Solve : `(2)/(x)+(3)/(y)=13 and (5)/(x)-(4)/(y)=-2`.

Answer

To solve the equations \( \frac{2}{x} + \frac{3}{y} = 13 \) and \( \frac{5}{x} - \frac{4}{y} = -2 \), we will follow these steps:

### Step 1: Substitute Variables
Let \( u = \frac{1}{x} \) and \( v = \frac{1}{y} \). Then we can rewrite the equations as:
1. \( 2u + 3v = 13 \) (Equation 1)
2. \( 5u - 4v = -2 \) (Equation 2)

### Step 2: Solve Equation 1 for \( v \)
From Equation 1, we can express \( v \) in terms of \( u \):
\[ 3v = 13 - 2u \implies v = \frac{13 - 2u}{3} \]

### Step 3: Substitute \( v \) in Equation 2
Now substitute \( v \) in Equation 2:
\[ 5u - 4\left(\frac{13 - 2u}{3}\right) = -2 \]
Multiply through by 3 to eliminate the fraction:
\[ 15u - 4(13 - 2u) = -6 \]
Distributing the -4:
\[ 15u - 52 + 8u = -6 \]
Combine like terms:
\[ 23u - 52 = -6 \]

### Step 4: Solve for \( u \)
Add 52 to both sides:
\[ 23u = 46 \]
Now divide by 23:
\[ u = 2 \]

### Step 5: Substitute \( u \) back to find \( v \)
Substituting \( u = 2 \) back into the expression for \( v \):
\[ v = \frac{13 - 2(2)}{3} = \frac{13 - 4}{3} = \frac{9}{3} = 3 \]

### Step 6: Find \( x \) and \( y \)
Since \( u = \frac{1}{x} \) and \( v = \frac{1}{y} \):
\[ x = \frac{1}{u} = \frac{1}{2}, \quad y = \frac{1}{v} = \frac{1}{3} \]

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