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{:(2x - 3y = k),(4x + 5y = 3):} Find the...

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

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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**.
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