`{:("Column A",x = 1//y,"Column B"),(x +1 + 1//x,,y + 1 + 1//y):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the expressions in Column A and Column B given that \( x = \frac{1}{y} \).

### Step-by-Step Solution:

1. **Identify the expressions in Column A and Column B:**
- Column A: \( x + 1 + \frac{1}{x} \)
- Column B: \( y + 1 + \frac{1}{y} \)

2. **Substitute \( x \) in terms of \( y \):**
- Since \( x = \frac{1}{y} \), we can express \( \frac{1}{x} \) as \( y \) (because \( \frac{1}{x} = y \)).
- Therefore, we can rewrite Column A:
\[ \text{Column A} = x + 1 + y \]

3. **Substitute \( y \) in terms of \( x \):**
- From the relationship \( y = \frac{1}{x} \), we can express \( \frac{1}{y} \) as \( x \).
- Thus, we can rewrite Column B:
\[ \text{Column B} = y + 1 + x \]

4. **Compare the two columns:**
- Now we have:
\[ \text{Column A} = x + 1 + y \]
\[ \text{Column B} = y + 1 + x \]
- Both expressions are identical, as the order of addition does not affect the result.

5. **Conclusion:**
- Since Column A and Column B are equal, we conclude that:
\[ \text{Column A} = \text{Column B} \]

### Final Answer:
Both columns are equal.