`{:("Column A",x = 1//y,"Column B"),((x^2 + 1)/(x),,(y^2 + 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 1: Write down the expressions for Column A and Column B
- Column A: \( \frac{x^2 + 1}{x} \)
- Column B: \( \frac{y^2 + 1}{y} \)

### Step 2: Simplify Column A
We start with Column A:
\[ \text{Column A} = \frac{x^2 + 1}{x} \]
This can be separated into two fractions:
\[ = \frac{x^2}{x} + \frac{1}{x} = x + \frac{1}{x} \]

### Step 3: Substitute \( y \) in Column A
Since we know that \( y = \frac{1}{x} \), we can substitute \( \frac{1}{x} \) with \( y \):
\[ \text{Column A} = x + y \]

### Step 4: Simplify Column B
Now, we simplify Column B:
\[ \text{Column B} = \frac{y^2 + 1}{y} \]
This can also be separated into two fractions:
\[ = \frac{y^2}{y} + \frac{1}{y} = y + \frac{1}{y} \]

### Step 5: Substitute \( x \) in Column B
Since we know that \( x = \frac{1}{y} \), we can substitute \( \frac{1}{y} \) with \( x \):
\[ \text{Column B} = y + x \]

### Step 6: Compare Column A and Column B
Now we have:
- Column A: \( x + y \)
- Column B: \( y + x \)

Since addition is commutative, we can conclude that:
\[ \text{Column A} = \text{Column B} \]

### Conclusion
Both columns are equal, so the answer is that Column A is equal to Column B.

### Final Answer
Column A = Column B

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