In the given questions, two equations (I) & (II) are given . You have to solve both the equations and mark the answer accordingly
I. ` x^(2) - 20 x + 91 = 0`
II. ` y^(2) + 16 y + 63 = 0 `

Options

A. ` x lt y`
B. ` x gt y `
C. ` x le y `
D. ` x ge y`

Answer

Let's solve the given equations step by step.

### Given Equations:
1. \( x^2 - 20x + 91 = 0 \) (Equation I)
2. \( y^2 + 16y + 63 = 0 \) (Equation II)

### Step 1: Solve Equation I
We start with the first equation:

\[ x^2 - 20x + 91 = 0 \]

To factor this equation, we need two numbers that multiply to \( 91 \) and add up to \( -20 \). The numbers that satisfy these conditions are \( -13 \) and \( -7 \).

Thus, we can rewrite the equation as:

\[ (x - 13)(x - 7) = 0 \]

Setting each factor to zero gives us:

1. \( x - 13 = 0 \) → \( x = 13 \)
2. \( x - 7 = 0 \) → \( x = 7 \)

So the solutions for \( x \) are \( x = 13 \) and \( x = 7 \).

### Step 2: Solve Equation II
Now, we move on to the second equation:

\[ y^2 + 16y + 63 = 0 \]

We need two numbers that multiply to \( 63 \) and add up to \( 16 \). The numbers that satisfy these conditions are \( 9 \) and \( 7 \).

Thus, we can rewrite the equation as:

\[ (y + 9)(y + 7) = 0 \]

Setting each factor to zero gives us:

1. \( y + 9 = 0 \) → \( y = -9 \)
2. \( y + 7 = 0 \) → \( y = -7 \)

So the solutions for \( y \) are \( y = -9 \) and \( y = -7 \).

### Step 3: Compare the Values of \( x \) and \( y \)
Now we have the following values:

- From Equation I: \( x = 13 \) and \( x = 7 \)
- From Equation II: \( y = -9 \) and \( y = -7 \)

We can compare these values:
- \( 13 > -9 \)
- \( 13 > -7 \)
- \( 7 > -9 \)
- \( 7 > -7 \)

### Conclusion
From the comparisons, we can conclude that both values of \( x \) are greater than both values of \( y \). Therefore, we can say:

\[ x > y \]

### Final Answer
The relation between \( x \) and \( y \) is \( x > y \).

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