Question

If pair of linear equations given by \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) has infinitely many solutions, then which of the following is definitely true ?

• A. \(\frac{a_1}{a_2} = \frac{c_2}{c_1}\)
• B. \(a_1a_2 \neq b_1b_2\)
• C. \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
• D. \(a_1b_2 = a_2b_1\)

Hint:

Remember the three algebraic conditions for a pair of linear equations:
1. Unique solution: \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
2. No solution: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
3. Infinitely many solutions: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a system of two linear equations in two variables. We need to determine the algebraic condition that is definitely true when the system has infinitely many solutions.

Step 2: Key Formula or Approach:
For a pair of linear equations to have infinitely many solutions (coincident lines), the ratios of their coefficients must be equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Step 3: Detailed Explanation:

• From the condition for infinitely many solutions, we have:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \]

• Cross-multiplying this equality yields:
\[ a_1 \cdot b_2 = a_2 \cdot b_1 \]

• Let us verify the options:
- Option (A) states \(\frac{a_1}{a_2} = \frac{c_2}{c_1}\), which is incorrect because \(\frac{a_1}{a_2} = \frac{c_1}{c_2}\).
- Option (B) states \(a_1a_2 \neq b_1b_2\), which is not related to this standard ratio condition.
- Option (C) states \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), which is the condition for a unique solution.
- Option (D) states \(a_1b_2 = a_2b_1\), which perfectly matches our cross-multiplication result.


Step 4: Final Answer:
The relation \(a_1b_2 = a_2b_1\) is definitely true.