I. ` x^(2) - 463 = 321`
II.` y^(2) - 421 = 308`

Options

A. if ` x gt y`
B. if `x ge y`
C. if ` x lt y`
D. if x = y or the relationship cannot be established

Explanation

To solve the given equations step by step, we will start with each equation separately.

### Step 1: Solve the first equation
The first equation is:
\[ x^2 - 463 = 321 \]

To isolate \( x^2 \), we will add 463 to both sides:
\[ x^2 = 321 + 463 \]
\[ x^2 = 784 \]

### Step 2: Find the value of \( x \)
Now, we will take the square root of both sides to find \( x \):
\[ x = \pm \sqrt{784} \]
\[ x = \pm 28 \]

### Step 3: Solve the second equation
The second equation is:
\[ y^2 - 421 = 308 \]

To isolate \( y^2 \), we will add 421 to both sides:
\[ y^2 = 308 + 421 \]
\[ y^2 = 729 \]

### Step 4: Find the value of \( y \)
Now, we will take the square root of both sides to find \( y \):
\[ y = \pm \sqrt{729} \]
\[ y = \pm 27 \]

### Step 5: Establish the relationship between \( x \) and \( y \)
Now we have:
- \( x = 28 \) or \( x = -28 \)
- \( y = 27 \) or \( y = -27 \)

We can compare the possible values:
- If \( x = 28 \), then \( y = 27 \)
- If \( x = -28 \), then \( y = -27 \)

However, we cannot establish a fixed relationship between \( x \) and \( y \) since both can take on positive and negative values independently.

### Conclusion
The relationship between \( x \) and \( y \) cannot be established as a fixed relation.

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