In the following questions two equations I. and II. are given. You have to solve both the equations and mark the appropriate options
I. ` x^(2) - 16 x + 63 = 0`
II. `y^(2) - 12 y + 35 = 0 `

Options

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

Answer

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

### Step 1: Solve the first equation \( I: x^2 - 16x + 63 = 0 \)

1. **Factor the quadratic equation**:
We need to express the quadratic in a factored form. We look for two numbers that multiply to \( 63 \) (the constant term) and add up to \( -16 \) (the coefficient of \( x \)).
- The numbers are \( -9 \) and \( -7 \).
- Thus, we can write:
\[ x^2 - 16x + 63 = (x - 9)(x - 7) = 0 \]

2. **Set each factor to zero**:
- \( x - 9 = 0 \) or \( x - 7 = 0 \)

3. **Solve for \( x \)**:
- From \( x - 9 = 0 \), we get \( x = 9 \).
- From \( x - 7 = 0 \), we get \( x = 7 \).

### Step 2: Solve the second equation \( II: y^2 - 12y + 35 = 0 \)

1. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 35 \) and add up to \( -12 \).
- The numbers are \( -7 \) and \( -5 \).
- Thus, we can write:
\[ y^2 - 12y + 35 = (y - 7)(y - 5) = 0 \]

2. **Set each factor to zero**:
- \( y - 7 = 0 \) or \( y - 5 = 0 \)

3. **Solve for \( y \)**:
- From \( y - 7 = 0 \), we get \( y = 7 \).
- From \( y - 5 = 0 \), we get \( y = 5 \).

### Step 3: Compare the values of \( x \) and \( y \)

We have the following values:
- From the first equation, \( x = 9 \) or \( x = 7 \).
- From the second equation, \( y = 7 \) or \( y = 5 \).

Now we will compare the values:
1. When \( x = 9 \):
- \( y = 5 \) → \( x > y \)
- \( y = 7 \) → \( x > y \)

2. When \( x = 7 \):
- \( y = 5 \) → \( x > y \)
- \( y = 7 \) → \( x = y \)

### Conclusion
From the comparisons, we can conclude that:
- In all cases, \( x \) is either greater than or equal to \( y \).

Thus, the final answer is:
\[ x \geq y \]