Question

If the pair of linear equations : \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) is consistent and dependent, then

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

Hint:

Remember the clear definitions:
- Consistent and Independent: Unique solution (Intersecting lines) \(\implies \frac{a_1}{a_2} \neq \frac{b_1}{b_2}\).
- Inconsistent: No solution (Parallel lines) \(\implies \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).
- Consistent and Dependent: Infinitely many solutions (Coincident lines) \(\implies \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).

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 identify the correct algebraic condition representing the case where the system is both "consistent" and "dependent".

Step 2: Key Formula or Approach:
For a system of two linear equations:
\[ a_1x + b_1y + c_1 = 0 \]
\[ a_2x + b_2y + c_2 = 0 \]
There are three possible geometric relationships and solvability conditions:
1.

Unique Solution (Consistent and Independent): The two lines intersect at a single point.
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
2.

No Solution (Inconsistent): The two lines are parallel and never meet.
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
3.

Infinitely Many Solutions (Consistent and Dependent): The two lines are coincident (lie on top of each other).
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Step 3: Detailed Explanation:
1. The question states that the system is "consistent and dependent".
2. "Consistent" means the system has at least one solution.
3. "Dependent" means that one equation is a scalar multiple of the other, representing the exact same line geometrically. This results in the lines overlapping entirely (coinciding).
4. Since the lines coincide, every point on the line is a solution, resulting in infinitely many solutions.
5. The mathematical condition for coincident lines requires all corresponding coefficients and constant ratios to be equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
This matches the expression in option (D).

Step 4: Final Answer:
The correct condition is \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), which corresponds to option (D).