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

Explanation

To solve the determinant equation \( | \begin{pmatrix} 8 & -5 & 1 \\ 5 & x & 1 \\ 6 & 3 & 1 \end{pmatrix} | = 2 \), we will calculate the determinant step by step.

### Step 1: Write the determinant
We start with the determinant:
\[ D = \begin{vmatrix} 8 & -5 & 1 \\ 5 & x & 1 \\ 6 & 3 & 1 \end{vmatrix} \]

### Step 2: Calculate the determinant using cofactor expansion
We can expand the determinant along 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. \( \begin{vmatrix} x & 1 \\ 3 & 1 \end{vmatrix} = x \cdot 1 - 1 \cdot 3 = x - 3 \)
2. \( \begin{vmatrix} 5 & 1 \\ 6 & 1 \end{vmatrix} = 5 \cdot 1 - 1 \cdot 6 = 5 - 6 = -1 \)
3. \( \begin{vmatrix} 5 & x \\ 6 & 3 \end{vmatrix} = 5 \cdot 3 - x \cdot 6 = 15 - 6x \)

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

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

### Step 6: Solve for \( x \)
Adding 14 to both sides:
\[ 2x = 16 \]
Dividing by 2:
\[ x = 8 \]

### Final Answer
Thus, the value of \( x \) is \( \boxed{8} \).

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