1. **State the problem:** Solve the system of equations using substitution: $$\begin{cases}-6x - 2y - z = -17 \\ 5x + y - 6z = 19 \\ -4x - 6y - 6z = -20 \end{cases}$$ 2. **Choose an equation to isolate a variable:** From the first equation, isolate $z$: $$-6x - 2y - z = -17 \implies -z = -17 + 6x + 2y \implies z = 17 - 6x - 2y$$ 3. **Substitute $z$ into the other two equations:** Second equation: $$5x + y - 6(17 - 6x - 2y) = 19$$ Simplify: $$5x + y - 102 + 36x + 12y = 19$$ Combine like terms: $$5x + 36x + y + 12y - 102 = 19 \implies 41x + 13y - 102 = 19$$ Add 102 to both sides: $$41x + 13y = 121$$ Fourth step, third equation: $$-4x - 6y - 6(17 - 6x - 2y) = -20$$ Simplify: $$-4x - 6y - 102 + 36x + 12y = -20$$ Combine like terms: $$(-4x + 36x) + (-6y + 12y) - 102 = -20 \implies 32x + 6y - 102 = -20$$ Add 102 to both sides: $$32x + 6y = 82$$ 4. **Solve the system of two equations with two variables:** $$\begin{cases}41x + 13y = 121 \\ 32x + 6y = 82 \end{cases}$$ Multiply the second equation by $-\frac{13}{6}$ to eliminate $y$: $$-\frac{13}{6} \times (32x + 6y) = -\frac{13}{6} \times 82$$ $$-\frac{416}{6}x - 13y = -\frac{1066}{3}$$ Rewrite first equation: $$41x + 13y = 121$$ Add the two equations: $$41x + 13y - \frac{416}{6}x - 13y = 121 - \frac{1066}{3}$$ Simplify left side: $$41x - \frac{416}{6}x = 121 - \frac{1066}{3}$$ Convert $41x$ to sixths: $$\frac{246}{6}x - \frac{416}{6}x = 121 - \frac{1066}{3}$$ Subtract: $$-\frac{170}{6}x = 121 - \frac{1066}{3}$$ Convert 121 to thirds: $$121 = \frac{363}{3}$$ Calculate right side: $$\frac{363}{3} - \frac{1066}{3} = -\frac{703}{3}$$ So: $$-\frac{170}{6}x = -\frac{703}{3}$$ Multiply both sides by $-\frac{6}{170}$: $$x = -\frac{703}{3} \times -\frac{6}{170} = \frac{703 \times 6}{3 \times 170}$$ Simplify: $$x = \frac{703 \times 2}{170} = \frac{1406}{170} = \frac{703}{85}$$ 5. **Substitute $x$ back to find $y$:** Use equation $32x + 6y = 82$: $$32 \times \frac{703}{85} + 6y = 82$$ Calculate: $$\frac{22496}{85} + 6y = 82$$ Convert 82 to fraction with denominator 85: $$82 = \frac{6970}{85}$$ Subtract: $$6y = \frac{6970}{85} - \frac{22496}{85} = -\frac{15526}{85}$$ Divide both sides by 6: $$y = -\frac{15526}{85} \times \frac{1}{6} = -\frac{15526}{510}$$ Simplify fraction by dividing numerator and denominator by 2: $$y = -\frac{7763}{255}$$ 6. **Substitute $x$ and $y$ back to find $z$:** $$z = 17 - 6x - 2y = 17 - 6 \times \frac{703}{85} - 2 \times \left(-\frac{7763}{255}\right)$$ Calculate each term: $$6 \times \frac{703}{85} = \frac{4218}{85}$$ $$2 \times -\frac{7763}{255} = -\frac{15526}{255}$$ Rewrite 17 as fraction with denominator 255: $$17 = \frac{4335}{255}$$ Rewrite $\frac{4218}{85}$ with denominator 255: $$\frac{4218}{85} = \frac{4218 \times 3}{255} = \frac{12654}{255}$$ Now: $$z = \frac{4335}{255} - \frac{12654}{255} + \frac{15526}{255} = \frac{4335 - 12654 + 15526}{255} = \frac{7207}{255}$$ Simplify numerator and denominator by dividing by 1 (no common factor): $$z = \frac{7207}{255}$$ **Final solution:** $$\boxed{x = \frac{703}{85}, \quad y = -\frac{7763}{255}, \quad z = \frac{7207}{255}}$$