For what value of k, the system of equations x - 3y = 5 and 2x - 6y = k has at most one solution?

Answer

To determine the value of \( k \) for which the system of equations

1. \( x - 3y = 5 \)
2. \( 2x - 6y = k \)

has at most one solution, we need to analyze the conditions under which the system can have a unique solution or no solution.

### Step 1: Identify the coefficients
The first equation can be rewritten in the standard form \( Ax + By = C \):
- For the first equation \( x - 3y = 5 \):
- \( a_1 = 1 \)
- \( b_1 = -3 \)
- \( c_1 = 5 \)

- For the second equation \( 2x - 6y = k \):
- \( a_2 = 2 \)
- \( b_2 = -6 \)
- \( c_2 = k \)

### Step 2: Determine the condition for unique solutions
For the system to have a unique solution, the ratio of the coefficients of \( x \) and \( y \) must not be equal to the ratio of the constants. This can be expressed as:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
Calculating these ratios:
\[ \frac{a_1}{a_2} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-3}{-6} = \frac{1}{2} \]
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \), we can conclude that the system does not have a unique solution.

### Step 3: Determine the condition for no solution
For the system to have no solution, the ratios of the coefficients must be equal, but the ratio of the constants must be different. This can be expressed as:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \quad \text{and} \quad \frac{a_1}{a_2} \neq \frac{c_1}{c_2} \]
From the previous calculations, we know:
\[ \frac{a_1}{a_2} = \frac{1}{2} \quad \text{and} \quad \frac{b_1}{b_2} = \frac{1}{2} \]
Now we need to check the ratio of the constants:
\[ \frac{c_1}{c_2} = \frac{5}{k} \]
Setting the condition for no solution:
\[ \frac{1}{2} = \frac{1}{2} \quad \text{and} \quad \frac{5}{k} \neq \frac{1}{2} \]
This leads to:
\[ 5 \neq \frac{k}{2} \quad \Rightarrow \quad k \neq 10 \]

### Conclusion
Thus, for the system of equations to have at most one solution, \( k \) can take any real value except \( k = 10 \).

### Final Answer
The value of \( k \) for which the system of equations has at most one solution is:
\[ k \in \mathbb{R}, \, k \neq 10 \]

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