1. We are given the system of equations: $$\frac{5x}{8} - \frac{y}{2} = \frac{1}{4}$$ $$\frac{2x}{3} - \frac{3y}{5} = \frac{2}{15}$$ 2. Our goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously. 3. First, clear denominators in each equation to simplify: Multiply the first equation by 8: $$8 \times \left(\frac{5x}{8} - \frac{y}{2}\right) = 8 \times \frac{1}{4}$$ $$5x - 4y = 2$$ Multiply the second equation by 15: $$15 \times \left(\frac{2x}{3} - \frac{3y}{5}\right) = 15 \times \frac{2}{15}$$ $$5 \times 2x - 3 \times 3y = 2$$ $$10x - 9y = 2$$ 4. Now we have the simpler system: $$5x - 4y = 2$$ $$10x - 9y = 2$$ 5. To eliminate $x$, multiply the first equation by 2: $$2(5x - 4y) = 2 \times 2$$ $$10x - 8y = 4$$ 6. Subtract the second equation from this result: $$\cancel{10x} - 8y - (\cancel{10x} - 9y) = 4 - 2$$ $$-8y + 9y = 2$$ $$y = 2$$ 7. Substitute $y=2$ back into the first simplified equation: $$5x - 4(2) = 2$$ $$5x - 8 = 2$$ $$5x = 10$$ $$x = 2$$ 8. Final solution: $$x = 2, \quad y = 2$$