Case Study Math Class 10 Pair of Linear Equations in Two Variables

Case Study Math Class 10 Pair of Linear Equations in Two Variables | Free Online Test

Case Study Math Class 10 Pair of Linear Equations in Two Variables

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Case Study: Pair of Linear Equations in Two Variables

Case Study 1: Pair of Linear Equations in Two Variables

Context: A mobile service provider offers two different monthly plans. Plan A charges a fixed rental of Rs.200 and Rs.1.5 per local call. Plan B has a fixed rental of Rs.300 but charges only Rs.1 per local call. Two friends, Aman and Ravi, subscribe to Plan A and Plan B respectively. They end up paying the same total monthly bill. Assuming they made the same number of local calls in a month, analyze this situation using graphical and algebraic methods of solving linear equations in two variables.

Theoretical Background:

  • General form of a pair of linear equations: \[ a_1x + b_1y = c_1, \quad a_2x + b_2y = c_2 \]
  • Graphical Interpretation:
    • Intersecting lines ⇒ Unique solution
    • Parallel lines ⇒ No solution
    • Coincident lines ⇒ Infinitely many solutions
  • Methods of Solution:
    • Substitution Method
    • Elimination Method
    • Cross-Multiplication Method

1. If the number of local calls made is \(x\), what equation represents Aman’s total bill under Plan A?

  • A) \(200 + x = 300\)
  • B) \(200 + 1.5x = \text{Total Bill}\)
  • C) \(x + 200 = 1.5\)
  • D) \(1.5x = 300\)
Answer: B) \(200 + 1.5x = \text{Total Bill}\)
Solution: Aman pays Rs.200 fixed and Rs.1.5 per call. Hence, his total bill is \(200 + 1.5x\).

2. What is the equation for Ravi’s bill under Plan B, if \(x\) calls were made?

  • A) \(300 + x\)
  • B) \(300x + 1\)
  • C) \(300 + 1x\)
  • D) \(x + 200\)
Answer: C) \(300 + 1x\)
Solution: Ravi pays Rs.300 fixed and Rs.1 per call. So, the bill is \(300 + x\).

3. If both friends pay the same total amount, what is the resulting equation?

  • A) \(200 + 1.5x = 300 + x\)
  • B) \(300 + 1.5x = 200 + x\)
  • C) \(200 + x = 300 + 1.5x\)
  • D) \(200x + 300x = x\)
Answer: A) \(200 + 1.5x = 300 + x\)
Solution: Set the total bills equal: \[ 200 + 1.5x = 300 + x \]

4. Solve: \(200 + 1.5x = 300 + x\)

  • A) \(x = 10\)
  • B) \(x = 50\)
  • C) \(x = 100\)
  • D) \(x = 200\)
Answer: D) \(x = 200\)
Solution: \[ 200 + 1.5x = 300 + x \Rightarrow 0.5x = 100 \Rightarrow x = 200 \]

5. Graphically, the equations \(y = 200 + 1.5x\) and \(y = 300 + x\) represent:

  • A) Parallel lines
  • B) Coincident lines
  • C) Intersecting lines
  • D) No lines
Answer: C) Intersecting lines
Solution: The lines have different slopes (1.5 and 1), so they intersect at one point ⇒ unique solution.

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