Solve the following pairs of equations by reducing them to a pair of linear equations:
`(1)/(2x) + (1)/(3y)= 2`
`(1)/(3x) + (1)/(2y)= (13)/(6)`

Answer

To solve the given pair of equations, we will first rewrite them in a more manageable form. The equations are:

1. \(\frac{1}{2x} + \frac{1}{3y} = 2\)
2. \(\frac{1}{3x} + \frac{1}{2y} = \frac{13}{6}\)

### Step 1: Substitute Variables
Let's introduce new variables:
- Let \(u = \frac{1}{x}\)
- Let \(v = \frac{1}{y}\)

Now, we can rewrite the equations in terms of \(u\) and \(v\):

1. \(\frac{u}{2} + \frac{v}{3} = 2\)
2. \(\frac{u}{3} + \frac{v}{2} = \frac{13}{6}\)

### Step 2: Clear Denominators
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators.

For the first equation, the LCM of 2 and 3 is 6:
\[ 6 \left(\frac{u}{2} + \frac{v}{3}\right) = 6 \cdot 2 \]
This simplifies to:
\[ 3u + 2v = 12 \quad \text{(Equation 1)} \]

For the second equation, the LCM of 3 and 2 is also 6:
\[ 6 \left(\frac{u}{3} + \frac{v}{2}\right) = 6 \cdot \frac{13}{6} \]
This simplifies to:
\[ 2u + 3v = 13 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now we have a system of linear equations:
1. \(3u + 2v = 12\)
2. \(2u + 3v = 13\)

We can solve these equations using the elimination method. First, we will multiply the first equation by 3 and the second equation by 2 to align the coefficients of \(v\):

\[ 9u + 6v = 36 \quad \text{(Equation 3)} \]
\[ 4u + 6v = 26 \quad \text{(Equation 4)} \]

### Step 4: Subtract Equations
Now, we subtract Equation 4 from Equation 3:
\[ (9u + 6v) - (4u + 6v) = 36 - 26 \]
This simplifies to:
\[ 5u = 10 \]
Thus, we find:
\[ u = 2 \]

### Step 5: Substitute Back to Find \(v\)
Now that we have \(u\), we can substitute it back into one of the original equations to find \(v\). Using Equation 1:
\[ 3(2) + 2v = 12 \]
This simplifies to:
\[ 6 + 2v = 12 \]
Subtracting 6 from both sides gives:
\[ 2v = 6 \]
Thus, we find:
\[ v = 3 \]

### Step 6: Convert Back to \(x\) and \(y\)
Recall that:
- \(u = \frac{1}{x}\) implies \(x = \frac{1}{u} = \frac{1}{2}\)
- \(v = \frac{1}{y}\) implies \(y = \frac{1}{v} = \frac{1}{3}\)

### Final Answer
The solution to the equations is:
\[ x = \frac{1}{2}, \quad y = \frac{1}{3} \]