In each of these questions two equations numbered I and II are given. You have to solve both the equations and give answer
I `x^2=81`
II `y^2-18y+81=0`

Options

A. if `xlty`
B. if `xley`
C. if `xgty`
D. if `xgey`

Answer

To solve the given equations step by step, let's start with the first equation.

### Step 1: Solve the first equation
The first equation is:
\[ I: x^2 = 81 \]
To find the value of \( x \), we take the square root of both sides:
\[ x = \sqrt{81} \]
This gives us:
\[ x = 9 \quad \text{or} \quad x = -9 \]

### Step 2: Solve the second equation
The second equation is:
\[ II: y^2 - 18y + 81 = 0 \]
This is a quadratic equation in the standard form \( ay^2 + by + c = 0 \). We can factor this equation. We need two numbers that multiply to \( 81 \) (the constant term) and add up to \( -18 \) (the coefficient of \( y \)).

The numbers are \( -9 \) and \( -9 \):
\[ (y - 9)(y - 9) = 0 \]
This simplifies to:
\[ (y - 9)^2 = 0 \]
Taking the square root of both sides gives:
\[ y - 9 = 0 \]
Thus, we find:
\[ y = 9 \]

### Step 3: Summarize the results
From the first equation, we found:
\[ x = 9 \quad \text{or} \quad x = -9 \]
From the second equation, we found:
\[ y = 9 \]

### Step 4: Determine the relationship between \( x \) and \( y \)
Now we compare the values of \( x \) and \( y \):
- If \( x = 9 \), then \( y = 9 \) (they are equal).
- If \( x = -9 \), then \( y = 9 \) (in this case, \( x < y \)).

### Conclusion
The relationship between \( x \) and \( y \) can be summarized as:
- \( x \) can be either \( 9 \) or \( -9 \).
- \( y \) is always \( 9 \).

Thus, we can conclude:
- If \( x = 9 \), then \( x = y \).
- If \( x = -9 \), then \( x < y \).

The answer to the question is:
- \( x \leq y \)