In each of the following questions, two equations (I)and (II) are given. Solve the equations and mark the correct options
I. ` x^(2) + 3 x = 0 `
II. ` x^(2) + y = 10`

Options

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

Answer

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

### Step 1: Solve the first equation
The first equation is:
\[ x^2 + 3x = 0 \]
We can factor this equation:
\[ x(x + 3) = 0 \]
Setting each factor to zero gives us:
1. \( x = 0 \)
2. \( x + 3 = 0 \) which simplifies to \( x = -3 \)

Thus, the solutions for \( x \) are:
\[ x = 0 \quad \text{and} \quad x = -3 \]

### Step 2: Solve the second equation for \( y \)
The second equation is:
\[ x^2 + y = 10 \]
We will substitute the values of \( x \) we found in Step 1 into this equation.

#### Case 1: When \( x = 0 \)
Substituting \( x = 0 \):
\[ 0^2 + y = 10 \implies y = 10 \]

#### Case 2: When \( x = -3 \)
Substituting \( x = -3 \):
\[ (-3)^2 + y = 10 \implies 9 + y = 10 \implies y = 1 \]

### Step 3: List the pairs of \( (x, y) \)
From the calculations above, we have the following pairs:
1. \( (0, 10) \)
2. \( (-3, 1) \)

### Step 4: Determine the relationship between \( x \) and \( y \)
Now we will compare the values of \( x \) and \( y \) for each pair:

- For the pair \( (0, 10) \):
\[ 0 < 10 \quad \text{(True)} \]

- For the pair \( (-3, 1) \):
\[ -3 < 1 \quad \text{(True)} \]

### Conclusion
In both cases, we find that \( x < y \). Therefore, we can conclude that:
\[ x < y \]

### Final Answer
The correct relationship is:
\[ \text{Option C: } x < y \]