Consider the system of linear equations
`a_(1)x+b_(1)y+ c_(1)z+d_(1)=0`,
` a_(2)x+b_(2)y+ c_(2)z+d_(2)= 0`,
`a_(3)x+b_(3)y +c_(3)z+d_(3)=0`, Let us denote by `Delta` (a,b,c) the determinant `|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|` , if `Delta (a,b,c) `# 0, then the value of x in the unique solution of the above equations is

Options

A. `(Delta(bcd))/(Delta(abc))`
B. `(-Delta(bcd))/(Delta(abc))`
C. `(Delta(acd))/(Delta(abc))`
D. `(Delta(abd))/(Delta(abc))`

Answer

To find the value of \( x \) in the unique solution of the given system of linear equations, we can follow these steps:

### Step 1: Write the System of Equations
The given system of linear equations is:
\[ \begin{align*} a_1 x + b_1 y + c_1 z + d_1 &= 0 \\ a_2 x + b_2 y + c_2 z + d_2 &= 0 \\ a_3 x + b_3 y + c_3 z + d_3 &= 0 \end{align*} \]

### Step 2: Rearranging the Equations
We can rearrange these equations to isolate the constants on one side:
\[ \begin{align*} a_1 x + b_1 y + c_1 z &= -d_1 \\ a_2 x + b_2 y + c_2 z &= -d_2 \\ a_3 x + b_3 y + c_3 z &= -d_3 \end{align*} \]

### Step 3: Form the Coefficient Matrix and the Determinants
Let \( \Delta \) be the determinant of the coefficients of \( x, y, z \):
\[ \Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \]
If \( \Delta \neq 0 \), the system has a unique solution.

### Step 4: Calculate the Determinants for \( x, y, z \)
We will compute the determinants to find \( x, y, z \):
- For \( x \):
\[ D_x = \begin{vmatrix} -d_1 & b_1 & c_1 \\ -d_2 & b_2 & c_2 \\ -d_3 & b_3 & c_3 \end{vmatrix} \]
- For \( y \):
\[ D_y = \begin{vmatrix} a_1 & -d_1 & c_1 \\ a_2 & -d_2 & c_2 \\ a_3 & -d_3 & c_3 \end{vmatrix} \]
- For \( z \):
\[ D_z = \begin{vmatrix} a_1 & b_1 & -d_1 \\ a_2 & b_2 & -d_2 \\ a_3 & b_3 & -d_3 \end{vmatrix} \]

### Step 5: Apply Cramer's Rule
According to Cramer's Rule, the solutions for \( x, y, z \) are given by:
\[ x = \frac{D_x}{\Delta}, \quad y = \frac{D_y}{\Delta}, \quad z = \frac{D_z}{\Delta} \]

### Step 6: Substitute to Find \( x \)
Thus, the value of \( x \) in the unique solution is:
\[ x = \frac{D_x}{\Delta} \]

### Summary of the Solution
The value of \( x \) in the unique solution of the given system of equations is:
\[ x = \frac{D_x}{\Delta} \]