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 the others.** From the first equation: $$4x - 3y + z = -10 \implies z = -10 - 4x + 3y$$ 3. **Substitute this expression for $z$ into the other two equations:** Second equation: $$2x + y + 3(-10 - 4x + 3y) = 0$$ Simplify: $$2x + y - 30 - 12x + 9y = 0$$ Combine like terms: $$-10x + 10y - 30 = 0$$ Add 30 to both sides: $$-10x + 10y = 30$$ Divide both sides by 10: $$\cancel{-10}x + \cancel{10}y = \cancel{30}3 \implies -x + y = 3$$ Fourth step: Third equation: $$-x + 2y - 5(-10 - 4x + 3y) = 17$$ Simplify: $$-x + 2y + 50 + 20x - 15y = 17$$ Combine like terms: $$19x - 13y + 50 = 17$$ Subtract 50 from both sides: $$19x - 13y = -33$$ 5. **Now solve the system:** $$\begin{cases} -x + y = 3 \\ 19x - 13y = -33 \end{cases}$$ From the first equation: $$y = x + 3$$ Substitute into the second: $$19x - 13(x + 3) = -33$$ Simplify: $$19x - 13x - 39 = -33$$ Combine like terms: $$6x - 39 = -33$$ Add 39 to both sides: $$6x = 6$$ Divide both sides by 6: $$\cancel{6}x = \cancel{6}1 \implies x = 1$$ 6. **Find $y$:** $$y = 1 + 3 = 4$$ 7. **Find $z$ using the expression from step 2:** $$z = -10 - 4(1) + 3(4) = -10 - 4 + 12 = -2$$ **Final solution:** $$\boxed{(x, y, z) = (1, 4, -2)}$$