Question

Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

Be careful with absolute terms like "always" in mathematical assertions and reasons.
Coincident lines also fall under the category of "not having a unique solution" (they have infinitely many), which invalidates the "always" condition of the Reason.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question consists of an Assertion (A) and a Reason (R).
We need to evaluate whether both statements are correct individually and whether the Reason correctly explains the Assertion.

Step 2: Key Formula or Approach:
For a system of two linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:
- The system is inconsistent (no solution) if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
This represents parallel lines.
- If a system has no unique solution, it can either have infinitely many solutions (coincident lines) or no solution (parallel lines).

Step 3: Detailed Explanation:

• Evaluate Assertion (A):
The given equations are:
\[ 3x - 5y + 7 = 0 \implies a_1 = 3, \quad b_1 = -5, \quad c_1 = 7 \]
\[ -6x + 10y + 14 = 0 \implies a_2 = -6, \quad b_2 = 10, \quad c_2 = 14 \]
Calculate the ratio of the coefficients:
\[ \frac{a_1}{a_2} = \frac{3}{-6} = -\frac{1}{2} \]
\[ \frac{b_1}{b_2} = \frac{-5}{10} = -\frac{1}{2} \]
\[ \frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2} \]
Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel. This means they have no common point of intersection, so the system is inconsistent.
Thus, Assertion (A) is true.

• Evaluate Reason (R):
The statement says: "When two linear equations don't have unique solution, they always represent parallel lines."
If a system does not have a unique solution, it could have no solution (parallel lines) OR infinitely many solutions (coincident lines).
Because they can also represent coincident lines, they do not "always" represent parallel lines.
Thus, Reason (R) is false.


Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false.