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 words like "always" in Reason statements. If there is even one exception (like coincident lines having infinite solutions), the "always" makes the statement false.

Answer 1

The Correct Option is C

To determine the truth of the assertion (A) and reason (R), let's analyze both separately and establish their relation.

Step 1: Analyze the Assertion (A)

The assertion (A) claims that the system of linear equations \(3x - 5y + 7 = 0\) and \(-6x + 10y + 14 = 0\) is inconsistent.

To ascertain this, let's compare these two equations:

• Equation 1: \(3x - 5y + 7 = 0\)
• Equation 2: \(-6x + 10y + 14 = 0\)

Observe that Equation 2 is a scalar multiple of Equation 1: \((-6x + 10y + 14 = 0)\) = \(-2 \times (3x - 5y + 7 = 0)\).

This implies that the two equations represent the same line, not two different parallel lines. Therefore, the system is consistent and does not lead to inconsistency.

Conclusion for Assertion (A): The assertion is false because it states the equations are inconsistent, which is incorrect as they represent the same line.

Step 2: Analyze the Reason (R)

The reason (R) states that when two linear equations don't have a unique solution, they always represent parallel lines.

The condition for two lines to not have a unique solution is when:

• They either have infinitely many solutions (coincident lines) or
• No solution (parallel lines).

Since not having a unique solution implies either coincident or parallel lines, the reason (R) is false because it only considers the case of parallel lines, omitting coincident lines.

Conclusion for Reason (R): The reason is false as it does not encompass all scenarios where a unique solution is lacking.

Final Conclusion: The correct answer to the problem is:
Assertion (A) is true, but Reason (R) is false.