Find the value of k for which the system of simultaneous equations `x+2y=5, 3x+ky+15=0` has no solution.

Options

A. `-6`
B. 6
C. 0
D. 3

Answer

To find the value of \( k \) for which the system of simultaneous equations \( x + 2y = 5 \) and \( 3x + ky + 15 = 0 \) has no solution, we can follow these steps:

### Step 1: Write the equations in standard form
The first equation is already in standard form:
\[ x + 2y = 5 \]

For the second equation, we rearrange it to standard form:
\[ 3x + ky + 15 = 0 \]
This can be rewritten as:
\[ 3x + ky = -15 \]

### Step 2: Identify the coefficients
From the equations, we can identify the coefficients of \( x \) and \( y \):
- For the first equation \( x + 2y = 5 \):
- Coefficient of \( x \) is \( 1 \)
- Coefficient of \( y \) is \( 2 \)

- For the second equation \( 3x + ky = -15 \):
- Coefficient of \( x \) is \( 3 \)
- Coefficient of \( y \) is \( k \)

### Step 3: Set up the determinant for no solution
For the system of equations to have no solution, the determinant \( D \) formed by the coefficients must be zero:
\[ D = \begin{vmatrix} 1 & 2 \\ 3 & k \end{vmatrix} \]

### Step 4: Calculate the determinant
The determinant is calculated as follows:
\[ D = (1)(k) - (2)(3) = k - 6 \]

### Step 5: Set the determinant to zero
To find the value of \( k \) for which there is no solution, we set the determinant equal to zero:
\[ k - 6 = 0 \]

### Step 6: Solve for \( k \)
Now, solve for \( k \):
\[ k = 6 \]

### Conclusion
Thus, the value of \( k \) for which the system of equations has no solution is:
\[ \boxed{6} \]