I.` (x+3)(y+2) = 12`
II. ` 2xy + 4x + 5y = 11`

Options

A. if ` x gt y`
B. if `x lt y`
C. if `x ge y`
D. if x = y or relation cannot be established between 'x' and 'y'.

Explanation

To solve the given equations step by step, we will follow these procedures:

### Given Equations:
1. \( (x + 3)(y + 2) = 12 \)
2. \( 2xy + 4x + 5y = 11 \)

### Step 1: Expand the First Equation
We start with the first equation and expand it:
\[ (x + 3)(y + 2) = xy + 2x + 3y + 6 = 12 \]
Rearranging gives us:
\[ xy + 2x + 3y + 6 - 12 = 0 \implies xy + 2x + 3y - 6 = 0 \]
This is our first equation in expanded form:
\[ xy + 2x + 3y - 6 = 0 \tag{1} \]

### Step 2: Write the Second Equation
The second equation remains as it is:
\[ 2xy + 4x + 5y = 11 \tag{2} \]

### Step 3: Multiply the First Equation by 2
To facilitate elimination, we multiply the first equation by 2:
\[ 2(xy + 2x + 3y - 6) = 0 \implies 2xy + 4x + 6y - 12 = 0 \]
This gives us:
\[ 2xy + 4x + 6y - 12 = 0 \tag{3} \]

### Step 4: Set Up for Elimination
Now we have:
- Equation (2): \( 2xy + 4x + 5y = 11 \)
- Equation (3): \( 2xy + 4x + 6y - 12 = 0 \)

### Step 5: Subtract Equation (2) from Equation (3)
Now, we subtract Equation (2) from Equation (3):
\[ (2xy + 4x + 6y - 12) - (2xy + 4x + 5y) = 0 - 11 \]
This simplifies to:
\[ 6y - 5y - 12 = -11 \implies y - 12 = -11 \implies y = 1 \]

### Step 6: Substitute \( y \) Back to Find \( x \)
Now that we have \( y = 1 \), we substitute this value back into the first equation to find \( x \):
\[ (x + 3)(1 + 2) = 12 \implies (x + 3)(3) = 12 \]
Dividing both sides by 3 gives:
\[ x + 3 = 4 \implies x = 4 - 3 = 1 \]

### Final Values
Thus, we have:
\[ x = 1, \quad y = 1 \]

### Step 7: Establish the Relation Between \( x \) and \( y \)
Since both \( x \) and \( y \) are equal, we conclude:
\[ x = y \]

### Summary of the Solution
The values of \( x \) and \( y \) are both 1, and the relation established is \( x = y \).

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