In each question two equations numbered I and II are given . you have to solve both the equations and mark the appropriate option
I `x^2-5x+6=0`
II `y^2-9y+20=0`

Options

A. `xgty`
B. `xlty`
C. The relation between x and y cannot be established.
D. `xgey`

Answer

To solve the given equations step by step, we will first solve each equation separately and then analyze the results.

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

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of \( x \)).

The factors of \( 6 \) that satisfy these conditions are \( -2 \) and \( -3 \).

So, we can factor the equation as:
\[ (x - 2)(x - 3) = 0 \]

### Step 2: Find the values of \( x \)

Setting each factor equal to zero gives us:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x - 3 = 0 \) → \( x = 3 \)

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

### Step 3: Solve the second equation \( y^2 - 9y + 20 = 0 \)

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 20 \) and add up to \( -9 \).

The factors of \( 20 \) that satisfy these conditions are \( -4 \) and \( -5 \).

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

### Step 4: Find the values of \( y \)

Setting each factor equal to zero gives us:
1. \( y - 4 = 0 \) → \( y = 4 \)
2. \( y - 5 = 0 \) → \( y = 5 \)

Thus, the solutions for \( y \) are:
\[ y = 4 \quad \text{and} \quad y = 5 \]

### Step 5: Compare the values of \( x \) and \( y \)

Now we have the values:
- For \( x \): \( 2, 3 \)
- For \( y \): \( 4, 5 \)

We can compare these values:
1. If \( x = 2 \), then \( y = 4 \) → \( x < y \)
2. If \( x = 2 \), then \( y = 5 \) → \( x < y \)
3. If \( x = 3 \), then \( y = 4 \) → \( x < y \)
4. If \( x = 3 \), then \( y = 5 \) → \( x < y \)

In all cases, \( x \) is less than \( y \).

### Conclusion

The relation between \( x \) and \( y \) is:
\[ x < y \]

### Final Answer

The correct option is the one that states \( x < y \).

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