Question

Assertion (A): The system of equations \[x+2y+2=0\] and \[3x+2y-2=0\] has a unique solution.

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

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 true and Reason (R) is false.

Explanation:

Step 1 – Assertion: A is true. Here \[\frac{a_{1}}{a_{2}}=\frac{1}{3}\] and \[\frac{b_{1}}{b_{2}}=\frac{2}{2}=1.\] Since these ratios are unequal, the system has a unique solution.

Step 2 – Reason: R is false. The stated condition \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}\] is the condition for no solutions, not for a unique solution.