In the following questions two equations numbered I and II are given. Solve both the equations and give answer
I. `x^2-5x+6=0` II. `3y^2+3y-18=0`

Options

A. `xgey`
B. `xlty`
C. `xley`
D. relationship can't be established

Answer

To solve the given equations step by step, we will start with each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 5x + 6 = 0 \]

This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -5 \), and \( c = 6 \).

#### Step 1.1: Factor the Equation
To factor the equation, we need two numbers that multiply to \( c \) (which is 6) and add up to \( b \) (which is -5). The numbers that satisfy these conditions are -2 and -3.

Thus, we can write:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \]

#### Step 1.2: Set Each Factor to Zero
Now we set each factor equal to zero:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x - 3 = 0 \) → \( x = 3 \)

So, the solutions for \( x \) are:
\[ x = 2 \quad \text{and} \quad x = 3 \]

### Step 2: Solve Equation II
The second equation is:
\[ 3y^2 + 3y - 18 = 0 \]

#### Step 2.1: Simplify the Equation
First, we can simplify the equation by dividing all terms by 3:
\[ y^2 + y - 6 = 0 \]

#### Step 2.2: Factor the Equation
Next, we need to factor this quadratic equation. We are looking for two numbers that multiply to -6 (the constant term) and add to 1 (the coefficient of \( y \)). The numbers that satisfy this are 3 and -2.

Thus, we can write:
\[ y^2 + y - 6 = (y + 3)(y - 2) = 0 \]

#### Step 2.3: Set Each Factor to Zero
Now we set each factor equal to zero:
1. \( y + 3 = 0 \) → \( y = -3 \)
2. \( y - 2 = 0 \) → \( y = 2 \)

So, the solutions for \( y \) are:
\[ y = -3 \quad \text{and} \quad y = 2 \]

### Summary of Solutions
- From Equation I, we have \( x = 2 \) and \( x = 3 \).
- From Equation II, we have \( y = -3 \) and \( y = 2 \).

### Final Comparison
Now we can compare the values of \( x \) and \( y \):
- \( x = 2 \) and \( y = -3 \) → \( x > y \)
- \( x = 3 \) and \( y = -3 \) → \( x > y \)
- \( x = 2 \) and \( y = 2 \) → \( x = y \)

Thus, in all cases, \( x \) is greater than or equal to \( y \).

### Final Answer
The correct answer is:
\[ x \geq y \]