In each of these questions, two equations are given . You have to solve both the equations and give answer
I. ` x^(2) + 24 x = - 119`
II. ` 3 y^(2) + 7 = - 10 y`

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, let's start with the first equation.

### Step 1: Solve the first equation \( x^2 + 24x = -119 \)

1. Rearrange the equation to set it to zero:
\[ x^2 + 24x + 119 = 0 \]

2. Now, we need to factor the quadratic equation. We look for two numbers that multiply to \( 119 \) and add up to \( 24 \). The numbers are \( 17 \) and \( 7 \):
\[ x^2 + 17x + 7x + 119 = 0 \]

3. Group the terms:
\[ (x^2 + 17x) + (7x + 119) = 0 \]

4. Factor by grouping:
\[ x(x + 17) + 7(x + 17) = 0 \]
\[ (x + 17)(x + 7) = 0 \]

5. Set each factor to zero:
\[ x + 17 = 0 \quad \text{or} \quad x + 7 = 0 \]

6. Solve for \( x \):
\[ x = -17 \quad \text{or} \quad x = -7 \]

### Step 2: Solve the second equation \( 3y^2 + 7 = -10y \)

1. Rearrange the equation to set it to zero:
\[ 3y^2 + 10y + 7 = 0 \]

2. Now, we need to factor this quadratic equation. We look for two numbers that multiply to \( 3 \times 7 = 21 \) and add up to \( 10 \). The numbers are \( 3 \) and \( 7 \):
\[ 3y^2 + 3y + 7y + 7 = 0 \]

3. Group the terms:
\[ (3y^2 + 3y) + (7y + 7) = 0 \]

4. Factor by grouping:
\[ 3y(y + 1) + 7(y + 1) = 0 \]
\[ (y + 1)(3y + 7) = 0 \]

5. Set each factor to zero:
\[ y + 1 = 0 \quad \text{or} \quad 3y + 7 = 0 \]

6. Solve for \( y \):
\[ y = -1 \quad \text{or} \quad y = -\frac{7}{3} \]

### Summary of Solutions

- The values of \( x \) are \( -17 \) and \( -7 \).
- The values of \( y \) are \( -1 \) and \( -\frac{7}{3} \).

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

- Compare \( x = -17 \) with \( y = -1 \):
\[ -17 < -1 \quad \text{(True)} \]

- Compare \( x = -7 \) with \( y = -1 \):
\[ -7 < -1 \quad \text{(True)} \]

- Compare \( x = -7 \) with \( y = -\frac{7}{3} \):
\[ -7 < -\frac{7}{3} \quad \text{(True)} \]

### Conclusion

In all cases, \( x \) is less than \( y \). Therefore, the relationship between \( x \) and \( y \) is:
\[ x < y \]