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

Options

A. `(2,3)`
B. `(1/2, 1/3)`
C. `(4,6)`
D. `(1/4, 1/6)`

Answer

To solve the equations

\[ \frac{2}{x} + \frac{3}{y} = 13 \quad \text{(1)} \]
\[ \frac{5}{x} - \frac{4}{y} = -2 \quad \text{(2)} \]

we will follow these steps:

### Step 1: Rewrite the equations with a common denominator

For equation (1), the common denominator is \(xy\):

\[ \frac{2y + 3x}{xy} = 13 \]

Multiplying both sides by \(xy\):

\[ 2y + 3x = 13xy \quad \text{(3)} \]

For equation (2), the common denominator is also \(xy\):

\[ \frac{5y - 4x}{xy} = -2 \]

Multiplying both sides by \(xy\):

\[ 5y - 4x = -2xy \quad \text{(4)} \]

### Step 2: Rearranging the equations

From equation (3):

\[ 2y + 3x - 13xy = 0 \quad \text{(5)} \]

From equation (4):

\[ 5y - 4x + 2xy = 0 \quad \text{(6)} \]

### Step 3: Multiply the equations to eliminate one variable

Multiply equation (5) by 4:

\[ 8y + 12x - 52xy = 0 \quad \text{(7)} \]

Multiply equation (6) by 3:

\[ 15y - 12x + 6xy = 0 \quad \text{(8)} \]

### Step 4: Add equations (7) and (8)

Adding (7) and (8):

\[ (8y + 15y) + (12x - 12x) + (-52xy + 6xy) = 0 \]

This simplifies to:

\[ 23y - 46xy = 0 \]

Factoring out \(y\):

\[ y(23 - 46x) = 0 \]

Since \(y \neq 0\), we have:

\[ 23 - 46x = 0 \implies 46x = 23 \implies x = \frac{23}{46} = \frac{1}{2} \]

### Step 5: Substitute \(x\) back to find \(y\)

Substituting \(x = \frac{1}{2}\) into equation (6):

\[ 5y - 4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right)y = 0 \]

This becomes:

\[ 5y - 2 + y = 0 \implies 6y = 2 \implies y = \frac{2}{6} = \frac{1}{3} \]

### Final Solution

Thus, the solution is:

\[ x = \frac{1}{2}, \quad y = \frac{1}{3} \]