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: Check Assertion (A) using a scalar-multiple test instead of computing all three ratios.
Multiply the first equation by $-2$: $-2(3x-5y+7) = -6x+10y-14$. Compare this with the second equation $-6x+10y+14=0$: the $x$ and $y$ terms match exactly, but the constant becomes $-14$ instead of $+14$. Equal coefficients with an unequal constant means the lines are parallel (never coincident), so the system is inconsistent. Assertion (A) is true.
Step 2: Test Reason (R) with a quick counterexample.
Consider $x+y=1$ and $2x+2y=2$. These have no unique solution too, but they represent the same line (coincident), not parallel lines.
Step 3: Conclude.
This counterexample shows "no unique solution" does not always mean "parallel", so Reason (R) is false, even though Assertion (A) is true.
\[ \boxed{\text{Assertion (A) is true, but Reason (R) is false.}} \]