Find the value of k, if the lines given by 4x + 5ky = 10 and 3x + y + 1 = 0 are parallel.

Options

A. 7
B. `3/8`
C. `4/15`
D. `-1`

Answer

To find the value of \( k \) such that the lines given by the equations \( 4x + 5ky = 10 \) and \( 3x + y + 1 = 0 \) are parallel, we can follow these steps:

### Step 1: Write the equations in standard form
The first equation is already in a form we can work with:
\[ 4x + 5ky - 10 = 0 \]

The second equation can be rearranged:
\[ 3x + y + 1 = 0 \]
This can be rewritten as:
\[ 3x + y + 1 = 0 \implies 3x + y = -1 \]

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For the first line \( 4x + 5ky - 10 = 0 \):
- \( a_1 = 4 \)
- \( b_1 = 5k \)
- \( c_1 = -10 \)

- For the second line \( 3x + y + 1 = 0 \):
- \( a_2 = 3 \)
- \( b_2 = 1 \)
- \( c_2 = 1 \)

### Step 3: Use the condition for parallel lines
For two lines to be parallel, the ratios of the coefficients of \( x \) and \( y \) must be equal, while the ratio of the constant terms must not be equal. Thus, we set up the following equation:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \]
Substituting the coefficients:
\[ \frac{4}{3} = \frac{5k}{1} \]

### Step 4: Solve for \( k \)
Now we can solve for \( k \):
\[ \frac{4}{3} = 5k \]
To isolate \( k \), we multiply both sides by \( \frac{1}{5} \):
\[ k = \frac{4}{15} \]

### Final Answer
The value of \( k \) is:
\[ \boxed{\frac{4}{15}} \]