In each of these questions, two equations (I) and (II) are given. Solve the equations and mark the correct option
I. ` x^(2) - 5 x + 6 = 0 `
II. ` y^(2) + 7 y + 6 = 0 `

Options

A. ` x gt y `
B. ` x ge y `
C. ` x lt y `
D. ` x ge y `

Answer

To solve the given equations and find the relationship between \( x \) and \( y \), we will proceed step by step.

### Step 1: Solve the first equation \( x^2 - 5x + 6 = 0 \)

We will factor the quadratic equation:

1. Identify two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of \( x \)). The numbers are \( -2 \) and \( -3 \).
2. Rewrite the equation as:
\[ (x - 2)(x - 3) = 0 \]
3. Set each factor to zero:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

Thus, the solutions for \( x \) are \( x = 2 \) and \( x = 3 \).

### Step 2: Solve the second equation \( y^2 + 7y + 6 = 0 \)

We will also factor this quadratic equation:

1. Identify two numbers that multiply to \( 6 \) and add up to \( 7 \). The numbers are \( 1 \) and \( 6 \).
2. Rewrite the equation as:
\[ (y + 1)(y + 6) = 0 \]
3. Set each factor to zero:
\[ y + 1 = 0 \quad \Rightarrow \quad y = -1 \]
\[ y + 6 = 0 \quad \Rightarrow \quad y = -6 \]

Thus, the solutions for \( y \) are \( y = -1 \) and \( y = -6 \).

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

Now we have the values:
- \( x = 2 \) and \( x = 3 \)
- \( y = -1 \) and \( y = -6 \)

We will compare each value of \( x \) with each value of \( y \):

1. For \( x = 2 \):
- Compare with \( y = -1 \): \( 2 > -1 \)
- Compare with \( y = -6 \): \( 2 > -6 \)

2. For \( x = 3 \):
- Compare with \( y = -1 \): \( 3 > -1 \)
- Compare with \( y = -6 \): \( 3 > -6 \)

### Conclusion

In all cases, we find that \( x > y \). Therefore, the final relationship is:
\[ x > y \]