1. **State the problem:** Solve the system of linear equations: $$\begin{cases}-6x - 2y - z = -17 \\ 5x + y - 6z = 19 \\ -4x - 6y - 6z = -20 \end{cases}$$ 2. **Use substitution or elimination method:** We will use elimination to find $x$, $y$, and $z$. 3. **Eliminate one variable:** Multiply the first equation by 3 to help eliminate $z$ with the second equation: $$3(-6x - 2y - z) = 3(-17) \Rightarrow -18x - 6y - 3z = -51$$ 4. **Add this to the second equation to eliminate $z$:** $$(-18x - 6y - 3z) + (5x + y - 6z) = -51 + 19$$ $$-18x + 5x - 6y + y - 3z - 6z = -32$$ $$-13x - 5y - 9z = -32$$ But this still has $z$, so instead multiply the second equation by 0.5 and add to the third to eliminate $z$: $$0.5(5x + y - 6z) = 0.5(19) \Rightarrow 2.5x + 0.5y - 3z = 9.5$$ Add to third equation: $$(-4x - 6y - 6z) + (2.5x + 0.5y - 3z) = -20 + 9.5$$ $$-4x + 2.5x - 6y + 0.5y - 6z - 3z = -10.5$$ $$-1.5x - 5.5y - 9z = -10.5$$ 5. **Now subtract the two new equations to eliminate $z$:** $$(-13x - 5y - 9z) - (-1.5x - 5.5y - 9z) = -32 - (-10.5)$$ $$-13x + 1.5x - 5y + 5.5y - 9z + 9z = -21.5$$ $$-11.5x + 0.5y = -21.5$$ 6. **Solve for $y$:** $$0.5y = 11.5x - 21.5$$ $$y = \frac{11.5x - 21.5}{0.5} = 23x - 43$$ 7. **Substitute $y$ back into the first equation:** $$-6x - 2(23x - 43) - z = -17$$ $$-6x - 46x + 86 - z = -17$$ $$-52x + 86 - z = -17$$ $$-z = -17 + 52x - 86$$ $$-z = 52x - 103$$ $$z = -52x + 103$$ 8. **Substitute $y$ and $z$ into the second equation:** $$5x + (23x - 43) - 6(-52x + 103) = 19$$ $$5x + 23x - 43 + 312x - 618 = 19$$ $$340x - 661 = 19$$ $$340x = 680$$ $$x = \frac{680}{340} = 2$$ 9. **Find $y$ and $z$ using $x=2$:** $$y = 23(2) - 43 = 46 - 43 = 3$$ $$z = -52(2) + 103 = -104 + 103 = -1$$ **Final answer:** $$\boxed{(x, y, z) = (2, 3, -1)}$$