Given below are two equations in each question, which you have to solve and give answer
I. ` x^(2) - 3 x = 10`
II. ` y^(2) + 7y + 10 = 0`

Options

A. A)If ` x gt y`
B. B)If ` x ge y `
C. C)If ` y gt x`
D. D)If ` y ge x `

Answer

To solve the given equations step by step, let's break down the process for each equation.

### Step 1: Solve the first equation \( x^2 - 3x = 10 \)

1. **Rearrange the equation**:
\[ x^2 - 3x - 10 = 0 \]

2. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(-5\) and \(2\) work because:
\[ -5 \times 2 = -10 \quad \text{and} \quad -5 + 2 = -3 \]
Thus, we can factor the equation as:
\[ (x - 5)(x + 2) = 0 \]

3. **Set each factor to zero**:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Step 2: Solve the second equation \( y^2 + 7y + 10 = 0 \)

1. **Factor the quadratic equation**:
We need to find two numbers that multiply to \(10\) and add to \(7\). The numbers \(5\) and \(2\) work because:
\[ 5 \times 2 = 10 \quad \text{and} \quad 5 + 2 = 7 \]
Thus, we can factor the equation as:
\[ (y + 5)(y + 2) = 0 \]

2. **Set each factor to zero**:
\[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]

### Step 3: Compare the values of \(x\) and \(y\)

Now we have the solutions:
- From the first equation: \( x = 5 \) or \( x = -2 \)
- From the second equation: \( y = -5 \) or \( y = -2 \)

### Step 4: Establish the relationship between \(x\) and \(y\)

1. **Compare \(x = 5\) with \(y = -5\)**:
\[ 5 > -5 \]

2. **Compare \(x = 5\) with \(y = -2\)**:
\[ 5 > -2 \]

3. **Compare \(x = -2\) with \(y = -5\)**:
\[ -2 > -5 \]

4. **Compare \(x = -2\) with \(y = -2\)**:
\[ -2 = -2 \]

### Conclusion

From the comparisons, we can conclude that:
- \(x\) is greater than \(y\) when \(x = 5\).
- \(x\) is greater than or equal to \(y\) when \(x = -2\).

Thus, the final relation is:
\[ x \geq y \]

### Final Answer

The relation between \(x\) and \(y\) is:
\[ \boxed{x \geq y} \]