For which value of ‘g’ the linear graph of `6x + 12y + 9 = 0 and 2x + gy + 3 = 0` has infinite number of solutions?

Options

A. 3
B. 4
C. 6
D. 9

Answer

To find the value of 'g' for which the linear equations \(6x + 12y + 9 = 0\) and \(2x + gy + 3 = 0\) have an infinite number of solutions, we need to determine when these two lines are coincident. This occurs when the ratios of the coefficients of \(x\), \(y\), and the constant terms are equal.

### Step 1: Write down the equations
The two equations are:
1. \(6x + 12y + 9 = 0\)
2. \(2x + gy + 3 = 0\)

### Step 2: Rearrange the equations
We can express both equations in the standard form \(Ax + By + C = 0\):
1. \(A_1 = 6, B_1 = 12, C_1 = 9\)
2. \(A_2 = 2, B_2 = g, C_2 = 3\)

### Step 3: Set up the condition for infinite solutions
For the lines to be coincident, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

### Step 4: Calculate the ratios
Calculating the ratios:
1. \(\frac{A_1}{A_2} = \frac{6}{2} = 3\)
2. \(\frac{B_1}{B_2} = \frac{12}{g}\)
3. \(\frac{C_1}{C_2} = \frac{9}{3} = 3\)

### Step 5: Set the ratios equal
Setting \(\frac{B_1}{B_2}\) equal to the other ratios:
\[ \frac{12}{g} = 3 \]

### Step 6: Solve for 'g'
Now, we solve for \(g\):
\[ 12 = 3g \implies g = \frac{12}{3} = 4 \]

### Conclusion
Thus, the value of \(g\) for which the two lines have an infinite number of solutions is \(g = 4\).