1. **State the problem:** Solve the system of equations: $$\begin{cases} 2x + 3y - 2z = -1 \\ x + 5y = 9 \\ 4z - 5x = 4 \end{cases}$$ 2. **Express variables from simpler equations:** From the second equation, solve for $x$: $$x + 5y = 9 \implies x = 9 - 5y$$ 3. **Express $z$ from the third equation:** $$4z - 5x = 4 \implies 4z = 4 + 5x \implies z = \frac{4 + 5x}{4}$$ 4. **Substitute $x$ and $z$ into the first equation:** $$2x + 3y - 2z = -1$$ Substitute $x = 9 - 5y$ and $z = \frac{4 + 5x}{4}$: $$2(9 - 5y) + 3y - 2\left(\frac{4 + 5(9 - 5y)}{4}\right) = -1$$ 5. **Simplify step-by-step:** $$18 - 10y + 3y - \frac{2}{4}(4 + 45 - 25y) = -1$$ $$18 - 7y - \frac{1}{2}(49 - 25y) = -1$$ 6. **Distribute and simplify:** $$18 - 7y - \frac{49}{2} + \frac{25y}{2} = -1$$ 7. **Multiply entire equation by 2 to clear denominators:** $$2(18 - 7y) - 49 + 25y = 2(-1)$$ $$36 - 14y - 49 + 25y = -2$$ 8. **Combine like terms:** $$36 - 49 + (-14y + 25y) = -2$$ $$-13 + 11y = -2$$ 9. **Solve for $y$:** $$11y = -2 + 13$$ $$11y = 11$$ $$y = \frac{11}{11} = 1$$ 10. **Find $x$ using $y=1$:** $$x = 9 - 5(1) = 9 - 5 = 4$$ 11. **Find $z$ using $x=4$:** $$z = \frac{4 + 5(4)}{4} = \frac{4 + 20}{4} = \frac{24}{4} = 6$$ **Final answer:** $$\boxed{(x, y, z) = (4, 1, 6)}$$