If `(xy)/(x-y)=1 and xne-y`, what is x in terms of y?

Options

A. `(y+1)/(y-1)`
B. `(y+1)/(y)`
C. `(y)/(y-1)`
D. `(y)/(y+1)`

Answer

To solve the equation \(\frac{xy}{x-y} = 1\) for \(x\) in terms of \(y\), we can follow these steps:

### Step 1: Multiply both sides by \(x - y\)
Starting with the equation:
\[ \frac{xy}{x-y} = 1 \]
We can eliminate the fraction by multiplying both sides by \(x - y\):
\[ xy = 1(x - y) \]
This simplifies to:
\[ xy = x - y \]

### Step 2: Rearrange the equation
Next, we want to isolate \(x\). We can rearrange the equation by moving all terms involving \(x\) to one side:
\[ xy - x = -y \]

### Step 3: Factor out \(x\)
Now, we can factor \(x\) out from the left side:
\[ x(y - 1) = -y \]

### Step 4: Solve for \(x\)
To isolate \(x\), we divide both sides by \(y - 1\) (noting that \(y \neq 1\) to avoid division by zero):
\[ x = \frac{-y}{y - 1} \]

### Step 5: Simplify the expression
We can simplify the expression further:
\[ x = \frac{y}{1 - y} \]

Thus, the final answer is:
\[ x = \frac{y}{1 - y} \]