For what value of k, the system of equations 3x + 4y = 6 and 6x + 8y = k represents coincident lines?

Answer

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

1. \( 3x + 4y = 6 \)
2. \( 6x + 8y = k \)

represents coincident lines, we need to apply the condition for coincident lines. The condition states that for two linear equations to represent coincident lines, the ratios of the coefficients of \( x \), \( y \), and the constant terms must be equal. This can be expressed as:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

### Step 1: Identify coefficients
From the first equation \( 3x + 4y = 6 \), we can identify:
- \( a_1 = 3 \)
- \( b_1 = 4 \)
- \( c_1 = -6 \) (moving 6 to the other side gives us \( 3x + 4y - 6 = 0 \))

From the second equation \( 6x + 8y = k \), we can identify:
- \( a_2 = 6 \)
- \( b_2 = 8 \)
- \( c_2 = -k \) (moving \( k \) to the other side gives us \( 6x + 8y - k = 0 \))

### Step 2: Set up the ratio
Now, we set up the ratios based on the coefficients:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Substituting the values we have:

\[ \frac{3}{6} = \frac{4}{8} = \frac{-6}{-k} \]

### Step 3: Simplify the ratios
Now, simplify the left-hand side:

\[ \frac{3}{6} = \frac{1}{2} \]
\[ \frac{4}{8} = \frac{1}{2} \]

Thus, we have:

\[ \frac{1}{2} = \frac{6}{k} \]

### Step 4: Cross-multiply to solve for \( k \)
Cross-multiplying gives us:

\[ 1 \cdot k = 2 \cdot 6 \]
\[ k = 12 \]

### Conclusion
Therefore, the value of \( k \) for which the system of equations represents coincident lines is:

\[ \boxed{12} \]