I.` x^(2) - 7x + 10 = 0" "II. Y^(2) - 12y + 35 = 0`

Options

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

Explanation

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

### Step 1: Solve the first equation \( x^2 - 7x + 10 = 0 \)

We will factor the quadratic equation:

1. Identify two numbers that multiply to \( 10 \) (the constant term) and add to \( -7 \) (the coefficient of \( x \)).
- The numbers are \( -2 \) and \( -5 \).

2. Rewrite the equation as:
\[ (x - 2)(x - 5) = 0 \]

3. Set each factor equal to zero:
\[ x - 2 = 0 \quad \text{or} \quad x - 5 = 0 \]

4. Solve for \( x \):
\[ x = 2 \quad \text{or} \quad x = 5 \]

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

Similarly, we will factor this quadratic equation:

1. Identify two numbers that multiply to \( 35 \) and add to \( -12 \).
- The numbers are \( -7 \) and \( -5 \).

2. Rewrite the equation as:
\[ (y - 7)(y - 5) = 0 \]

3. Set each factor equal to zero:
\[ y - 7 = 0 \quad \text{or} \quad y - 5 = 0 \]

4. Solve for \( y \):
\[ y = 7 \quad \text{or} \quad y = 5 \]

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

Now we have the values:
- For \( x \): \( 2, 5 \)
- For \( y \): \( 5, 7 \)

We can analyze the relationships:
- When \( x = 2 \), \( y \) can be \( 5 \) or \( 7 \) (both are greater than \( 2 \)).
- When \( x = 5 \), \( y \) can be \( 5 \) (equal) or \( 7 \) (greater).

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

### Final Conclusion

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