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 the Assertion by comparing coefficient ratios.
For $3x-5y+7=0$ and $-6x+10y+14=0$: $\frac{a_1}{a_2}=\frac{3}{-6}=-\frac12$, $\frac{b_1}{b_2}=\frac{-5}{10}=-\frac12$, $\frac{c_1}{c_2}=\frac{7}{14}=\frac12$.
Step 2: Interpret the ratios.
Since $\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$, the lines are parallel with no common point, so the system is inconsistent. Assertion (A) is true.
Step 3: Test the Reason with a counterexample.
Take equations $x+y=2$ and $2x+2y=4$: these have no unique solution but are coincident lines, not parallel ones. So the claim "always parallel" fails, making Reason (R) false.
Step 4: Conclude.
Assertion (A) is true but Reason (R) is false, matching option (3).
\[ \boxed{\text{Option (3)}} \]