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) - 14x + 13 = 0`
II. `y^(2) - 12 y + 11 = 0`

Options

A. `y gt x`
B. `x gt y`
C. `x ge y`
D. x = y or no relation can be established

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:
\[ x^2 - 14x + 13 = 0 \]
We can factor this quadratic equation. We need two numbers that multiply to \( 13 \) and add up to \( -14 \). These numbers are \( -1 \) and \( -13 \).

So we can rewrite the equation as:
\[ (x - 1)(x - 13) = 0 \]
Setting each factor to zero gives us:
\[ x - 1 = 0 \quad \text{or} \quad x - 13 = 0 \]
Thus, the solutions for \( x \) are:
\[ x = 1 \quad \text{or} \quad x = 13 \]

### Step 2: Solve the second equation for \( y \)
The second equation is:
\[ y^2 - 12y + 11 = 0 \]
Similarly, we need two numbers that multiply to \( 11 \) and add up to \( -12 \). These numbers are \( -1 \) and \( -11 \).

So we can rewrite the equation as:
\[ (y - 1)(y - 11) = 0 \]
Setting each factor to zero gives us:
\[ y - 1 = 0 \quad \text{or} \quad y - 11 = 0 \]
Thus, the solutions for \( y \) are:
\[ y = 1 \quad \text{or} \quad y = 11 \]

### Step 3: Determine the relationship between \( x \) and \( y \)
Now we have the possible values for \( x \) and \( y \):
- \( x = 1 \) or \( x = 13 \)
- \( y = 1 \) or \( y = 11 \)

Now we can analyze the relationships:
1. If \( x = 1 \) and \( y = 1 \), then \( x = y \).
2. If \( x = 1 \) and \( y = 11 \), then \( x < y \).
3. If \( x = 13 \) and \( y = 1 \), then \( x > y \).
4. If \( x = 13 \) and \( y = 11 \), then \( x > y \).

From this analysis, we can conclude that:
- \( x \) can be equal to \( y \) (when both are 1).
- \( x \) can be greater than \( y \) (when \( x = 13 \) and \( y = 1 \) or \( y = 11 \)).
- Therefore, the relationship can be summarized as \( x \geq y \).

### Final Conclusion
The relationship between \( x \) and \( y \) is:
\[ x \geq y \]