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: Test the Assertion by trying to make the equations identical.
Multiply the first equation $3x-5y+7=0$ by $-2$: $-6x+10y-14=0$. The left-hand sides now match the second equation exactly, but the constant terms are $-14$ and $+14$, which are different.
Step 2: Conclude what this means geometrically.
Since the equations become identical on the left but disagree on the constant, the two lines are parallel and distinct, so they never meet. This confirms the system is inconsistent, so Assertion (A) is true.
Step 3: Test the Reason with a counterexample.
The Reason claims that not having a unique solution always means the lines are parallel. But take $x+y=1$ and $2x+2y=2$: here both constants scale consistently, so the lines are coincident (infinitely many common points), not parallel, yet there is still no unique solution. This counterexample shows Reason (R) is false.
Step 4: State the final match.
Assertion (A) is true, but Reason (R) is false.
\[ \boxed{\text{Assertion (A) is true, but Reason (R) is false.}} \]