In the following question two equations are given in variables x and y. you have to solve these equations and determine the relation between x and y.
I. `x^(2) - 15x + 56 = 0`
II. `y^(2) - 17y + 72 = 0`

Options

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

Answer

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

### Step 1: Solve the first equation for \( x \)
The first equation is:
\[ x^2 - 15x + 56 = 0 \]
We need to factor this quadratic equation. We look for two numbers that multiply to \( 56 \) and add up to \( 15 \).

The factors of \( 56 \) that satisfy this condition are \( 7 \) and \( 8 \). Thus, we can rewrite the equation as:
\[ (x - 7)(x - 8) = 0 \]

Setting each factor to zero gives us:
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]
\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \]

So, the possible values of \( x \) are \( 7 \) and \( 8 \).

### Step 2: Solve the second equation for \( y \)
The second equation is:
\[ y^2 - 17y + 72 = 0 \]
Similarly, we need to factor this quadratic equation. We look for two numbers that multiply to \( 72 \) and add up to \( 17 \).

The factors of \( 72 \) that satisfy this condition are \( 8 \) and \( 9 \). Thus, we can rewrite the equation as:
\[ (y - 8)(y - 9) = 0 \]

Setting each factor to zero gives us:
\[ y - 8 = 0 \quad \Rightarrow \quad y = 8 \]
\[ y - 9 = 0 \quad \Rightarrow \quad y = 9 \]

So, the possible values of \( y \) are \( 8 \) and \( 9 \).

### Step 3: Determine the relationship between \( x \) and \( y \)
Now we have the possible values:
- For \( x \): \( 7, 8 \)
- For \( y \): \( 8, 9 \)

We can analyze the relationships:
1. If \( x = 7 \), then \( y \) can be \( 8 \) or \( 9 \). In both cases, \( y > x \).
2. If \( x = 8 \), then \( y \) can be \( 8 \) or \( 9 \). Here, \( y \) can be equal to \( x \) (when \( y = 8 \)) or \( y > x \) (when \( y = 9 \)).

Thus, we conclude that:
\[ x \leq y \]

### Final Answer
The relationship between \( x \) and \( y \) is:
\[ x \leq y \]