The system ` x - 2y = 3 and 3x + ky = 1 ` has a unique solution, when

Options

A. ` k = - 6`
B. ` k ne - 6`
C. ` k = 0 `
D. ` k ne 0 `

Explanation

To determine the value of \( k \) for which the system of equations has a unique solution, we need to analyze the given equations:

1. \( x - 2y = 3 \) (Equation 1)
2. \( 3x + ky = 1 \) (Equation 2)

### Step 1: Identify the coefficients
From the equations, we can identify the coefficients:
- For Equation 1:
- Coefficient of \( x \) (let's call it \( a_1 \)) = 1
- Coefficient of \( y \) (let's call it \( b_1 \)) = -2
- Constant term (let's call it \( c_1 \)) = 3

- For Equation 2:
- Coefficient of \( x \) (let's call it \( a_2 \)) = 3
- Coefficient of \( y \) (let's call it \( b_2 \)) = \( k \)
- Constant term (let's call it \( c_2 \)) = 1

### Step 2: Condition for unique solution
For the system of equations to have a unique solution, the following condition must hold:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

### Step 3: Substitute the coefficients
Substituting the coefficients into the condition:
\[ \frac{1}{3} \neq \frac{-2}{k} \]

### Step 4: Cross-multiply to solve the inequality
Cross-multiplying gives us:
\[ 1 \cdot k \neq 3 \cdot (-2) \]
\[ k \neq -6 \]

### Conclusion
Thus, the value of \( k \) must not be equal to \(-6\) for the system to have a unique solution.

### Final Answer
The system has a unique solution when \( k \neq -6 \).

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