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 need to determine the slopes of both lines and set them equal to each other.

### Step 1: Rearranging the first equation

We start with the first equation:

\[ 4x + 5ky = 10 \]

We need to express \( y \) in terms of \( x \):

\[ 5ky = 10 - 4x \]

Now, divide both sides by \( 5k \):

\[ y = \frac{10 - 4x}{5k} \]

### Step 2: Finding the slope of the first line

The equation is now in the slope-intercept form \( y = mx + c \), where \( m \) is the slope. Thus, the slope \( M_1 \) of the first line is:

\[ M_1 = -\frac{4}{5k} \]

### Step 3: Rearranging the second equation

Now, we take the second equation:

\[ 3x + y + 1 = 0 \]

Rearranging it to express \( y \):

\[ y = -3x - 1 \]

### Step 4: Finding the slope of the second line

From the equation \( y = -3x - 1 \), we can see that the slope \( M_2 \) of the second line is:

\[ M_2 = -3 \]

### Step 5: Setting the slopes equal

Since the lines are parallel, their slopes must be equal:

\[ M_1 = M_2 \]

Substituting the values we found:

\[ -\frac{4}{5k} = -3 \]

### Step 6: Solving for \( k \)

Removing the negative sign from both sides gives:

\[ \frac{4}{5k} = 3 \]

Now, cross-multiply to solve for \( k \):

\[ 4 = 15k \]

Dividing both sides by 15:

\[ k = \frac{4}{15} \]

Thus, the value of \( k \) is:

\[ \boxed{\frac{4}{15}} \]