In the following questions, two equations numberded I and II are given. You have to solve both the equations and give answer If:
I. `x^2 - 13x+40=0` II. `y^2-16y+63=0`

Options

A. `x lt y`
B. `x ge y`
C. `x gt y`
D. x= y or the relationship cannot be established

Answer

To solve the given equations step by step, we will first tackle each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 13x + 40 = 0 \]

To solve this quadratic equation, we will factor it. We need two numbers that multiply to \( 40 \) (the constant term) and add up to \( -13 \) (the coefficient of \( x \)).

The factors of \( 40 \) that satisfy these conditions are \( -8 \) and \( -5 \):
\[ (x - 8)(x - 5) = 0 \]

Now, we set each factor to zero:
1. \( x - 8 = 0 \) → \( x = 8 \)
2. \( x - 5 = 0 \) → \( x = 5 \)

So, the solutions for \( x \) are:
\[ x = 5 \quad \text{and} \quad x = 8 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 16y + 63 = 0 \]

Similarly, we will factor this equation. We need two numbers that multiply to \( 63 \) and add up to \( -16 \).

The factors of \( 63 \) that satisfy these conditions are \( -9 \) and \( -7 \):
\[ (y - 9)(y - 7) = 0 \]

Now, we set each factor to zero:
1. \( y - 9 = 0 \) → \( y = 9 \)
2. \( y - 7 = 0 \) → \( y = 7 \)

So, the solutions for \( y \) are:
\[ y = 9 \quad \text{and} \quad y = 7 \]

### Step 3: Compare Values of x and y
Now we have the values:
- For \( x \): \( 5 \) and \( 8 \)
- For \( y \): \( 9 \) and \( 7 \)

We will compare the values of \( x \) and \( y \):
1. If \( x = 5 \), then \( y = 7 \) → \( x < y \)
2. If \( x = 5 \), then \( y = 9 \) → \( x < y \)
3. If \( x = 8 \), then \( y = 7 \) → \( x > y \)
4. If \( x = 8 \), then \( y = 9 \) → \( x < y \)

From this, we can see that:
- When \( x = 5 \), \( y \) can be either \( 7 \) or \( 9 \) (in both cases \( x < y \)).
- When \( x = 8 \), \( y \) can be \( 7 \) (in this case \( x > y \)) or \( 9 \) (in this case \( x < y \)).

### Conclusion
Since there are instances where \( x < y \) and instances where \( x > y \), we cannot establish a consistent relationship between \( x \) and \( y \).

Thus, the answer is:
**The relationship cannot be established.**