Question

The pair of equations \(3x+4y=k \)and \(9x+12y=6\) has infinitely many solutions if \(k=\)

• A. 3
• B. 2
• C. 6
• D. 5

Answer 1

The Correct Option is B

To solve the problem, we need to determine the value of \( k \) for which the pair of equations \( 3x + 4y = k \) and \( 9x + 12y = 6 \) has infinitely many solutions.

1. Understanding the Condition for Infinite Solutions:
Two linear equations have infinitely many solutions if they are equivalent, i.e., one is a scalar multiple of the other.

So, we must have:
\( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

2. Identify Coefficients:
From the equations:
First equation: \( 3x + 4y = k \) ⇒ \( a_1 = 3, b_1 = 4, c_1 = k \)
Second equation: \( 9x + 12y = 6 \) ⇒ \( a_2 = 9, b_2 = 12, c_2 = 6 \)

3. Apply the Proportionality Condition:
\[ \frac{3}{9} = \frac{4}{12} = \frac{k}{6} \]
\[ \frac{1}{3} = \frac{1}{3} = \frac{k}{6} \]
\[ \Rightarrow \frac{k}{6} = \frac{1}{3} \Rightarrow k = 2 \]

Final Answer:
The value of \( k \) is \( 2 \) for the system to have infinitely many solutions.