Question

Assertion (A): The system of linear equations \[x+y-9=0\], \[x-y+3=0\] 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 \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}.\]

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 true and Reason (R) is false.

Explanation:

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

Step 2 – Reason: R is false. Equal ratios of the coefficients of x and y alone do not guarantee a unique solution; the third coefficient ratio must also be considered. Here, the coefficient ratios are unequal.