Solve for x and y :
` (xy )/(x + y ) = ( 6)/(5), (xy)/( y - x) = 6 (x ne 0, y ne 0 and x ne y )`.

Options

A. `x=1 , y=3`
B. `x=2 ,y=2`
C. `x=3 , y=3`
D. `x=2 , y=3`

Explanation

To solve the equations given in the problem, we will follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \(\frac{xy}{x + y} = \frac{6}{5}\)
2. \(\frac{xy}{y - x} = 6\)

### Step 2: Cross multiply the first equation
From the first equation, we can cross-multiply:
\[ 5xy = 6(x + y) \]
This simplifies to:
\[ 5xy = 6x + 6y \]

### Step 3: Cross multiply the second equation
From the second equation, we cross-multiply:
\[ xy = 6(y - x) \]
This simplifies to:
\[ xy = 6y - 6x \]

### Step 4: Rearranging both equations
Now we can rearrange both equations:
1. From \(5xy = 6x + 6y\), we can write:
\[ 5xy - 6x - 6y = 0 \quad (1) \]
2. From \(xy = 6y - 6x\), we can write:
\[ xy + 6x - 6y = 0 \quad (2) \]

### Step 5: Divide equation (1) by equation (2)
Now, we will divide equation (1) by equation (2):
\[ \frac{5xy - 6x - 6y}{xy + 6x - 6y} = 0 \]
This simplifies to:
\[ 5xy - 6x - 6y = 0 \quad \text{and} \quad xy + 6x - 6y \neq 0 \]

### Step 6: Rearranging the divided equation
From the division, we can cross-multiply:
\[ 5(y - x) = x + y \]
This simplifies to:
\[ 5y - 5x = x + y \]

### Step 7: Rearranging to isolate variables
Rearranging gives:
\[ 5y - y = x + 5x \]
This simplifies to:
\[ 4y = 6x \quad \Rightarrow \quad y = \frac{3}{2}x \]

### Step 8: Substitute \(y\) back into one of the original equations
Substituting \(y = \frac{3}{2}x\) into the first equation:
\[ 5x\left(\frac{3}{2}x\right) = 6\left(x + \frac{3}{2}x\right) \]
This simplifies to:
\[ \frac{15}{2}x^2 = 6\left(\frac{5}{2}x\right) \]
\[ \frac{15}{2}x^2 = 15x \]

### Step 9: Solve for \(x\)
Dividing both sides by \(15\) gives:
\[ \frac{1}{2}x^2 = x \]
Rearranging gives:
\[ \frac{1}{2}x^2 - x = 0 \]
Factoring out \(x\):
\[ x\left(\frac{1}{2}x - 1\right) = 0 \]
Thus, \(x = 0\) or \(\frac{1}{2}x - 1 = 0\) leading to:
\[ x = 2 \]

### Step 10: Find \(y\)
Substituting \(x = 2\) back into \(y = \frac{3}{2}x\):
\[ y = \frac{3}{2} \cdot 2 = 3 \]

### Final Solution
Thus, the solution is:
\[ x = 2, \quad y = 3 \]