Suppose the system of equations
`a_(1)x+b_(1)y+c_(1)z=d_(1)`
`a_(2)x+b_(2)y+c_(2)z=d_(2)`
`a_(3)x+b_(3)y+c_(3)z=d_(3)`
has a unique solution `(x_(0),y_(0),z_(0))`. If `x_(0) = 0`, then which one of the following is correct ?

Options

A. `|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0`
B. `|{:(d_(1),b_(1),c_(1)),(d_(2),b_(2),c_(2)),(d_(3),b_(3),c_(3)):}|=0`
C. `|{:(d_(1),a_(1),c_(1)),(d_(2),a_(2),c_(2)),(d_(3),a_(3),c_(3)):}|=0`
D. `|{:(d_(1),a_(1),b_(1)),(d_(2),a_(2),b_(2)),(d_(3),a_(3),b_(3)):}|=0`

Explanation

To solve the problem, we need to analyze the given system of equations and the implications of having a unique solution with \( x_0 = 0 \).

### Step-by-Step Solution:

1. **Understanding the System of Equations**:
We have the following system of equations:
\[ a_1x + b_1y + c_1z = d_1 \]
\[ a_2x + b_2y + c_2z = d_2 \]
\[ a_3x + b_3y + c_3z = d_3 \]
This can be represented in matrix form as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix.

2. **Matrix Representation**:
The coefficient matrix \( A \) is:
\[ A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} \]
The constant matrix \( B \) is:
\[ B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix} \]

3. **Determinant and Unique Solution**:
For the system to have a unique solution, the determinant of the coefficient matrix \( A \) (denoted as \( \Delta \)) must be non-zero:
\[ \Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \neq 0 \]

4. **Using Cramer's Rule**:
According to Cramer's Rule, the solution for \( x \), \( y \), and \( z \) can be expressed as:
\[ x_0 = \frac{\Delta_x}{\Delta}, \quad y_0 = \frac{\Delta_y}{\Delta}, \quad z_0 = \frac{\Delta_z}{\Delta} \]
where \( \Delta_x \), \( \Delta_y \), and \( \Delta_z \) are the determinants formed by replacing the respective column of \( A \) with the matrix \( B \).

5. **Given Condition**:
We are given that \( x_0 = 0 \). Therefore, we have:
\[ x_0 = \frac{\Delta_x}{\Delta} = 0 \]
Since \( \Delta \neq 0 \), this implies:
\[ \Delta_x = 0 \]

6. **Finding \( \Delta_x \)**:
The determinant \( \Delta_x \) is given by:
\[ \Delta_x = \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{vmatrix} \]
Since \( \Delta_x = 0 \), it indicates that the constants \( d_1, d_2, d_3 \) are linearly dependent with the coefficients of \( y \) and \( z \).

### Conclusion:
Thus, if \( x_0 = 0 \), it implies that \( \Delta_x = 0 \), which means that the constants \( d_1, d_2, d_3 \) must be such that they do not provide a unique contribution to the solution for \( x \).