In this question two equations numbered I and II are given. You have to solve both the equations and mark the appropriate option. Give answer.
I `x^2-22x+117=0`
II`y^2+4y-96=0`

Options

A. `xgty`
B. `xlty`
C. `xley`
D. `xgey`

Answer

To solve the given equations step by step, we will start with the first equation and then move on to the second one.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 22x + 117 = 0 \]

To solve this quadratic equation, we can use the factorization method. We need to find two numbers that multiply to \(117\) (the constant term) and add up to \(-22\) (the coefficient of \(x\)).

1. We can rewrite the middle term:
\[ x^2 - 13x - 9x + 117 = 0 \]

2. Now, group the terms:
\[ (x^2 - 13x) + (-9x + 117) = 0 \]

3. Factor by grouping:
\[ x(x - 13) - 9(x - 13) = 0 \]

4. Factor out the common term:
\[ (x - 13)(x - 9) = 0 \]

5. Set each factor to zero:
\[ x - 13 = 0 \quad \text{or} \quad x - 9 = 0 \]

6. Solve for \(x\):
\[ x = 13 \quad \text{or} \quad x = 9 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 + 4y - 96 = 0 \]

Again, we will use the factorization method. We need to find two numbers that multiply to \(-96\) and add up to \(4\).

1. Rewrite the middle term:
\[ y^2 + 12y - 8y - 96 = 0 \]

2. Group the terms:
\[ (y^2 + 12y) + (-8y - 96) = 0 \]

3. Factor by grouping:
\[ y(y + 12) - 8(y + 12) = 0 \]

4. Factor out the common term:
\[ (y + 12)(y - 8) = 0 \]

5. Set each factor to zero:
\[ y + 12 = 0 \quad \text{or} \quad y - 8 = 0 \]

6. Solve for \(y\):
\[ y = -12 \quad \text{or} \quad y = 8 \]

### Summary of Solutions
From the first equation, we have:
- \(x = 13\) or \(x = 9\)

From the second equation, we have:
- \(y = -12\) or \(y = 8\)

### Step 3: Compare the Values
Now we need to compare the values of \(x\) and \(y\):
- If \(x = 13\), then \(x > y\) for both \(y = -12\) and \(y = 8\).
- If \(x = 9\), then \(x > y\) for \(y = -12\) and \(x < y\) for \(y = 8\).

Thus, in both cases, we can conclude that:
- \(x\) is greater than \(y\) when \(y = -12\) and \(x\) is greater than \(y\) when \(y = 8\).

### Final Answer
The correct answer is:
\[ x > y \]