Directions: Solve both equations and determine the relationship between x and y.

$\begin{array}{l}\text{I.}{x}^2-6x+5=0\\ \text{II.}{y}^2-8y+7=0\end{array}$

The correct option is If x = y or no relation can be established.

To find the relationship between $x$ and $y$, we solve both quadratic equations step-by-step.

Step 1: Solve Equation I for $x$

$x^2-6x+5=0$

Splitting the middle term to factorize the equation:

$x^2-5x-x+5=0$

$x(x-5)-1(x-5)=0$

$(x-1)(x-5)=0$

This gives the roots for $x$:

$x=1$ or $x=5$

Step 2: Solve Equation II for $y$

$y^2-8y+7=0$

Splitting the middle term to factorize the equation:

$y^2-7y-y+7=0$

$y(y-7)-1(y-7)=0$

$(y-1)(y-7)=0$

This gives the roots for $y$:

$y=1$ or $y=7$

Step 3: Compare the values of $x$ and $y$

We compare each calculated value of $x$ with each value of $y$:

1. When $x=1$ and $y=1$: $x=y$
2. When $x=1$ and $y=7$: $x<y$
3. When $x=5$ and $y=1$: $x>y$
4. When $x=5$ and $y=7$: $x<y$

Since we obtain both $x>y$ and $x<y$, no single relationship holds true for all possible root combinations.

Thus, no relation can be established between $x$ and $y$.