Solve the following pairs of equations by reducing them to a pair of linear equations:
`(2)/(sqrtx) + (3)/(sqrty)= 2`
`(4)/(sqrtx)- (9)/(sqrty)= -1`

Answer

To solve the given pair of equations by reducing them to a pair of linear equations, we will follow these steps:

### Step 1: Substitute Variables
We start with the equations:
1. \(\frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2\)
2. \(\frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = -1\)

Let:
- \(a = \frac{1}{\sqrt{x}}\)
- \(b = \frac{1}{\sqrt{y}}\)

Now we can rewrite the equations in terms of \(a\) and \(b\):
1. \(2a + 3b = 2\) (Equation 1)
2. \(4a - 9b = -1\) (Equation 2)

### Step 2: Solve the Linear Equations
Now we have a pair of linear equations:
1. \(2a + 3b = 2\) (1)
2. \(4a - 9b = -1\) (2)

We can use the elimination method to solve these equations.

### Step 3: Eliminate One Variable
To eliminate \(a\), we can multiply Equation (1) by 2:
\[ 4a + 6b = 4 \quad \text{(Equation 3)} \]

Now we have:
1. \(4a + 6b = 4\) (Equation 3)
2. \(4a - 9b = -1\) (Equation 2)

### Step 4: Subtract the Equations
Now, we subtract Equation (2) from Equation (3):
\[ (4a + 6b) - (4a - 9b) = 4 - (-1) \]
This simplifies to:
\[ 6b + 9b = 4 + 1 \]
\[ 15b = 5 \]
\[ b = \frac{5}{15} = \frac{1}{3} \]

### Step 5: Substitute Back to Find \(a\)
Now that we have \(b\), we can substitute \(b = \frac{1}{3}\) back into Equation (1):
\[ 2a + 3\left(\frac{1}{3}\right) = 2 \]
\[ 2a + 1 = 2 \]
\[ 2a = 2 - 1 \]
\[ 2a = 1 \]
\[ a = \frac{1}{2} \]

### Step 6: Find \(x\) and \(y\)
Now we can find \(x\) and \(y\) using the definitions of \(a\) and \(b\):
\[ a = \frac{1}{\sqrt{x}} \implies \sqrt{x} = \frac{1}{a} = \frac{1}{\frac{1}{2}} = 2 \implies x = 2^2 = 4 \]
\[ b = \frac{1}{\sqrt{y}} \implies \sqrt{y} = \frac{1}{b} = \frac{1}{\frac{1}{3}} = 3 \implies y = 3^2 = 9 \]

### Final Solution
Thus, the solution to the given equations is:
\[ x = 4, \quad y = 9 \]