If `|(8,-5,1),(5,x,1),(6,3,1)| = 2`, then what is the value of x ?

Options

A. 4
B. 5
C. 6
D. 8

Answer

To solve the determinant \( |(8, -5, 1), (5, x, 1), (6, 3, 1)| = 2 \), we will follow these steps:

### Step 1: Write the Determinant
The determinant can be expressed as:
\[ D = \begin{vmatrix} 8 & -5 & 1 \\ 5 & x & 1 \\ 6 & 3 & 1 \end{vmatrix} \]

### Step 2: Expand the Determinant
We will expand the determinant using the first row:
\[ D = 8 \begin{vmatrix} x & 1 \\ 3 & 1 \end{vmatrix} - (-5) \begin{vmatrix} 5 & 1 \\ 6 & 1 \end{vmatrix} + 1 \begin{vmatrix} 5 & x \\ 6 & 3 \end{vmatrix} \]

### Step 3: Calculate the 2x2 Determinants
Now we calculate the 2x2 determinants:
1. For \( \begin{vmatrix} x & 1 \\ 3 & 1 \end{vmatrix} \):
\[ = x \cdot 1 - 1 \cdot 3 = x - 3 \]

2. For \( \begin{vmatrix} 5 & 1 \\ 6 & 1 \end{vmatrix} \):
\[ = 5 \cdot 1 - 1 \cdot 6 = 5 - 6 = -1 \]

3. For \( \begin{vmatrix} 5 & x \\ 6 & 3 \end{vmatrix} \):
\[ = 5 \cdot 3 - x \cdot 6 = 15 - 6x \]

### Step 4: Substitute Back into the Determinant
Substituting these values back into the determinant:
\[ D = 8(x - 3) + 5(-1) + (15 - 6x) \]
\[ = 8x - 24 - 5 + 15 - 6x \]
\[ = (8x - 6x) + (-24 - 5 + 15) \]
\[ = 2x - 14 \]

### Step 5: Set the Determinant Equal to 2
Now we set the determinant equal to 2:
\[ 2x - 14 = 2 \]

### Step 6: Solve for \( x \)
To solve for \( x \):
\[ 2x = 2 + 14 \]
\[ 2x = 16 \]
\[ x = \frac{16}{2} = 8 \]

### Conclusion
Thus, the value of \( x \) is \( 8 \).