For what value of k, the following system of equations will represent the coincident lines?
3x - 2y = 5 and 12x - ky = 20

Answer

To find the value of \( k \) for which the system of equations represents coincident lines, we need to analyze the given equations:

1. **Write the equations**:
\[ 3x - 2y = 5 \quad \text{(Equation 1)} \]
\[ 12x - ky = 20 \quad \text{(Equation 2)} \]

2. **Convert the equations into standard form**:
We can rewrite both equations in the form \( Ax + By + C = 0 \).

For Equation 1:
\[ 3x - 2y - 5 = 0 \quad \Rightarrow \quad A_1 = 3, \, B_1 = -2, \, C_1 = -5 \]

For Equation 2:
\[ 12x - ky - 20 = 0 \quad \Rightarrow \quad A_2 = 12, \, B_2 = -k, \, C_2 = -20 \]

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

Substituting the values we found:
\[ \frac{3}{12} = \frac{-2}{-k} = \frac{-5}{-20} \]

4. **Simplify the fractions**:
\[ \frac{3}{12} = \frac{1}{4}, \quad \frac{-5}{-20} = \frac{1}{4} \]

Thus, we have:
\[ \frac{1}{4} = \frac{2}{k} \]

5. **Cross-multiply to solve for \( k \)**:
\[ 1 \cdot k = 4 \cdot 2 \]
\[ k = 8 \]

6. **Conclusion**:
The value of \( k \) for which the given system of equations represents coincident lines is:
\[ \boxed{8} \]