`x^(2) -365= 364, y- sqrt324= sqrt81`

Options

A. if `x gt y`
B. if `x gt= y`
C. if `x lt=y`
D. if `x lt y`

Answer

To solve the equations given in the question, we will break it down into two parts: finding the value of \( x \) from the first equation and finding the value of \( y \) from the second equation.

### Step 1: Solve for \( x \)

The first equation is:
\[ x^2 - 365 = 364 \]

**Step 1.1:** Add 365 to both sides of the equation to isolate \( x^2 \):
\[ x^2 = 364 + 365 \]
\[ x^2 = 729 \]

**Step 1.2:** Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{729} \quad \text{or} \quad x = -\sqrt{729} \]
\[ x = 27 \quad \text{or} \quad x = -27 \]

### Step 2: Solve for \( y \)

The second equation is:
\[ y - \sqrt{324} = \sqrt{81} \]

**Step 2.1:** First, calculate \( \sqrt{324} \) and \( \sqrt{81} \):
\[ \sqrt{324} = 18 \quad \text{and} \quad \sqrt{81} = 9 \]

**Step 2.2:** Substitute these values back into the equation:
\[ y - 18 = 9 \]

**Step 2.3:** Add 18 to both sides to solve for \( y \):
\[ y = 9 + 18 \]
\[ y = 27 \]

### Summary of Values
- The values we found are:
- \( x = 27 \) or \( x = -27 \)
- \( y = 27 \)

### Step 3: Compare \( x \) and \( y \)

Now, we need to compare the values of \( x \) and \( y \):
- Since \( x \) can be both \( 27 \) and \( -27 \), we have:
- If \( x = 27 \), then \( x = y \).
- If \( x = -27 \), then \( x < y \).

### Conclusion
The relationship between \( x \) and \( y \) can be established as:
- \( x \) can be equal to \( y \) when \( x = 27 \).
- \( x \) is less than \( y \) when \( x = -27 \).