What is the value of k for which the pair of linear equations kx-2y=3 and 3x+y=5 has a unique solution.

Options

A. k=6
B. `kne-6`
C. k=-6
D. None of these

Answer

To find the value of \( k \) for which the pair of linear equations \( kx - 2y = 3 \) and \( 3x + y = 5 \) has a unique solution, we can use the concept of determinants from linear algebra.

### Step 1: Write the equations in standard form
The given equations are:
1. \( kx - 2y = 3 \) (Equation 1)
2. \( 3x + y = 5 \) (Equation 2)

### Step 2: Identify coefficients
From these equations, we can identify the coefficients:
- For Equation 1: \( a_1 = k \), \( b_1 = -2 \), \( c_1 = 3 \)
- For Equation 2: \( a_2 = 3 \), \( b_2 = 1 \), \( c_2 = 5 \)

### Step 3: Set up the determinant
For the system of equations to have a unique solution, the determinant of the coefficients must be non-zero. The determinant \( D \) is given by:

\[ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = \begin{vmatrix} k & -2 \\ 3 & 1 \end{vmatrix} \]

### Step 4: Calculate the determinant
Calculating the determinant:

\[ D = k \cdot 1 - (-2) \cdot 3 = k + 6 \]

### Step 5: Set the determinant not equal to zero
For a unique solution, we need:

\[ D \neq 0 \]

This gives us:

\[ k + 6 \neq 0 \]

### Step 6: Solve for \( k \)
Solving the inequality:

\[ k \neq -6 \]

### Conclusion
Thus, the value of \( k \) for which the pair of linear equations has a unique solution is:

\[ k \neq -6 \]