In the following questions, two equations numbered I and II are given. You have to solve both the equations and
Give answer If
`x^2-21x+38=0`
`y^2-11y+28=0`

Options

A. `xgty`
B. `xley`
C. `xgey`
D. x=y or the relationship cannot be established

Answer

To solve the given equations step by step, we will start with the first equation and then move on to the second equation.

### Step 1: Solve the first equation \(x^2 - 21x + 38 = 0\)

1. **Identify the coefficients**: The equation is in the standard quadratic form \(ax^2 + bx + c = 0\) where:
- \(a = 1\)
- \(b = -21\)
- \(c = 38\)

2. **Factor the equation**: We need to find two numbers that multiply to \(c\) (which is \(38\)) and add up to \(b\) (which is \(-21\)). The factors of \(38\) that satisfy this condition are \(-19\) and \(-2\).

3. **Rewrite the equation**: We can express the equation as:
\[ (x - 19)(x - 2) = 0 \]

4. **Set each factor to zero**:
- \(x - 19 = 0 \Rightarrow x = 19\)
- \(x - 2 = 0 \Rightarrow x = 2\)

Thus, the solutions for \(x\) are \(x = 19\) and \(x = 2\).

### Step 2: Solve the second equation \(y^2 - 11y + 28 = 0\)

1. **Identify the coefficients**: The equation is also in the standard quadratic form where:
- \(a = 1\)
- \(b = -11\)
- \(c = 28\)

2. **Factor the equation**: We need to find two numbers that multiply to \(c\) (which is \(28\)) and add up to \(b\) (which is \(-11\)). The factors of \(28\) that satisfy this condition are \(-7\) and \(-4\).

3. **Rewrite the equation**: We can express the equation as:
\[ (y - 7)(y - 4) = 0 \]

4. **Set each factor to zero**:
- \(y - 7 = 0 \Rightarrow y = 7\)
- \(y - 4 = 0 \Rightarrow y = 4\)

Thus, the solutions for \(y\) are \(y = 7\) and \(y = 4\).

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

We have the following pairs of values:
- \(x = 2\) and \(y = 4\)
- \(x = 2\) and \(y = 7\)
- \(x = 19\) and \(y = 4\)
- \(x = 19\) and \(y = 7\)

Now we will compare the values:

1. For \(x = 2\) and \(y = 4\): \(2 < 4\) (so \(x < y\))
2. For \(x = 2\) and \(y = 7\): \(2 < 7\) (so \(x < y\))
3. For \(x = 19\) and \(y = 4\): \(19 > 4\) (so \(x > y\))
4. For \(x = 19\) and \(y = 7\): \(19 > 7\) (so \(x > y\))

### Conclusion:

Since \(x\) can be both less than \(y\) and greater than \(y\) depending on the values chosen, we cannot establish a consistent relationship between \(x\) and \(y\). Therefore, the final answer is that the relationship cannot be established.