In each of these questions, two equation (I) and (II) are given. You have to solve both the equations and give answer.

I: $x^2-24x+143=0$
II: $y^2-13y+22=0$

To determine the correct relationship between $x$ and $y$, we need to solve both given quadratic equations.

Step 1: Solve Equation (I)
The first equation is:

$x^2-24x+143=0$

To solve this quadratic equation, we find two numbers that multiply to $143$ and add up to $-24$.
These two numbers are $-11$ and $-13$.
Thus, we can factor the equation as:

$(x-11)(x-13)=0$

Therefore, the roots for $x$ are:

$x=11$

and

$x=13$

Step 2: Solve Equation (II)
The second equation is:

$y^2-13y+22=0$

Similarly, we find two numbers that multiply to $22$ and add up to $-13$.
These two numbers are $-11$ and $-2$.
Thus, we can factor the equation as:

$(y-11)(y-2)=0$

Therefore, the roots for $y$ are:

$y=11$

and

$y=2$

Step 3: Compare the values of $x$ and $y$
Let us compare each value of $x$ with the values of $y$:
- For $x=13$, comparing with $y=11$ and $y=2$, we have $x>y$.
- For $x=11$, comparing with $y=11$, we have $x=y$; comparing with $y=2$, we have $x>y$.
Combining these observations, we conclude that $x$ is always greater than or equal to $y$.
Thus, the relationship is $x\ge y$.