Question

Assertion (A): The system of equations \[2x+3y-6=0\] and \[4x+6y-12=0\] has infinitely many solutions.

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 infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}.\]

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

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 coefficient ratios are \[\frac{a_{1}}{a_{2}}=\frac{2}{4}=\frac{1}{2},\quad \frac{b_{1}}{b_{2}}=\frac{3}{6}=\frac{1}{2},\quad \frac{c_{1}}{c_{2}}=\frac{-6}{-12}=\frac{1}{2}.\] All three ratios are equal, so the system has infinitely many solutions.

Step 2 – Reason: R is true. It states that a pair of linear equations has infinitely many solutions when all three coefficient ratios are equal.

Step 3 – Link: The ratios for the given equations meet this criterion, so R correctly explains A.