Analyze the two equations labeled (I) and (II) to establish the mathematical relationship between x and y. Solve both equations to find your answer.

$\text{I.}4x^2+3x-10=0$

$\text{II.}{y}^2+10y-119=0$

Correct Answer: If x is equal to y, or if no definite relation can be determined between x and y

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

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

The first equation is:

$4x^2+3x-10=0$

We can solve this quadratic equation by splitting the middle term. We look for two numbers that multiply to give $4\times (-10)=-40$ and add up to $3$. These numbers are $8$ and $-5$.

$4x^2+8x-5x-10=0$

Factor by grouping:

$4x(x+2)-5(x+2)=0$

$(4x-5)(x+2)=0$

Setting each factor to zero gives:

$4x-5=0\Rightarrow x=\frac{5}{4}=1.25$

$x+2=0\Rightarrow x=-2$

Thus, the roots for $x$ are $x_1=1.25$ and $x_2=-2$.

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

The second equation is:

$y^2+10y-119=0$

We look for two numbers that multiply to give $-119$ and add up to $10$. These numbers are $17$ and $-7$.

$y^2+17y-7y-119=0$

Factor by grouping:

$y(y+17)-7(y+17)=0$

$(y-7)(y+17)=0$

Setting each factor to zero gives:

$y-7=0\Rightarrow y=7$

$y+17=0\Rightarrow y=-17$

Thus, the roots for $y$ are $y_1=7$ and $y_2=-17$.

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

Now we compare each value of $x$ with each value of $y$:

1. Comparing $x=1.25$ and $y=7$: Here, $x<y$.

2. Comparing $x=1.25$ and $y=-17$: Here, $x>y$.

3. Comparing $x=-2$ and $y=7$: Here, $x<y$.

4. Comparing $x=-2$ and $y=-17$: Here, $x>y$.

Since we obtain both $x<y$ and $x>y$ depending on which roots are selected, no single, consistent relationship can be established between $x$ and $y$.

Therefore, no definite relation can be determined between x and y.