Class 7 Notes

Class 7 Mathematics Notes Chapter 12 (Algebraic Expressions) – Mathematics Book

Philoid

Philoid

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Mathematics
Alright class, let's get started with a crucial chapter for your foundational mathematics: Algebraic Expressions. Understanding this chapter well is vital, not just for Class 7, but it forms the basis for much of higher mathematics you'll encounter, including in various government exams.

Chapter 12: Algebraic Expressions - Detailed Notes

1. Introduction: What are Algebraic Expressions?

  • Think of arithmetic. We use numbers (like 5, -10, 3/4) and operations (+, -, ×, ÷).
  • Algebra introduces variables – symbols (usually letters like x, y, a, b) that can represent any number.
  • Constants are fixed numerical values (like 7, -2, 0).
  • An Algebraic Expression is a combination of constants and variables connected by one or more mathematical operations (+, -, ×, ÷).
    • Examples: 5x, 2x - 3, 4xy + 7, a² + 2ab + b²

2. Building Blocks of Expressions:

  • Terms: Parts of an expression separated by '+' or '-' signs.
    • In 4x + 5y - 3, the terms are 4x, 5y, and -3. (Remember to include the sign!)
  • Factors: Each term is a product of its factors. Factors can be numerical or algebraic (variables).
    • The term 4x has factors 4 and x.
    • The term 5xy has factors 5, x, and y.
    • The term -3ab² has factors -3, a, b, and b.
  • Coefficients: The numerical factor of a term is called its numerical coefficient (or simply coefficient).
    • In 7xy, the coefficient is 7.
    • In -5a²b, the coefficient is -5.
    • In x², the coefficient is 1 (since x² = 1 × x²).
    • In -y, the coefficient is -1 (since -y = -1 × y).
    • Sometimes, you might be asked for the coefficient of a specific variable part. E.g., in 7xy, the coefficient of y is 7x, and the coefficient of x is 7y. However, usually, "coefficient" refers to the numerical part.

3. Like and Unlike Terms:

  • Like Terms: Terms that have the same algebraic factors (same variables raised to the same powers). The numerical coefficients can be different.
    • Examples: 2x and -5x (same variable x to power 1)
    • 7a²b and (1/2)a²b (same variables a to power 2, b to power 1)
    • 3p and 10p
  • Unlike Terms: Terms that have different algebraic factors.
    • Examples: 2x and 3y (different variables)
    • 7a²b and 7ab² (different powers of a and b)
    • 5p and 5 (one has variable p, the other is a constant)
  • Importance: We can only add or subtract like terms.

4. Types of Algebraic Expressions (Based on Number of Terms):

  • Monomial: An expression with only one term.
    • Examples: 7x, -5m², 10, 2abc
  • Binomial: An expression with two unlike terms.
    • Examples: x + y, 2a - 3b, p² + 4q, 5 - 3xy
  • Trinomial: An expression with three unlike terms.
    • Examples: a + b + c, x² + 2x + 1, 3p - 5q + 7r
  • Polynomial: An expression with one or more terms (with non-negative integer exponents for variables). Monomials, binomials, and trinomials are all types of polynomials.
    • Example: 3x³ - 2x² + 5x - 1 is a polynomial.

5. Addition and Subtraction of Algebraic Expressions:

  • The Rule: Combine (add or subtract) only the like terms. The coefficients of like terms are added/subtracted, while the variable part remains the same. Unlike terms are kept as they are.
  • Addition:
    • Method 1 (Horizontal): Write expressions in a line, group like terms together, then add their coefficients.
      • Example: Add (2x + 3y) and (5x - y + 2)
        = 2x + 3y + 5x - y + 2
        = (2x + 5x) + (3y - y) + 2
        = (2+5)x + (3-1)y + 2
        = 7x + 2y + 2
    • Method 2 (Vertical): Write expressions one below the other such that like terms are aligned vertically. Then add column-wise.
      • Example: Add (2x + 3y) and (5x - y + 2)
          2x + 3y
+ 5x -  y + 2
----------------
  7x + 2y + 2
  • Subtraction:
    • Key Step: Change the sign of each term in the expression being subtracted, and then add the resulting expression to the first one.
    • Example: Subtract (3a - 2b + c) from (5a + 4b - 2c)
      Means: (5a + 4b - 2c) - (3a - 2b + c)
      Change signs of the second expression: -(3a - 2b + c) = -3a + 2b - c
      Now add: (5a + 4b - 2c) + (-3a + 2b - c)
      Horizontal Method:
      = 5a + 4b - 2c - 3a + 2b - c
      = (5a - 3a) + (4b + 2b) + (-2c - c)
      = (5-3)a + (4+2)b + (-2-1)c
      = 2a + 6b - 3c
      Vertical Method:
         5a + 4b - 2c
- (3a - 2b +  c)
-----------------
Change signs & Add:
   5a + 4b - 2c
+ -3a + 2b -  c
-----------------
   2a + 6b - 3c

