Question

Assertion (A): The system of linear equations \[2x-3y+1=0\], \[4x-6y+3=0\] is inconsistent.

Reason (R): The system of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has a unique solution, if \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. The ratios are \[\frac{a_{1}}{a_{2}}=\frac{2}{4}=\frac{1}{2},\quad \frac{b_{1}}{b_{2}}=\frac{-3}{-6}=\frac{1}{2},\quad \frac{c_{1}}{c_{2}}=\frac{1}{3}.\] The first two ratios are equal but differ from the third, so the system is inconsistent.

Step 2 – Reason: R is true. It states that a pair of linear equations has a unique solution when the ratios of the coefficients of x are unequal.

Step 3 – Link: The reason's condition for a unique solution does not establish the inconsistency asserted here; the inconsistent-system criterion does. Thus, R does not explain A.