1. **State the problem:** Solve the system of equations using the substitution method: $$\begin{cases} 4x - 3y + z = -10 \\ 2x + y + 3z = 0 \\ -x + 2y - 5z = 17 \end{cases}$$ 2. **Choose one equation to express one variable in terms of others.** From the first equation: $$4x - 3y + z = -10 \implies z = -10 - 4x + 3y$$ 3. **Substitute $z$ into the other two equations:** - Second equation: $$2x + y + 3(-10 - 4x + 3y) = 0$$ Simplify: $$2x + y - 30 - 12x + 9y = 0$$ $$\cancel{2x} - 12x + \cancel{y} + 9y - 30 = 0$$ $$-10x + 10y - 30 = 0$$ 4. **Simplify the above:** $$-10x + 10y = 30$$ Divide both sides by 10: $$\cancel{-10}x + \cancel{10}y = \cancel{30}3$$ $$-x + y = 3 \implies y = x + 3$$ 5. **Substitute $z$ and $y$ into the third equation:** $$-x + 2y - 5z = 17$$ Substitute $y = x + 3$ and $z = -10 - 4x + 3y$: $$-x + 2(x + 3) - 5(-10 - 4x + 3(x + 3)) = 17$$ Simplify inside the parentheses: $$-x + 2x + 6 - 5(-10 - 4x + 3x + 9) = 17$$ $$-x + 2x + 6 - 5(-10 - x + 9) = 17$$ $$-x + 2x + 6 - 5(-1 - x) = 17$$ $$x + 6 - 5(-1 - x) = 17$$ Distribute: $$x + 6 + 5 + 5x = 17$$ $$6x + 11 = 17$$ 6. **Solve for $x$:** $$6x = 17 - 11$$ $$6x = 6$$ $$x = 1$$ 7. **Find $y$ using $y = x + 3$:** $$y = 1 + 3 = 4$$ 8. **Find $z$ using $z = -10 - 4x + 3y$:** $$z = -10 - 4(1) + 3(4) = -10 - 4 + 12 = -2$$ **Final solution:** $$\boxed{(x, y, z) = (1, 4, -2)}$$