Class 7 Maths Algebraic Expressions Notes

Introduction:

The chapter “Algebraic Expressions” in Class 7 Maths introduces students to the concept of algebraic expressions, a fundamental topic in algebra. Algebraic expressions are mathematical phrases that involve numbers, variables, and operators like addition, subtraction, multiplication, and division. Understanding algebraic expressions is crucial because they form the basis for solving equations and more complex algebraic problems. In this chapter, students will learn to identify, simplify, and evaluate algebraic expressions.

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Key Concepts Covered:

1. What is an Algebraic Expression?

• An algebraic expression is a mathematical expression that consists of variables (letters representing numbers), constants (fixed numbers), and operators (+, −, ×, ÷). It does not contain an equality sign (=).

Example:
$2x + 3$2x+3, $4a – 5b + 7$4a−5b+7, and $3xy – 2y$3xy−2y are algebraic expressions.

In the expression $2x + 3$2x+3, 2 is the coefficient of the variable $x$x, and 3 is the constant.

2. Terms in an Algebraic Expression:

• The terms of an algebraic expression are the parts that are added or subtracted.
  • Example: In $4x + 5y – 7$4x+5y−7, the terms are $4x$4x, $5y$5y, and $-7$−7.
  • A monomial has only one term (e.g., $3x$3x).
  • A binomial has two terms (e.g., $2x + 3$2x+3).
  • A trinomial has three terms (e.g., $x + 2y – 5$x+2y−5).

3. Variables and Constants:

• Variables: A symbol (usually a letter) that represents an unknown value.
  • Example: In $3x + 4$3x+4, $x$x is the variable.
• Constants: A fixed numerical value that does not change.
  • Example: In $3x + 4$3x+4, 4 is the constant.

4. Coefficients:

• The coefficient is the numerical factor in front of a variable.
  • Example: In $5x + 2$5x+2, 5 is the coefficient of the variable $x$x.

The coefficient shows how many times the variable is multiplied.

5. Like and Unlike Terms:

• Like terms are terms that have the same variables raised to the same powers.
  • Example: $3x$3x and $5x$5x are like terms because both have the variable $x$x raised to the same power (1).
• Unlike terms are terms that have different variables or the same variable raised to different powers.
  • Example: $3x$3x and $4y$4y are unlike terms because $x$x and $y$y are different variables.

6. Simplifying Algebraic Expressions:

• To simplify an algebraic expression, combine like terms by adding or subtracting their coefficients.

Example:
Simplify $3x + 4x – 2y + 5y$3x+4x−2y+5y:$$= (3x + 4x) + (-2y + 5y) = 7x + 3y$$=(3x+4x)+(−2y+5y)=7x+3y

7. Evaluating Algebraic Expressions:

• To evaluate an algebraic expression, substitute the values of the variables and perform the operations.

Example:
If $x = 3$x=3 and $y = 2$y=2, evaluate the expression $2x + 3y$2x+3y:$$2x + 3y = 2(3) + 3(2) = 6 + 6 = 12$$2x+3y=2(3)+3(2)=6+6=12

8. Addition and Subtraction of Algebraic Expressions:

• To add or subtract algebraic expressions, first combine like terms.

Example:
Add $(2x + 3y)$(2x+3y) and $(4x – 5y)$(4x−5y):$$(2x + 3y) + (4x – 5y) = (2x + 4x) + (3y – 5y) = 6x – 2y$$(2x+3y)+(4x−5y)=(2x+4x)+(3y−5y)=6x−2y

9. Multiplication of Algebraic Expressions:

• To multiply algebraic expressions, apply the distributive property, which means multiplying each term in one expression by each term in the other expression.

Example:
Multiply $(2x + 3)$(2x+3) by $(x – 1)$(x−1):$$(2x + 3)(x – 1) = 2x(x – 1) + 3(x – 1) = 2x^2 – 2x + 3x – 3 = 2x^2 + x – 3$$(2x+3)(x−1)=2x(x−1)+3(x−1)=2×2−2x+3x−3=2×2+x−3

10. Identity in Algebra:

• An algebraic identity is an equation that is true for all values of the variables.

Example:
The identity $(a + b)^2 = a^2 + 2ab + b^2$(a+b)2=a2+2ab+b2 is valid for all values of $a$a and $b$b.

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Important Questions with Answers:

1. What is an algebraic expression?
   • Answer: An algebraic expression is a mathematical expression involving variables, constants, and operators (e.g., $3x + 4$3x+4).
2. How do you simplify 5x+3y−2x+4y5x + 3y – 2x + 4y5x+3y−2x+4y?
   • Answer: Simplify by combining like terms: $$(5x – 2x) + (3y + 4y) = 3x + 7y$$(5x−2x)+(3y+4y)=3x+7y
3. What is the coefficient of xxx in the expression 7x+57x + 57x+5?
   • Answer: The coefficient of $x$x is 7.
4. How do you evaluate 2x+3y2x + 3y2x+3y if x=2x = 2x=2 and y=3y = 3y=3?
   • Answer: $$2x + 3y = 2(2) + 3(3) = 4 + 9 = 13$$2x+3y=2(2)+3(3)=4+9=13
5. What are like terms?
   • Answer: Like terms are terms that have the same variable raised to the same power, e.g., $2x$2x and $5x$5x.
6. How do you multiply (x+4)(x + 4)(x+4) by (x−2)(x – 2)(x−2)?
   • Answer: $$(x + 4)(x – 2) = x(x – 2) + 4(x – 2) = x^2 – 2x + 4x – 8 = x^2 + 2x – 8$$(x+4)(x−2)=x(x−2)+4(x−2)=x2−2x+4x−8=x2+2x−8