If `(a_(1)//x)+(b_(1)//y)=c_(1),(a_(2)//x)+(b_(2)//y)=c_(2) Delta_(1)=|{:(a_(1),b_(1)),(a_(2),b_(2)):}|,Delta_(2)=|{:(b_(1),c_(1)),(b_(2),c_(2)):}|," "Delta_(3)=|{:(c_(1),a_(1)),(c_(2),a_(2)):}|`, then (x, y) is equal to which one of the following ?

Options

A. `(Delta_(2)//Delta_(1),Delta_(3)//Delta_(1))`
B. `(Delta_(3)//Delta_(1),Delta_(2)//Delta_(1))`
C. `(-Delta_(1)//Delta_(2),-Delta_(1)//Delta_(3))`
D. `(-Delta_(1)//Delta_(2),-Delta_(1)//Delta_(3))`

Explanation

To solve the given problem, we need to find the values of \( x \) and \( y \) based on the equations provided. The equations are:

\[ \frac{a_1}{x} + \frac{b_1}{y} = c_1 \]
\[ \frac{a_2}{x} + \frac{b_2}{y} = c_2 \]

We can rewrite these equations in terms of new variables \( u \) and \( v \):

Let:
\[ u = \frac{1}{x} \quad \text{and} \quad v = \frac{1}{y} \]

Then the equations become:

\[ a_1 u + b_1 v = c_1 \tag{1} \]
\[ a_2 u + b_2 v = c_2 \tag{2} \]

### Step 1: Set up the determinant equations

We can express these equations in matrix form and use determinants to solve for \( u \) and \( v \).

The determinant for \( u \) can be expressed as:

\[ \Delta_1 = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1 b_2 - a_2 b_1 \]

The determinant for \( v \) can be expressed as:

\[ \Delta_2 = \begin{vmatrix} b_1 & c_1 \\ b_2 & c_2 \end{vmatrix} = b_1 c_2 - b_2 c_1 \]

### Step 2: Solve for \( u \) and \( v \)

Using Cramer's rule, we can find \( u \) and \( v \):

For \( u \):
\[ u = \frac{\Delta_1}{\Delta_2} = \frac{c_1 b_2 - c_2 b_1}{b_1 c_2 - b_2 c_1} \]

For \( v \):
\[ v = \frac{\Delta_3}{\Delta_1} = \frac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1} \]

### Step 3: Substitute back to find \( x \) and \( y \)

Now, since \( u = \frac{1}{x} \) and \( v = \frac{1}{y} \), we can find \( x \) and \( y \):

\[ \frac{1}{x} = u \implies x = \frac{1}{u} \]
\[ \frac{1}{y} = v \implies y = \frac{1}{v} \]

Thus:
\[ x = \frac{\Delta_2}{\Delta_1} \quad \text{and} \quad y = \frac{\Delta_1}{\Delta_3} \]

### Step 4: Final values of \( x \) and \( y \)

From the determinants, we can express:

\[ x = -\frac{\Delta_1}{\Delta_2} \quad \text{and} \quad y = -\frac{\Delta_1}{\Delta_3} \]

### Conclusion

Thus, the final values of \( (x, y) \) are:

\[ x = -\frac{\Delta_1}{\Delta_2}, \quad y = -\frac{\Delta_1}{\Delta_3} \]