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

Step 1: Understanding the Concept:
A system of linear equations is inconsistent if it has no solution. This occurs when the lines are parallel. For equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), the condition for no solution is:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
Step 2: Detailed Explanation:
Evaluating Assertion (A):
Eq 1: \(3x - 5y + 7 = 0 \implies a_1=3, b_1=-5, c_1=7\)
Eq 2: \(-6x + 10y + 14 = 0 \implies a_2=-6, b_2=10, c_2=14\)
Check the ratios:
\[ \frac{a_1}{a_2} = \frac{3}{-6} = -\frac{1}{2} \]
\[ \frac{b_1}{b_2} = \frac{-5}{10} = -\frac{1}{2} \]
\[ \frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2} \]
Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) (\(-\frac{1}{2} = -\frac{1}{2} \neq \frac{1}{2}\)), the system has no solution.
Thus, the system is inconsistent. Assertion (A) is True.
Evaluating Reason (R):
A system "doesn't have a unique solution" when it has either no solution (parallel lines) or infinitely many solutions (coincident lines). Therefore, it is incorrect to say it \textit{always} represents parallel lines.
Reason (R) is False.
Step 3: Final Answer:
Assertion (A) is true and Reason (R) is false.