In the following question two equations are given in variables X and Y. You have to solve these equations and determine the relation between X and Y.
A) `x^2 – 7x + 12 = 0`
B ) `y^2 – 9y + 20 = 0`

Options

A. `X gt Y`
B. `X ge Y`
C. `X lt Y`
D. `X le Y`

Explanation

To solve the equations and determine the relationship between \( X \) and \( Y \), we will follow these steps:

### Step 1: Solve the first equation \( x^2 - 7x + 12 = 0 \)

1. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 12 \) (the constant term) and add up to \( -7 \) (the coefficient of \( x \)).
The numbers that satisfy this condition are \( -3 \) and \( -4 \).

Therefore, we can factor the equation as:
\[ (x - 3)(x - 4) = 0 \]

2. **Set each factor to zero**:
\[ x - 3 = 0 \quad \text{or} \quad x - 4 = 0 \]
This gives us:
\[ x = 3 \quad \text{or} \quad x = 4 \]

### Step 2: Solve the second equation \( y^2 - 9y + 20 = 0 \)

1. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 20 \) (the constant term) and add up to \( -9 \) (the coefficient of \( y \)).
The numbers that satisfy this condition are \( -4 \) and \( -5 \).

Therefore, we can factor the equation as:
\[ (y - 4)(y - 5) = 0 \]

2. **Set each factor to zero**:
\[ y - 4 = 0 \quad \text{or} \quad y - 5 = 0 \]
This gives us:
\[ y = 4 \quad \text{or} \quad y = 5 \]

### Step 3: Determine the relationship between \( X \) and \( Y \)

Now we have the possible values for \( X \) and \( Y \):
- \( X \) can be \( 3 \) or \( 4 \)
- \( Y \) can be \( 4 \) or \( 5 \)

We will compare the values:
- If \( X = 3 \), then \( Y \) can be \( 4 \) or \( 5 \) (so \( X < Y \)).
- If \( X = 4 \), then \( Y \) can be \( 4 \) or \( 5 \) (so \( X = Y \) or \( X < Y \)).

### Conclusion

The relationship between \( X \) and \( Y \) can be summarized as:
- \( X \leq Y \)