In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct options :
I. ` x^(2) - 31 x + 238 = 0`
II. ` y^(2) - 37 y + 342 = 0 `

Options

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

Answer

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

### Step 1: Solve the first equation
The first equation is:
\[ x^2 - 31x + 238 = 0 \]

We will factor this quadratic equation. We need to find two numbers that multiply to \( 238 \) and add up to \( -31 \).

After checking the factors of \( 238 \), we find:
- \( -14 \) and \( -17 \)

Thus, we can rewrite the equation as:
\[ x^2 - 14x - 17x + 238 = 0 \]
\[ (x - 14)(x - 17) = 0 \]

Setting each factor to zero gives us:
1. \( x - 14 = 0 \) → \( x = 14 \)
2. \( x - 17 = 0 \) → \( x = 17 \)

### Step 2: Solve the second equation
The second equation is:
\[ y^2 - 37y + 342 = 0 \]

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 342 \) and add up to \( -37 \).

After checking the factors of \( 342 \), we find:
- \( -9 \) and \( -38 \)

Thus, we can rewrite the equation as:
\[ y^2 - 9y - 38y + 342 = 0 \]
\[ (y - 9)(y - 38) = 0 \]

Setting each factor to zero gives us:
1. \( y - 9 = 0 \) → \( y = 9 \)
2. \( y - 38 = 0 \) → \( y = 38 \)

### Step 3: Compare the values of x and y
Now we have the values:
- For \( x \): \( 14 \) and \( 17 \)
- For \( y \): \( 9 \) and \( 38 \)

We can compare these values:
- The smallest \( x \) is \( 14 \) and the largest \( y \) is \( 38 \).
- Therefore, \( 14 < 38 \) and \( 17 > 9 \).

### Step 4: Determine the relationship between x and y
From the comparisons:
- The smallest value of \( x \) (which is \( 14 \)) is less than the smallest value of \( y \) (which is \( 9 \)).
- The largest value of \( x \) (which is \( 17 \)) is greater than the smallest value of \( y \) (which is \( 9 \)).

Thus, we can conclude:
- \( x < y \)

### Final Answer
The correct relationship is:
\[ x < y \]