In each of the following questions, two equations (I) and (II) are given . Solve the equations and mark the correct option :
I. ` x^(2) - 26 x + 168 = 0 `
II. ` y^(2) - 29 y + 208 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y `
C. if ` x lt y `
D. If `x = y` or no relation can be established between `x` and `y`

Answer

To solve the given equations and determine the relationship between the values of \( x \) and \( y \), we will follow these steps:

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 26x + 168 = 0 \]

To solve this quadratic equation, we will factor it. We need to find two numbers that multiply to \( 168 \) and add up to \( 26 \).

The factors of \( 168 \) that satisfy these conditions are \( 14 \) and \( 12 \) because:
\[ 14 + 12 = 26 \]
\[ 14 \times 12 = 168 \]

Now we can rewrite the equation as:
\[ (x - 14)(x - 12) = 0 \]

Setting each factor to zero gives us:
1. \( x - 14 = 0 \) → \( x = 14 \)
2. \( x - 12 = 0 \) → \( x = 12 \)

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 29y + 208 = 0 \]

Again, we will factor this quadratic equation. We need to find two numbers that multiply to \( 208 \) and add up to \( 29 \).

The factors of \( 208 \) that satisfy these conditions are \( 13 \) and \( 16 \) because:
\[ 13 + 16 = 29 \]
\[ 13 \times 16 = 208 \]

Now we can rewrite the equation as:
\[ (y - 16)(y - 13) = 0 \]

Setting each factor to zero gives us:
1. \( y - 16 = 0 \) → \( y = 16 \)
2. \( y - 13 = 0 \) → \( y = 13 \)

### Step 3: Compare Values of \( x \) and \( y \)
Now we have the following values:
- From Equation I: \( x = 14 \) or \( x = 12 \)
- From Equation II: \( y = 16 \) or \( y = 13 \)

We will compare these values:
1. For \( x = 14 \):
- \( 14 < 16 \) (True)
- \( 14 > 13 \) (True)

2. For \( x = 12 \):
- \( 12 < 16 \) (True)
- \( 12 < 13 \) (True)

### Conclusion
From the comparisons:
- \( x = 14 \) is less than \( y = 16 \) and greater than \( y = 13 \).
- \( x = 12 \) is less than both \( y = 16 \) and \( y = 13 \).

Since there is no consistent relationship established between \( x \) and \( y \) across all values, we conclude that:
**No relation can be established between \( x \) and \( y \).**

### Final Answer
The correct option is: **No relation can be established between \( x \) and \( y \)**.

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