For what value of k, the system of equations 5x - ky = 3 and 10x - 3y = 6 is independent?

Answer

To determine the value of \( k \) for which the system of equations is independent, we need to analyze the given equations:

1. **Write the equations**:
\[ 5x - ky = 3 \quad \text{(Equation 1)} \]
\[ 10x - 3y = 6 \quad \text{(Equation 2)} \]

2. **Identify coefficients**:
From Equation 1, we have:
- \( a_1 = 5 \)
- \( b_1 = -k \)
- \( c_1 = 3 \)

From Equation 2, we have:
- \( a_2 = 10 \)
- \( b_2 = -3 \)
- \( c_2 = 6 \)

3. **Calculate the ratios of coefficients**:
We need to find the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \):
\[ \frac{a_1}{a_2} = \frac{5}{10} = \frac{1}{2} \]
\[ \frac{b_1}{b_2} = \frac{-k}{-3} = \frac{k}{3} \]
\[ \frac{c_1}{c_2} = \frac{3}{6} = \frac{1}{2} \]

4. **Set up the condition for independence**:
For the system of equations to be independent (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 we calculated:
\[ \frac{1}{2} \neq \frac{k}{3} \]

5. **Solve the inequality**:
To find the values of \( k \) that satisfy this inequality, we can set up the equation:
\[ \frac{1}{2} = \frac{k}{3} \]
Cross-multiplying gives:
\[ 1 \cdot 3 = 2 \cdot k \implies 3 = 2k \implies k = \frac{3}{2} \]

6. **Conclusion**:
The system of equations will be independent for all real values of \( k \) except \( k = \frac{3}{2} \).

Thus, the final answer is:
\[ \text{The system of equations is independent for all real values of } k \text{ except } k = \frac{3}{2}. \]