Solve the System of Equations with Fractions: Find the Values of X and Y

To solve this system of linear equations, follow these steps:

• Step 1: Examine and rearrange the first equation $\frac{3}{x} + \frac{3}{y} = 2$.
• Step 2: Express one variable in terms of the other. We can isolate $\frac{3}{x}$ to get $\frac{3}{x} = 2 - \frac{3}{y}$.
• Step 3: Substitute $\frac{3}{x} = 2 - \frac{3}{y}$ into the second equation $\frac{9}{x} - \frac{4}{y} = -7$.
• Step 4: Replace $\frac{9}{x}$ with $3\left(2 - \frac{3}{y}\right)$ leading to $3\left(2 - \frac{3}{y}\right) - \frac{4}{y} = -7$.
• Step 5: Simplify the equation: Distribute $3$ into $2 - \frac{3}{y}$ to obtain $(6 - \frac{9}{y}) - \frac{4}{y} = -7$.
• Step 6: Combine the terms to get $6 - \frac{13}{y} = -7$.
• Step 7: Isolate $\frac{13}{y}$: $\frac{13}{y} = 13$.
• Step 8: Solve for $y$: Multiply both sides by $y$, leading to $13 = 13y$, hence $y = 1$.
• Step 9: Substitute $y = 1$ back into $\frac{3}{x} = 2 - \frac{3}{y}$, resulting in $\frac{3}{x} = 2 - 3$, or $\frac{3}{x} = -1$.
• Step 10: Solve for $x$: Multiply both sides by $x$, leading to $3 = -x$, thus $x = -3$.

Therefore, the solution to the system of equations is $\boldsymbol{x = -3}$ and $\boldsymbol{y = 1}$.

The correct choice among the given answer choices is the third option: $\boldsymbol{x = -3, y = 1}$.