L2. Suppose the length of the square is 10cm, so its area will be 10x10= 100.

Q1: If m = 2, find the value of

(i) m – 2

Answer:- Substitute the value of m=2 in the question.

= 2 -2

= 0

(ii) 3m – 5

Answer:- Substitute the value of m=2 in the question.

= (3 × 2) – 5

= 6 – 5

= 1

(iii) 9 – 5m

Answer:- Substitute the value of m=2 in the question.

= 9 – (5 × 2)

= 9 – 10

= – 1

(iv) 3m2 – 2m – 7

Answer:- Substitute the value of m=2 in the question.

= (3 × 22) – (2 × 2) – 7

= (3 × 4) – (4) – 7

= 12 – 4 -7

= 12 – 11

= 1

An algebraic expression can be used to write mathematical formulas and rules in a concise form. 

1. Perimeter formulas

Equilateral triangle: with 3 sides of length l.
Then its perimeter can be denoted by the formula 3l.

Square: If the length of the 4 sides of a square is l, its perimeter will be 4l

Pentagon: Perimeter of a regular pentagon with 5 sides will be 5l. 

2. Area formulas

Area of a square with side l = l2
Area of a rectangle with length l and breadth b = lb
Area of a triangle with base side b and height h = bxh/2

3. Rules for number patterns

1. If a number is denoted by n its successor will be (n+1).
   for example n=10 then n+1 =11

2. If n denotes a natural number, 2n will be even numbers and (2n+1) will be odd numbers.
   For example, n=2, 2n=4 and (2n+1)= 5. 

3. If the multiples of 3 are arranged in increasing order, the number at the nth position is denoted by 3n.
   For example, the number at 10th position will be 3x10= 30. 

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