` (15)/sqrtx +9/sqrtx = 11sqrtx, (sqrty/4) + ((5sqrty)/12) = (1/sqrty)`

Options

A. if ` x gt y`
B. if ` x lt y`
C. if ` x ge y`
D. if ` x le y`

Explanation

To solve the given equations step by step, we will break down each equation and find the values of \( x \) and \( y \).

### Step 1: Solve the first equation

The first equation is:

\[ \frac{15}{\sqrt{x}} + \frac{9}{\sqrt{x}} = 11\sqrt{x} \]

Combine the fractions on the left side:

\[ \frac{15 + 9}{\sqrt{x}} = 11\sqrt{x} \]

This simplifies to:

\[ \frac{24}{\sqrt{x}} = 11\sqrt{x} \]

### Step 2: Eliminate the fraction

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

\[ 24 = 11x \]

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

Now, divide both sides by 11:

\[ x = \frac{24}{11} \]

Calculating this gives:

\[ x \approx 2.18 \]

### Step 4: Solve the second equation

The second equation is:

\[ \frac{\sqrt{y}}{4} + \frac{5\sqrt{y}}{12} = \frac{1}{\sqrt{y}} \]

To combine the left side, find a common denominator, which is 12:

\[ \frac{3\sqrt{y}}{12} + \frac{5\sqrt{y}}{12} = \frac{1}{\sqrt{y}} \]

This simplifies to:

\[ \frac{8\sqrt{y}}{12} = \frac{1}{\sqrt{y}} \]

### Step 5: Simplify the left side

Reducing the fraction gives:

\[ \frac{2\sqrt{y}}{3} = \frac{1}{\sqrt{y}} \]

### Step 6: Cross-multiply

Cross-multiply to eliminate the fractions:

\[ 2\sqrt{y} \cdot \sqrt{y} = 3 \]

This simplifies to:

\[ 2y = 3 \]

### Step 7: Solve for \( y \)

Now, divide both sides by 2:

\[ y = \frac{3}{2} \]

Calculating this gives:

\[ y = 1.5 \]

### Step 8: Determine the relationship between \( x \) and \( y \)

Now we have:

\[ x \approx 2.18 \quad \text{and} \quad y = 1.5 \]

Since \( x > y \), we conclude that:

\[ x > y \]

### Summary of Results

- \( x \approx 2.18 \)
- \( y = 1.5 \)
- Relationship: \( x > y \)