(b) (i) Simplify $$ \frac{1+x y^{-1}}{3-x y^{-1}} $$

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1. Rewrite the expression by replacing $$ y^{-1} $$ with $$ \frac{1}{y} $$:
   $$
   \frac{1+x y^{-1}}{3-x y^{-1}} = \frac{1+\frac{x}{y}}{3-\frac{x}{y}}.
   $$

2. Combine terms in the numerator and denominator by getting a common denominator:
   - Numerator:
     $$
     1+\frac{x}{y} = \frac{y}{y}+\frac{x}{y}=\frac{x+y}{y}.
     $$
   - Denominator:
     $$
     3-\frac{x}{y} = \frac{3y}{y}-\frac{x}{y}=\frac{3y-x}{y}.
     $$

3. Write the expression as a single fraction:
   $$
   \frac{\frac{x+y}{y}}{\frac{3y-x}{y}}.
   $$

4. Simplify by canceling the common factor $$\frac{1}{y}$$:
   $$
   \frac{x+y}{3y-x}.
   $$

The simplified form is:
$$
\frac{x+y}{3y-x}.
$$
The simplified form is $$ \frac{x + y}{3y - x} $$.

(b)
(i) Simplify

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