`(4)/(sqrtx) + (7)/(sqrtx) = sqrtx, y^(2)-((11)^(5/2))/(sqrty) = 0`

Options

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

Answer

To solve the given problem, we need to find the values of \( x \) and \( y \) from the equations provided and then compare these values.

### Step 1: Solve for \( x \)

The first equation is:

\[ \frac{4}{\sqrt{x}} + \frac{7}{\sqrt{x}} = \sqrt{x} \]

Combine the fractions on the left side:

\[ \frac{4 + 7}{\sqrt{x}} = \sqrt{x} \]

This simplifies to:

\[ \frac{11}{\sqrt{x}} = \sqrt{x} \]

### Step 2: Cross-multiply

Cross-multiply to eliminate the fraction:

\[ 11 = \sqrt{x} \cdot \sqrt{x} \]

This simplifies to:

\[ 11 = x \]

### Step 3: Solve for \( y \)

Now, we move to the second equation:

\[ y^2 - \frac{11^{5/2}}{\sqrt{y}} = 0 \]

Rearranging gives:

\[ y^2 = \frac{11^{5/2}}{\sqrt{y}} \]

### Step 4: Multiply both sides by \( \sqrt{y} \)

To eliminate the fraction, multiply both sides by \( \sqrt{y} \):

\[ y^2 \cdot \sqrt{y} = 11^{5/2} \]

This can be rewritten as:

\[ y^{2.5} = 11^{5/2} \]

### Step 5: Solve for \( y \)

Taking both sides to the power of \( \frac{2}{5} \):

\[ y = \left(11^{5/2}\right)^{\frac{2}{5}} = 11^{\frac{5 \cdot 2}{2 \cdot 5}} = 11 \]

### Step 6: Compare \( x \) and \( y \)

Now we have:

\[ x = 11 \quad \text{and} \quad y = 11 \]

### Conclusion

Since \( x = y \), the correct option is:

**Option 4: \( x \) is equal to \( y \)**.

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