In each of these questions, two equations are given . You have to solve both the equations and give answer
I. ` ( x + y) ^(2) = 361 `
II. ` 92442 = y^(2) + 92361`

Options

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

Answer

To solve the given equations step by step, we will first address each equation separately and then find the relationship between \( x \) and \( y \).

### Step 1: Solve the first equation
The first equation is:
\[ (x + y)^2 = 361 \]
To solve for \( x + y \), we take the square root of both sides:
\[ x + y = \pm \sqrt{361} \]
Calculating the square root:
\[ \sqrt{361} = 19 \]
Thus, we have two cases:
1. \( x + y = 19 \)
2. \( x + y = -19 \)

### Step 2: Solve the second equation
The second equation is:
\[ 92442 = y^2 + 92361 \]
To isolate \( y^2 \), we subtract \( 92361 \) from both sides:
\[ y^2 = 92442 - 92361 \]
Calculating the right side:
\[ y^2 = 81 \]
Now, taking the square root of both sides gives:
\[ y = \pm \sqrt{81} \]
Calculating the square root:
\[ \sqrt{81} = 9 \]
Thus, we have two cases for \( y \):
1. \( y = 9 \)
2. \( y = -9 \)

### Step 3: Find values of \( x \) for each case of \( y \)
Now we will substitute the values of \( y \) back into the equations we derived from the first equation.

#### Case 1: \( y = 9 \)
Using \( x + y = 19 \):
\[ x + 9 = 19 \implies x = 19 - 9 = 10 \]

Using \( x + y = -19 \):
\[ x + 9 = -19 \implies x = -19 - 9 = -28 \]

#### Case 2: \( y = -9 \)
Using \( x + y = 19 \):
\[ x - 9 = 19 \implies x = 19 + 9 = 28 \]

Using \( x + y = -19 \):
\[ x - 9 = -19 \implies x = -19 + 9 = -10 \]

### Summary of Solutions
From the above calculations, we have the following pairs of \( (x, y) \):
1. \( (10, 9) \)
2. \( (-28, 9) \)
3. \( (28, -9) \)
4. \( (-10, -9) \)

### Step 4: Compare values of \( x \) and \( y \)
Now we compare \( x \) and \( y \) for each pair:
1. For \( (10, 9) \): \( x > y \)
2. For \( (-28, 9) \): \( y > x \)
3. For \( (28, -9) \): \( x > y \)
4. For \( (-10, -9) \): \( y > x \)

### Conclusion
The relationship between \( x \) and \( y \) varies depending on the specific values derived from the equations. Therefore, we cannot establish a consistent relationship between \( x \) and \( y \).