`x^(2)-32=112, y- sqrt169= 0`

Options

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

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve for \( x \)
The first equation is:
\[ x^2 - 32 = 112 \]
To isolate \( x^2 \), we add 32 to both sides:
\[ x^2 = 112 + 32 \]
\[ x^2 = 144 \]

### Step 2: Find the values of \( x \)
Next, we take the square root of both sides:
\[ x = \pm \sqrt{144} \]
Calculating the square root:
\[ x = \pm 12 \]
So, the possible values for \( x \) are \( 12 \) and \( -12 \).

### Step 3: Solve for \( y \)
Now, we move to the second equation:
\[ y - \sqrt{169} = 0 \]
To find \( y \), we add \( \sqrt{169} \) to both sides:
\[ y = \sqrt{169} \]
Calculating the square root:
\[ y = 13 \]

### Step 4: Compare \( x \) and \( y \)
Now we have:
- \( x = 12 \) or \( x = -12 \)
- \( y = 13 \)

We need to compare the values of \( x \) and \( y \):
1. If \( x = 12 \):
- \( 12 < 13 \) (so \( x < y \))
2. If \( x = -12 \):
- \( -12 < 13 \) (so \( x < y \))

In both cases, \( x < y \).

### Conclusion
Since \( x \) can either be \( 12 \) or \( -12 \) and in both scenarios \( x < y \), we conclude that:
\[ \text{The relationship between } x \text{ and } y \text{ is } x < y. \]