Solve : `(2)/(sqrt(x)) + (3)/(sqrt(y)) = 2` and `(4)/(sqrt(x)) - (9)/(sqrt(y)) = -1`

Options

A. x = 1/2, y = 1/3
B. x = 4, y = 9
C. X = 4, y = 1/3
D. X = 2, y = 3

Answer

To solve the equations

\[ \frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2 \quad \text{(1)} \]
\[ \frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = -1 \quad \text{(2)} \]

we can follow these steps:

### Step 1: Multiply the first equation by 3
Multiply both sides of equation (1) by 3 to eliminate the fraction:

\[ 3 \left( \frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} \right) = 3 \cdot 2 \]

This simplifies to:

\[ \frac{6}{\sqrt{x}} + \frac{9}{\sqrt{y}} = 6 \quad \text{(3)} \]

### Step 2: Write down the second equation
We can keep the second equation (2) as it is:

\[ \frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = -1 \quad \text{(2)} \]

### Step 3: Add equations (3) and (2)
Now, we add equations (3) and (2):

\[ \left(\frac{6}{\sqrt{x}} + \frac{9}{\sqrt{y}}\right) + \left(\frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}}\right) = 6 - 1 \]

This simplifies to:

\[ \frac{6 + 4}{\sqrt{x}} + \frac{9 - 9}{\sqrt{y}} = 5 \]

Which results in:

\[ \frac{10}{\sqrt{x}} = 5 \]

### Step 4: Solve for \(\sqrt{x}\)
To isolate \(\sqrt{x}\), multiply both sides by \(\sqrt{x}\):

\[ 10 = 5\sqrt{x} \]

Now, divide both sides by 5:

\[ \sqrt{x} = 2 \]

### Step 5: Solve for \(x\)
Now square both sides to find \(x\):

\[ x = 2^2 = 4 \]

### Step 6: Substitute \(x\) back into equation (1)
Substitute \(x = 4\) back into equation (1):

\[ \frac{2}{\sqrt{4}} + \frac{3}{\sqrt{y}} = 2 \]

This simplifies to:

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

Which simplifies to:

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

### Step 7: Isolate \(\frac{3}{\sqrt{y}}\)
Subtract 1 from both sides:

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

### Step 8: Solve for \(\sqrt{y}\)
Multiply both sides by \(\sqrt{y}\):

\[ 3 = \sqrt{y} \]

### Step 9: Solve for \(y\)
Now square both sides to find \(y\):

\[ y = 3^2 = 9 \]

### Final Answer
Thus, the solution is:

\[ x = 4, \quad y = 9 \]