I.` x^(2) - 4x+3=0`
II.` y^(2) - 11y + 18 = 0`

Options

A. If x = y or no relation can be established
B. If`x gt y`
C. If ` x lt y`
D. If`x ge y`

Explanation

To solve the given equations and find the relationship between the values of x and y, we will follow these steps:

### Step 1: Solve the first equation \(x^2 - 4x + 3 = 0\)

To factor the quadratic equation, we look for two numbers that multiply to \(3\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)).

The numbers that satisfy this are \(-3\) and \(-1\).

So, we can factor the equation as:
\[ (x - 3)(x - 1) = 0 \]

Setting each factor to zero gives us:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

Thus, the solutions for \(x\) are \(x = 3\) and \(x = 1\).

### Step 2: Solve the second equation \(y^2 - 11y + 18 = 0\)

Similarly, we look for two numbers that multiply to \(18\) and add up to \(-11\).

The numbers that satisfy this are \(-9\) and \(-2\).

So, we can factor the equation as:
\[ (y - 9)(y - 2) = 0 \]

Setting each factor to zero gives us:
\[ y - 9 = 0 \quad \Rightarrow \quad y = 9 \]
\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]

Thus, the solutions for \(y\) are \(y = 9\) and \(y = 2\).

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

We have the following pairs of solutions:
- For \(x\): \(3\) and \(1\)
- For \(y\): \(9\) and \(2\)

Now we will compare:
1. For \(x = 3\) and \(y = 2\):
\[ 3 > 2 \quad \Rightarrow \quad x > y \]

2. For \(x = 1\) and \(y = 9\):
\[ 1 < 9 \quad \Rightarrow \quad x < y \]

### Conclusion

From the comparisons, we see that:
- In one case, \(x\) is greater than \(y\) (when \(x = 3\) and \(y = 2\)).
- In the other case, \(x\) is less than \(y\) (when \(x = 1\) and \(y = 9\)).

Thus, we cannot establish a consistent relationship between \(x\) and \(y\) based on the given equations.

### Final Answer
Since we cannot establish a consistent relationship, the answer is that there is no definitive relation between \(x\) and \(y\).

---