`{:(2x - 3y = k),(4x + 5y = 3):}` Find the value of `k` for which given lines are intersecting.

Answer

To find the value of \( k \) for which the lines represented by the equations \( 2x - 3y = k \) and \( 4x + 5y = 3 \) are intersecting, we will use the condition for the existence of a unique solution for a system of linear equations.

### Step-by-Step Solution:

1. **Rewrite the equations in standard form:**
- The first equation is already in standard form:
\[ 2x - 3y - k = 0 \quad \text{(Equation 1)} \]
- The second equation can be rewritten as:
\[ 4x + 5y - 3 = 0 \quad \text{(Equation 2)} \]

2. **Identify coefficients:**
- From Equation 1, the coefficients are:
- \( a_1 = 2 \)
- \( b_1 = -3 \)
- \( c_1 = -k \)
- From Equation 2, the coefficients are:
- \( a_2 = 4 \)
- \( b_2 = 5 \)
- \( c_2 = -3 \)

3. **Use the condition for unique solutions:**
- For the lines to intersect (i.e., have a unique solution), the following condition must hold:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
- Substituting the values:
\[ \frac{2}{4} \neq \frac{-3}{5} \]
- Simplifying the left side:
\[ \frac{1}{2} \neq \frac{-3}{5} \]

4. **Verify the inequality:**
- Since \( \frac{1}{2} \) is not equal to \( \frac{-3}{5} \), the lines will intersect for any value of \( k \) as long as the coefficients maintain this relationship.

5. **Conclusion:**
- The value of \( k \) can be any real number for the lines to intersect.

### Final Answer:
The value of \( k \) for which the given lines are intersecting is **any real number**.