If A, x and y are all distinct numbers , and `A =(xy-2)/(x-y)` which of the following represents x , in terms of A and y ?

Options

A. `(Ay-2)/(A-y)`
B. `(A-2)/(x-1)`
C. `(A-y)/(x-y)`
D. `(Ay-2)/(A+y)`

Answer

To solve the equation \( A = \frac{xy - 2}{x - y} \) for \( x \) in terms of \( A \) and \( y \), we will follow these steps:

### Step 1: Start with the given equation
We have:
\[ A = \frac{xy - 2}{x - y} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ A(x - y) = xy - 2 \]

### Step 3: Distribute \( A \) on the left side
Distributing \( A \) results in:
\[ Ax - Ay = xy - 2 \]

### Step 4: Rearrange the equation
Now, we want to get all terms involving \( x \) on one side and the constant terms on the other side. We can rearrange it as follows:
\[ Ax - xy = Ay - 2 \]

### Step 5: Factor out \( x \)
Now, we can factor \( x \) out from the left side:
\[ x(A - y) = Ay - 2 \]

### Step 6: Solve for \( x \)
Finally, we can solve for \( x \) by dividing both sides by \( (A - y) \):
\[ x = \frac{Ay - 2}{A - y} \]

### Final Answer
Thus, the expression for \( x \) in terms of \( A \) and \( y \) is:
\[ x = \frac{Ay - 2}{A - y} \]

---