Solve ` (2)/(x) + (3)/(y) = 13, (5)/(x) - ( 4)/(y) = - 2`, where ` x ne 0 and y ne 0`.

Options

A. `x = (1)/(2), y = (1)/(3)`
B. `x = (1)/(4), y = (1)/(6)`
C. `x = (1)/(6), y = (1)/(9)`
D. `x = (1)/(8), y = (1)/(12)`

Explanation

To solve the system of equations

\[ \frac{2}{x} + \frac{3}{y} = 13 \]
\[ \frac{5}{x} - \frac{4}{y} = -2 \]

we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can multiply the first equation by \(4\) and the second equation by \(3\):

1. Multiply the first equation by \(4\):
\[ 4 \left(\frac{2}{x} + \frac{3}{y}\right) = 4 \cdot 13 \]
This simplifies to:
\[ \frac{8}{x} + \frac{12}{y} = 52 \]

2. Multiply the second equation by \(3\):
\[ 3 \left(\frac{5}{x} - \frac{4}{y}\right) = 3 \cdot -2 \]
This simplifies to:
\[ \frac{15}{x} - \frac{12}{y} = -6 \]

### Step 2: Add the two equations
Now we have:
\[ \frac{8}{x} + \frac{12}{y} = 52 \quad \text{(1)} \]
\[ \frac{15}{x} - \frac{12}{y} = -6 \quad \text{(2)} \]

Next, we will add these two equations to eliminate \(y\):
\[ \left(\frac{8}{x} + \frac{12}{y}\right) + \left(\frac{15}{x} - \frac{12}{y}\right) = 52 - 6 \]
This simplifies to:
\[ \frac{8}{x} + \frac{15}{x} = 46 \]
\[ \frac{23}{x} = 46 \]

### Step 3: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{23}{46} = \frac{1}{2} \]

### Step 4: Substitute \(x\) back to find \(y\)
Now that we have \(x\), we can substitute it back into one of the original equations to find \(y\). Let's use the first equation:
\[ \frac{2}{\frac{1}{2}} + \frac{3}{y} = 13 \]
This simplifies to:
\[ 4 + \frac{3}{y} = 13 \]
Subtract \(4\) from both sides:
\[ \frac{3}{y} = 9 \]

### Step 5: Solve for \(y\)
Now, we can solve for \(y\):
\[ y = \frac{3}{9} = \frac{1}{3} \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = \frac{1}{2}, \quad y = \frac{1}{3} \]