If `1/x - 1/y=4`, find the value of `(2x+4xy-2y)/(x-y-2xy)`.

Answer

To solve the equation \( \frac{1}{x} - \frac{1}{y} = 4 \) and find the value of \( \frac{2x + 4xy - 2y}{x - y - 2xy} \), we can follow these steps:

### Step 1: Rewrite the given equation
Starting with the equation:
\[ \frac{1}{x} - \frac{1}{y} = 4 \]
We can find a common denominator:
\[ \frac{y - x}{xy} = 4 \]
This implies:
\[ y - x = 4xy \]

### Step 2: Rearranging the equation
From \( y - x = 4xy \), we can express \( y \) in terms of \( x \):
\[ y = x + 4xy \]
Rearranging gives:
\[ y(1 - 4x) = x \quad \Rightarrow \quad y = \frac{x}{1 - 4x} \]

### Step 3: Substitute \( y \) into the expression
Now, we need to substitute \( y \) into the expression \( \frac{2x + 4xy - 2y}{x - y - 2xy} \).

First, calculate \( 4xy \):
\[ 4xy = 4x \cdot \frac{x}{1 - 4x} = \frac{4x^2}{1 - 4x} \]

Now, calculate \( 2y \):
\[ 2y = 2 \cdot \frac{x}{1 - 4x} = \frac{2x}{1 - 4x} \]

### Step 4: Substitute into the numerator
The numerator becomes:
\[ 2x + 4xy - 2y = 2x + \frac{4x^2}{1 - 4x} - \frac{2x}{1 - 4x} \]
Combine the terms:
\[ = 2x + \frac{4x^2 - 2x}{1 - 4x} = 2x + \frac{2x(2x - 1)}{1 - 4x} \]

### Step 5: Substitute into the denominator
Now for the denominator:
\[ x - y - 2xy = x - \frac{x}{1 - 4x} - 2 \cdot \frac{4x^2}{1 - 4x} \]
Combine the terms:
\[ = x - \frac{x + 8x^2}{1 - 4x} = \frac{x(1 - 4x) - (x + 8x^2)}{1 - 4x} \]
Simplifying gives:
\[ = \frac{x - 4x^2 - x - 8x^2}{1 - 4x} = \frac{-12x^2}{1 - 4x} \]

### Step 6: Combine the results
Now we can combine the numerator and denominator:
\[ \frac{2x + \frac{2x(2x - 1)}{1 - 4x}}{\frac{-12x^2}{1 - 4x}} = \frac{(2x(1 - 4x) + 2x(2x - 1))}{-12x^2} \]
This simplifies to:
\[ \frac{2x(1 - 4x + 2x - 1)}{-12x^2} = \frac{2x(-2x)}{-12x^2} = \frac{4x^2}{12x^2} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( \frac{2x + 4xy - 2y}{x - y - 2xy} \) is:
\[ \frac{1}{3} \]