The given equation is:

$\frac{x}{3} + \frac{3}{y} = \frac{x}{4} - \frac{4}{y} = 1$

Let’s solve for $x$ and $y$.

Step 1: Separate the equation into two parts

We can split the problem into two equations:

1. $\frac{x}{3} + \frac{3}{y} = 1$
2. $\frac{x}{4} - \frac{4}{y} = 1$

Step 2: Solve the first equation

$\frac{x}{3} + \frac{3}{y} = 1$

Multiply through by $3y$ (the common denominator):

$y \cdot x + 9 = 3y$

Rearrange to express $x$ in terms of $y$:

$x = \frac{3y - 9}{y}$

Step 3: Solve the second equation

$\frac{x}{4} - \frac{4}{y} = 1$

Multiply through by $4y$ (the common denominator):

$y \cdot x - 16 = 4y$

Rearrange to express $x$ in terms of $y$:

$x = \frac{4y + 16}{y}$

Step 4: Equate the two expressions for $x$

From the first equation: $x = \frac{3y - 9}{y}$

From the second equation: $x = \frac{4y + 16}{y}$

Equate the two expressions for $x$:

$\frac{3y - 9}{y} = \frac{4y + 16}{y}$

Since the denominators are the same, equate the numerators:

$3y - 9 = 4y + 16$

Simplify:

$-9 = y + 16$

$y = -25$

Step 5: Solve for $x$

Substitute $y = -25$ into either equation for $x$. Using the first equation:

$x = \frac{3y - 9}{y}$

$x = \frac{3(-25) - 9}{-25}$

$x = \frac{-75 - 9}{-25}$

$x = \frac{-84}{-25} = \frac{84}{25}$

Final Answer:

$x = \frac{84}{25}, \, y = -25$

Would you like me to verify the solution or provide additional details?

Related Questions:

1. How do you simplify fractions in equations?
2. What are other methods to solve systems of equations involving fractions?
3. How can we verify the solution of this equation?
4. How do you handle equations with multiple variables?
5. What are practical applications of such equations?

Tip:

Always check your solution by substituting back into the original equations to ensure it satisfies all parts of the problem.