`((x-y)/(y))/(y^(-1)-x^(-1))`
The expression above is equivalent to

Options

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

Answer

To simplify the expression \(\frac{(x-y)}{y} \div (y^{-1} - x^{-1})\), we can follow these steps:

### Step 1: Rewrite the expression
The original expression can be rewritten as:
\[ \frac{(x-y)}{y} \div (y^{-1} - x^{-1}) \]

### Step 2: Simplify \(y^{-1} - x^{-1}\)
Using the property of negative exponents, we can express \(y^{-1}\) and \(x^{-1}\) as:
\[ y^{-1} = \frac{1}{y} \quad \text{and} \quad x^{-1} = \frac{1}{x} \]
Thus, we have:
\[ y^{-1} - x^{-1} = \frac{1}{y} - \frac{1}{x} \]

### Step 3: Find a common denominator
To combine the fractions \(\frac{1}{y}\) and \(\frac{1}{x}\), we find a common denominator, which is \(xy\):
\[ \frac{1}{y} - \frac{1}{x} = \frac{x}{xy} - \frac{y}{xy} = \frac{x - y}{xy} \]

### Step 4: Substitute back into the expression
Now we can substitute this back into our original expression:
\[ \frac{(x-y)}{y} \div \left(\frac{x - y}{xy}\right) \]

### Step 5: Change division to multiplication
Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[ \frac{(x-y)}{y} \times \frac{xy}{(x - y)} \]

### Step 6: Cancel common terms
Notice that \((x - y)\) in the numerator and denominator cancels out:
\[ \frac{xy}{y} \]

### Step 7: Simplify the expression
Now, we simplify \(\frac{xy}{y}\):
\[ \frac{xy}{y} = x \]

### Final Answer
Thus, the expression is equivalent to:
\[ \boxed{x} \]