If the lines given by `3x+2ky =2` and `2x+5y =1` are parallel, then the value of k is

Answer

To find the value of \( k \) such that the lines given by the equations \( 3x + 2ky = 2 \) and \( 2x + 5y = 1 \) are parallel, we can follow these steps:

### Step 1: Write the equations in standard form
We need to rewrite both equations in the standard form \( Ax + By + C = 0 \).

1. For the first equation:
\[ 3x + 2ky - 2 = 0 \]
Here, \( A_1 = 3 \), \( B_1 = 2k \), and \( C_1 = -2 \).

2. For the second equation:
\[ 2x + 5y - 1 = 0 \]
Here, \( A_2 = 2 \), \( B_2 = 5 \), and \( C_2 = -1 \).

### Step 2: Use the condition for parallel lines
For two lines to be parallel, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \quad \text{and} \quad \frac{A_1}{A_2} \neq \frac{C_1}{C_2} \]

### Step 3: Set up the equations based on the condition
From our equations, we can set up the following ratio:
\[ \frac{3}{2} = \frac{2k}{5} \]

### Step 4: Solve for \( k \)
Now we can solve for \( k \):
1. Cross-multiply:
\[ 3 \cdot 5 = 2 \cdot 2k \]
This simplifies to:
\[ 15 = 4k \]

2. Divide both sides by 4:
\[ k = \frac{15}{4} \]

### Step 5: Verify the condition for \( C_1 \) and \( C_2 \)
Now we check the second part of the condition:
\[ \frac{C_1}{C_2} = \frac{-2}{-1} = 2 \]
Since \( \frac{3}{2} \neq 2 \), the condition for parallel lines is satisfied.

### Final Answer
Thus, the value of \( k \) is:
\[ k = \frac{15}{4} \]

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