Question

Assertion (A): The system of linear equations \[3x+5y-4=0\], \[15x+25y-25=0\] 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 \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\ne\frac{c_{1}}{c_{2}}.\]

पर्याय

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

Explanation:

Step 1 – Assertion: A is true. The ratios are \[\frac{a_{1}}{a_{2}}=\frac{3}{15}=\frac{1}{5},\quad \frac{b_{1}}{b_{2}}=\frac{5}{25}=\frac{1}{5},\quad \frac{c_{1}}{c_{2}}=\frac{-4}{-25}=\frac{4}{25}.\] The first two ratios are equal and differ from the third, so the system is inconsistent.

Step 2 – Reason: R is true. It gives the condition for a pair of linear equations to be inconsistent.

Step 3 – Link: The given ratios satisfy the stated condition, so R correctly explains A.