Question

Assertion (A): The system of linear equations \[5x-9y+6=0\], \[10x-18y+12=0\] is inconsistent.

Reason (R): The pair of linear equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] is inconsistent, 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 false and Reason (R) is true.

Explanation:

Step 1 – Assertion: A is false. The ratios are \[\frac{a_{1}}{a_{2}}=\frac{5}{10}=\frac{1}{2},\quad \frac{b_{1}}{b_{2}}=\frac{-9}{-18}=\frac{1}{2},\quad \frac{c_{1}}{c_{2}}=\frac{6}{12}=\frac{1}{2}.\] All three ratios are equal, so the equations have infinitely many solutions, not an inconsistent system.

Step 2 – Reason: R is true. It correctly states the condition for inconsistency: the first two ratios must be equal and different from the third.