6. Finding the Value of an Expression:

  • To find the value of an expression for specific values of its variables, substitute the given numerical values in place of the variables and simplify using the order of operations (BODMAS/PEMDAS).
  • Example: Find the value of 3x² - 2y + 1 if x = 2 and y = -1.
    Substitute x=2 and y=-1:
    = 3(2)² - 2(-1) + 1
    = 3(4) - (-2) + 1
    = 12 + 2 + 1
    = 15

7. Using Algebraic Expressions - Formulas and Rules:

  • Many formulas in geometry and other areas are expressed using algebraic expressions.
    • Perimeter of a square with side s: P = 4s (Monomial)
    • Area of a rectangle with length l and breadth b: A = lb (Monomial)
    • Perimeter of a rectangle: P = 2(l + b) or P = 2l + 2b (Binomial)

Key Takeaways for Exams:

  • Clearly distinguish between variables and constants.
  • Be precise in identifying terms, factors, and coefficients (especially the sign!).
  • Master the identification of like and unlike terms – this is fundamental for operations.
  • Practice addition and subtraction thoroughly, paying close attention to signs during subtraction.
  • Be careful with substitution and order of operations when evaluating expressions.
  • Recognize the different types of expressions (Monomial, Binomial, Trinomial).

Multiple Choice Questions (MCQs)

  1. What is the numerical coefficient of the term -5x²y?
    (a) 5
    (b) -5
    (c) x²y
    (d) -5x²

  2. Which of the following pairs contains like terms?
    (a) 7x, 7y
    (b) 2a²b, 3ab²
    (c) 4p²q, -p²q
    (d) 5, 5x

  3. The expression 3a - 2b + 5 is a:
    (a) Monomial
    (b) Binomial
    (c) Trinomial
    (d) Constant

  4. What is the sum of (2x - 3y) and (x + 5y)?
    (a) 3x - 8y
    (b) 3x + 2y
    (c) 2x + 2y
    (d) 3x - 2y

  5. What should be subtracted from a + 2b - 3 to get a - 2b + 3?
    (a) 4b - 6
    (b) 4b + 6
    (c) -4b + 6
    (d) 2a - 6

  6. If m = 2, what is the value of 3m - 5?
    (a) 1
    (b) -1
    (c) 6
    (d) 11

  7. The factors of the term -xy² are:
    (a) -x, y, y
    (b) -1, x, y, y
    (c) -1, x, y²
    (d) x, y²

  8. Identify the expression which is NOT a polynomial.
    (a) x² + 2x + 1
    (b) 5y³ - 2
    (c) 7a + 3/b
    (d) 10

  9. What is the result of simplifying (5p - 3q + 2r) - (2p + q - r)?
    (a) 3p - 4q + 3r
    (b) 3p - 2q + r
    (c) 7p - 2q + r
    (d) 3p - 4q + r

  10. The terms of the expression 4x² - 3xy + 7 are:
    (a) 4x², 3xy, 7
    (b) 4x², -3xy, 7
    (c) 4x², -3xy
    (d) 4x², 3xy

Answer Key for MCQs:

  1. (b)
  2. (c)
  3. (c)
  4. (b)
  5. (a) [Let the required expression be P. Then (a + 2b - 3) - P = (a - 2b + 3). So, P = (a + 2b - 3) - (a - 2b + 3) = a + 2b - 3 - a + 2b - 3 = 4b - 6]
  6. (a) [3(2) - 5 = 6 - 5 = 1]
  7. (b) [Remember the numerical factor -1]
  8. (c) [Because the term 3/b is 3b⁻¹, which has a negative exponent for the variable]
  9. (a) [(5p - 3q + 2r) - 2p - q + r = (5p-2p) + (-3q-q) + (2r+r) = 3p - 4q + 3r]
  10. (b) [Remember to include the sign with the term]

Study these notes carefully, practice identifying the components, and work through addition/subtraction problems. Good luck!

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