The system of the linear equations `x + y – z = 6, x + 2y – 3z = 14 and 2x + 5y – lambdaz = 9 (lambda in R)` has a unique solution if

Options

A. `lambda=8`
B. `lambdane8`
C. `lambda=7`
D. `lambdane7`

Answer

To determine the condition under which the system of linear equations has a unique solution, we need to analyze the given equations and compute the determinant of the coefficient matrix. The equations provided are:

1. \( x + y - z = 6 \)
2. \( x + 2y - 3z = 14 \)
3. \( 2x + 5y - \lambda z = 9 \)

### Step 1: Write the equations in matrix form

The system can be represented in the form \( AX = B \), where:

\[ A = \begin{bmatrix} 1 & 1 & -1 \\ 1 & 2 & -3 \\ 2 & 5 & -\lambda \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ 14 \\ 9 \end{bmatrix} \]

### Step 2: Calculate the determinant of matrix \( A \)

To find the condition for a unique solution, we need to calculate the determinant of matrix \( A \) and set it not equal to zero.

\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & -1 \\ 1 & 2 & -3 \\ 2 & 5 & -\lambda \end{vmatrix} \]

### Step 3: Expand the determinant

Using the first row for expansion, we have:

\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 2 & -3 \\ 5 & -\lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & -3 \\ 2 & -\lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 2 \\ 2 & 5 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \(\begin{vmatrix} 2 & -3 \\ 5 & -\lambda \end{vmatrix} = 2(-\lambda) - (-3)(5) = -2\lambda + 15\)

2. \(\begin{vmatrix} 1 & -3 \\ 2 & -\lambda \end{vmatrix} = 1(-\lambda) - (-3)(2) = -\lambda + 6\)

3. \(\begin{vmatrix} 1 & 2 \\ 2 & 5 \end{vmatrix} = 1(5) - 2(2) = 5 - 4 = 1\)

### Step 4: Substitute back into the determinant

Now substituting these results back into the determinant expression:

\[ \text{det}(A) = 1(-2\lambda + 15) - 1(-\lambda + 6) - 1(1) \]

This simplifies to:

\[ \text{det}(A) = -2\lambda + 15 + \lambda - 6 - 1 \]
\[ = -\lambda + 8 \]

### Step 5: Set the determinant not equal to zero

For the system to have a unique solution, we require:

\[ -\lambda + 8 \neq 0 \]

This implies:

\[ \lambda \neq 8 \]

### Conclusion

The system of linear equations has a unique solution if \( \lambda \) is not equal to 8.

---