This is a system of equations:

$\frac{3}{x} - \frac{3}{y} = 3$ $\frac{2}{x} + \frac{1}{y} = 5$

Steps to solve:

1. Simplify the first equation: $\frac{3}{x} - \frac{3}{y} = 3$ Divide through by 3: \frac{1}{x} - \frac{1}{y} = 1 \tag{1}

2. Keep the second equation as is: \frac{2}{x} + \frac{1}{y} = 5 \tag{2}

3. Introduce substitutions: Let $a = \frac{1}{x}$ and $b = \frac{1}{y}$. The equations become: a - b = 1 \tag{3} 2a + b = 5 \tag{4}

4. Solve the system of linear equations: From equation (3): b = a - 1 \tag{5}

   Substitute $b = a - 1$ into equation (4): $2a + (a - 1) = 5$ Simplify: $3a - 1 = 5$ $3a = 6$ $a = 2$

   Substitute $a = 2$ into equation (5): $b = 2 - 1 = 1$

5. Back-substitute $a$ and $b$: Recall $a = \frac{1}{x}$ and $b = \frac{1}{y}$: $\frac{1}{x} = 2 \implies x = \frac{1}{2}$ $\frac{1}{y} = 1 \implies y = 1$

Final Solution:

$x = \frac{1}{2}, \, y = 1$

Would you like a deeper explanation or further assistance? Here are some related questions:

1. Can you solve similar equations with three variables?
2. How can substitution help in solving nonlinear systems of equations?
3. Could graphing the equations visually confirm the solution?
4. What are other methods, like elimination, to solve these equations?
5. How can we verify the solution in the original equations?

Tip: Always verify your solution by substituting back into the original equations!