Q. \(x + y + x y = 1\) expression \(x y + \frac{1}{x} – \frac{y}{x} – \frac{x}{y}\).

Problem:
Given the equation \(x+y+xy=1\), compute the value of the expression:
\(E=xy+\dfrac{1}{x}-\dfrac{y}{x}-\dfrac{x}{y}\)

1. Step 1 – Analyze the given equation:

   We are given \(x+y+xy=1\). We can rearrange this to express \(1-y\) in terms of \(x\). Subtract \(y\) from both sides: \(x+xy=1-y\). Factor out \(x\) on the left side: \(x(1+y)=1-y\).

2. Step 2 – Simplify the first part of the expression:

   The expression \(E\) contains the term \(\dfrac{1}{x}-\dfrac{y}{x}\). Since they have a common denominator, we can combine them: \(\dfrac{1-y}{x}\).

3. Step 3 – Substitute the relationship into the simplified term:

   From Step 1, we know that \(1-y=x(1+y)\). Substitute this into our combined fraction: \(\dfrac{1-y}{x}=\dfrac{x(1+y)}{x}\). Assuming \(x\neq 0\), the \(x\) terms cancel out, leaving us with \(1+y\).

4. Step 4 – Update the expression (E):

   Substitute \(1+y\) back into the original expression for the terms we combined: \(E=xy+(1+y)-\dfrac{x}{y}\). This can be rewritten by grouping terms: \(E=1+y+xy-\dfrac{x}{y}\).

5. Step 5 – Use the original equation to simplify further:

   The original equation is \(x+y+xy=1\). If we rearrange it to \(y+xy=1-x\), we can substitute this into our updated expression for \(E\): \(E=1+(1-x)-\dfrac{x}{y}=2-x-\dfrac{x}{y}\).

6. Step 6 – Express \(y\) in terms of \(x\):

   From Step 1, we have \(x(1+y)=1-y\). Expanding this gives \(x+xy=1-y\). Grouping \(y\) terms: \(y+xy=1-x\), then \(y(1+x)=1-x\). Therefore, \(y=\dfrac{1-x}{1+x}\).

7. Step 7 – Find the value of \(\dfrac{x}{y}\):

   Using the result from Step 6: \(\dfrac{x}{y}=x\div\dfrac{1-x}{1+x}=x\cdot\dfrac{1+x}{1-x}=\dfrac{x+x^2}{1-x}\).

8. Step 8 – Perform final substitution and simplification:

   Substitute \(\dfrac{x}{y}\) back into the expression for \(E\): \(E=2-x-\dfrac{x+x^2}{1-x}\). To combine these, find a common denominator:
   \(E=\dfrac{(2-x)(1-x)-(x+x^2)}{1-x}=\dfrac{(2-3x+x^2)-x-x^2}{1-x}=\dfrac{2-4x}{1-x}\).

9. Final Answer:

   \(\dfrac{2(1-2x)}{1-x}\)