Question

Assertion (A): The system of linear equations \[x-2y+6=0\], \[2x-4y+9=0\] has a unique solution.

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

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

Explanation:

Step 1 – Assertion: A is false. The ratios are \[\frac{a_{1}}{a_{2}}=\frac{1}{2},\quad \frac{b_{1}}{b_{2}}=\frac{-2}{-4}=\frac{1}{2},\quad \frac{c_{1}}{c_{2}}=\frac{6}{9}=\frac{2}{3}.\] The first two are equal and differ from the third, so the system has no solution.

Step 2 – Reason: R is true. It states the condition for a unique solution: the coefficient ratios for x and y must be unequal.