I.` 3x^(2) - 22x + 7 = 0`
II. ` y^(2) - 20y + 91 = 0`

Options

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

Explanation

To solve the given equations step by step, we will first solve for \( x \) in the equation \( 3x^2 - 22x + 7 = 0 \) and then for \( y \) in the equation \( y^2 - 20y + 91 = 0 \).

### Step 1: Solve for \( x \)

We start with the equation:
\[ 3x^2 - 22x + 7 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \( 3 \times 7 = 21 \) and add up to \( -22 \). The numbers that satisfy this condition are \( -21 \) and \( -1 \).

Now, we can rewrite the equation:
\[ 3x^2 - 21x - x + 7 = 0 \]

Next, we group the terms:
\[ (3x^2 - 21x) + (-x + 7) = 0 \]

Factoring out the common terms:
\[ 3x(x - 7) - 1(x - 7) = 0 \]

Now we can factor by grouping:
\[ (3x - 1)(x - 7) = 0 \]

Setting each factor to zero gives us the solutions for \( x \):
1. \( 3x - 1 = 0 \) → \( x = \frac{1}{3} \)
2. \( x - 7 = 0 \) → \( x = 7 \)

Thus, the values of \( x \) are:
\[ x = \frac{1}{3} \quad \text{and} \quad x = 7 \]

### Step 2: Solve for \( y \)

Now we solve the equation:
\[ y^2 - 20y + 91 = 0 \]

We need to find two numbers that multiply to \( 91 \) and add up to \( -20 \). The numbers that satisfy this condition are \( -13 \) and \( -7 \).

Rewriting the equation:
\[ y^2 - 13y - 7y + 91 = 0 \]

Grouping the terms:
\[ (y^2 - 13y) + (-7y + 91) = 0 \]

Factoring out the common terms:
\[ y(y - 13) - 7(y - 13) = 0 \]

Factoring by grouping gives us:
\[ (y - 13)(y - 7) = 0 \]

Setting each factor to zero gives us the solutions for \( y \):
1. \( y - 13 = 0 \) → \( y = 13 \)
2. \( y - 7 = 0 \) → \( y = 7 \)

Thus, the values of \( y \) are:
\[ y = 7 \quad \text{and} \quad y = 13 \]

### Step 3: Analyze the Results

Now we have the values:
- \( x = \frac{1}{3}, 7 \)
- \( y = 7, 13 \)

### Step 4: Determine the Relationship Between \( x \) and \( y \)

We can represent these values on a number line:
- \( x = \frac{1}{3} \)
- \( x = 7 \)
- \( y = 7 \)
- \( y = 13 \)

From the number line:
- The largest value is \( 13 \) (which is \( y \)).
- The next largest is \( 7 \) (which is both \( x \) and \( y \)).
- The smallest value is \( \frac{1}{3} \) (which is \( x \)).

Thus, we can conclude that:
\[ y \geq x \]

### Final Conclusion

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