For what value of k, the system of equation `kx + 2y = 2 and 3 x + y = 1 ` will be coincident ?

Options

A. 2
B. 3
C. 5
D. 6

Answer

To determine the value of \( k \) for which the system of equations

1. \( kx + 2y = 2 \)
2. \( 3x + y = 1 \)

is coincident, we need to understand that coincident lines are essentially the same line represented in different forms. This means that the ratios of the coefficients of \( x \), \( y \), and the constant terms must be equal.

### Step-by-step Solution:

1. **Write down the equations**:
\[ kx + 2y = 2 \quad \text{(Equation 1)} \]
\[ 3x + y = 1 \quad \text{(Equation 2)} \]

2. **Identify the coefficients**:
From Equation 1, the coefficients are:
- Coefficient of \( x \): \( k \)
- Coefficient of \( y \): \( 2 \)
- Constant term: \( 2 \)

From Equation 2, the coefficients are:
- Coefficient of \( x \): \( 3 \)
- Coefficient of \( y \): \( 1 \)
- Constant term: \( 1 \)

3. **Set up the ratio of coefficients**:
For the lines to be coincident, the following ratios must hold:
\[ \frac{k}{3} = \frac{2}{1} = \frac{2}{1} \]

4. **Solve for \( k \)**:
From the ratio \( \frac{k}{3} = \frac{2}{1} \):
\[ k = 3 \cdot 2 = 6 \]

5. **Conclusion**:
The value of \( k \) for which the system of equations is coincident is:
\[ k = 6 \]