Two equations (I) and (II) are given in each question. On the basis of these equations you have to decide the relation between x and y and give answer
I. ` 7 x^(2) + 16 x - 15 = 0 `
II. ` y^(2) - 6y - 7 = 0 `

Options

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

Answer

To solve the given problem, we need to find the values of \( x \) and \( y \) from the provided equations and then determine the relationship between them.

### Step 1: Solve Equation I
The first equation is:
\[ 7x^2 + 16x - 15 = 0 \]

To solve this quadratic equation, we will use the factorization method. We need to find two numbers that multiply to \( 7 \times (-15) = -105 \) and add to \( 16 \).

The numbers that satisfy these conditions are \( 21 \) and \( -5 \) because:
\[ 21 \times (-5) = -105 \]
\[ 21 + (-5) = 16 \]

Now we can rewrite the middle term:
\[ 7x^2 + 21x - 5x - 15 = 0 \]

Next, we group the terms:
\[ (7x^2 + 21x) + (-5x - 15) = 0 \]

Factoring each group gives:
\[ 7x(x + 3) - 5(x + 3) = 0 \]

Now we can factor out \( (x + 3) \):
\[ (7x - 5)(x + 3) = 0 \]

Setting each factor to zero gives us:
1. \( 7x - 5 = 0 \) → \( x = \frac{5}{7} \)
2. \( x + 3 = 0 \) → \( x = -3 \)

So, the solutions for \( x \) are:
\[ x = -3 \quad \text{and} \quad x = \frac{5}{7} \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 6y - 7 = 0 \]

Again, we will use the factorization method. We need two numbers that multiply to \( -7 \) and add to \( -6 \).

The numbers that satisfy these conditions are \( -7 \) and \( 1 \) because:
\[ -7 \times 1 = -7 \]
\[ -7 + 1 = -6 \]

Now we can rewrite the equation:
\[ (y - 7)(y + 1) = 0 \]

Setting each factor to zero gives us:
1. \( y - 7 = 0 \) → \( y = 7 \)
2. \( y + 1 = 0 \) → \( y = -1 \)

So, the solutions for \( y \) are:
\[ y = 7 \quad \text{and} \quad y = -1 \]

### Step 3: Determine the Relationship between \( x \) and \( y \)
Now we have the values:
- For \( x \): \( -3 \) and \( \frac{5}{7} \)
- For \( y \): \( 7 \) and \( -1 \)

Now we can compare:
1. For \( x = -3 \):
- \( -3 < 7 \) (true)
- \( -3 > -1 \) (false)

2. For \( x = \frac{5}{7} \):
- \( \frac{5}{7} < 7 \) (true)
- \( \frac{5}{7} > -1 \) (true)

### Conclusion
From the comparisons:
- \( x = -3 \) is less than \( y = 7 \) but greater than \( y = -1 \).
- \( x = \frac{5}{7} \) is less than \( y = 7 \) and greater than \( y = -1 \).

Thus, we can conclude that:
- \( x < y \) when \( y = 7 \)
- \( x > y \) when \( y = -1 \)

Since we have two different scenarios, we cannot establish a definitive relationship between \( x \) and \( y \).

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