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

Answer

To find the value of \( k \) for which the system of equations \( 3x + 4y = 6 \) and \( 6x + 8y = k \) represents coincident lines, we can follow these steps:

### Step 1: Write the equations in standard form
The given equations are:
1. \( 3x + 4y - 6 = 0 \) (Equation 1)
2. \( 6x + 8y - k = 0 \) (Equation 2)

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For Equation 1:
- \( a_1 = 3 \), \( b_1 = 4 \), \( c_1 = -6 \)
- For Equation 2:
- \( a_2 = 6 \), \( b_2 = 8 \), \( c_2 = -k \)

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

### Step 4: Simplify the ratios
Now, simplify the ratios:
1. \( \frac{3}{6} = \frac{1}{2} \)
2. \( \frac{4}{8} = \frac{1}{2} \)

Thus, we have:
\[ \frac{1}{2} = \frac{6}{k} \]

### Step 5: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ 1 \cdot k = 2 \cdot 6 \]
\[ k = 12 \]

### Conclusion
The value of \( k \) for which the system of equations represents coincident lines is \( k = 12 \).

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