` y^(2) -x^(2) = 32, y - x = 2`

Options

A. ` x gt y`
B. ` x lt y`
C. ` x ge y`
D. ` x le y`

Explanation

To solve the equations \( y^2 - x^2 = 32 \) and \( y - x = 2 \), we will follow these steps:

### Step 1: Rewrite the first equation using the difference of squares
The equation \( y^2 - x^2 = 32 \) can be factored using the difference of squares formula:
\[ y^2 - x^2 = (y - x)(y + x) \]
Thus, we can rewrite the equation as:
\[ (y - x)(y + x) = 32 \]

### Step 2: Substitute the value of \( y - x \)
From the second equation, we know:
\[ y - x = 2 \]
Now we can substitute this value into the factored form:
\[ 2(y + x) = 32 \]

### Step 3: Solve for \( y + x \)
To find \( y + x \), we can divide both sides of the equation by 2:
\[ y + x = \frac{32}{2} = 16 \]

### Step 4: Set up a system of equations
Now we have a system of two equations:
1. \( y - x = 2 \)
2. \( y + x = 16 \)

### Step 5: Add the two equations
Adding these two equations will help us eliminate \( x \):
\[ (y - x) + (y + x) = 2 + 16 \]
This simplifies to:
\[ 2y = 18 \]

### Step 6: Solve for \( y \)
Now, divide both sides by 2:
\[ y = \frac{18}{2} = 9 \]

### Step 7: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). We can use \( y - x = 2 \):
\[ 9 - x = 2 \]
Solving for \( x \):
\[ -x = 2 - 9 \implies -x = -7 \implies x = 7 \]

### Step 8: Conclusion
We have found \( x = 7 \) and \( y = 9 \). Now we can state the relationship between \( x \) and \( y \):
\[ y > x \quad \text{or} \quad x < y \]