I. ` x^(2) - 10x+24 = 0`
II. `y^(2) - 9y + 20 = 0`

Options

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

Explanation

To solve the given equations step by step, we will first solve for \( x \) in the first equation and then for \( y \) in the second equation. Finally, we will analyze the relationship between \( x \) and \( y \).

### Step 1: Solve the first equation \( x^2 - 10x + 24 = 0 \)

1. **Identify the equation**:
\[ x^2 - 10x + 24 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 24 \) (the constant term) and add up to \( -10 \) (the coefficient of \( x \)). The numbers are \( -4 \) and \( -6 \).
\[ (x - 4)(x - 6) = 0 \]

3. **Set each factor to zero**:
\[ x - 4 = 0 \quad \text{or} \quad x - 6 = 0 \]

4. **Solve for \( x \)**:
\[ x = 4 \quad \text{or} \quad x = 6 \]

### Step 2: Solve the second equation \( y^2 - 9y + 20 = 0 \)

1. **Identify the equation**:
\[ y^2 - 9y + 20 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 20 \) and add up to \( -9 \). The numbers are \( -4 \) and \( -5 \).
\[ (y - 4)(y - 5) = 0 \]

3. **Set each factor to zero**:
\[ y - 4 = 0 \quad \text{or} \quad y - 5 = 0 \]

4. **Solve for \( y \)**:
\[ y = 4 \quad \text{or} \quad y = 5 \]

### Step 3: Analyze the relationship between \( x \) and \( y \)

1. **Values of \( x \)**:
- \( x = 4 \)
- \( x = 6 \)

2. **Values of \( y \)**:
- \( y = 4 \)
- \( y = 5 \)

3. **Compare the values**:
- When \( x = 4 \), \( y = 4 \) → \( x = y \)
- When \( x = 6 \), \( y = 5 \) → \( x > y \)

### Conclusion
The relationship between \( x \) and \( y \) can be summarized as follows:
- \( x \) can be equal to \( y \) or greater than \( y \).