Determine the relationship between the two variables by solving the given quadratic equations (I) and (II).

$\text{I.}{x}^2-17x+70=0$

$\text{II.}{y}^2-14y+48=0$

Correct Answer: If $x=y$ or no relationship can be established between $x$ and $y$

To determine the relationship between $x$ and $y$, we need to solve both quadratic equations step-by-step and compare their roots.

Step 1: Solve Equation (I) for $x$

The given equation is:

$x^2-17x+70=0$

We need to find two numbers whose product is 70 and sum is -17. These numbers are -10 and -7.
Factoring the equation:

$x^2-10x-7x+70=0$

$x(x-10)-7(x-10)=0$

$(x-10)(x-7)=0$

Setting each factor to zero gives:

$x=10$ or $x=7$

Step 2: Solve Equation (II) for $y$

The given equation is:

$y^2-14y+48=0$

We need to find two numbers whose product is 48 and sum is -14. These numbers are -8 and -6.
Factoring the equation:

$y^2-8y-6y+48=0$

$y(y-8)-6(y-8)=0$

$(y-8)(y-6)=0$

Setting each factor to zero gives:

$y=8$ or $y=6$

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

Now, let's compare each value of $x$ with every value of $y$:

• When $x=7$ and $y=6$, we have $x>y$.
• When $x=7$ and $y=8$, we have $x<y$.
• When $x=10$ and $y=6$, we have $x>y$.
• When $x=10$ and $y=8$, we have $x>y$.

Since we obtain both $x>y$ and $x<y$, no relationship can be established between $x$ and $y$.