1. **State the problem:** Solve the system of equations using row operations: $$\begin{cases}-6x - 5y - 3z = 17 \\ 3x - 4y + 4z = -19 \\ -6x + 3y - 6z = 30 \end{cases}$$ 2. **Write the augmented matrix:** $$\left[\begin{array}{ccc|c}-6 & -5 & -3 & 17 \\ 3 & -4 & 4 & -19 \\ -6 & 3 & -6 & 30 \end{array}\right]$$ 3. **Perform row operations to get row echelon form:** - Multiply row 2 by 2 and add to row 1: $$R_1 \to R_1 + 2R_2: [-6 + 6, -5 + (-8), -3 + 8, 17 + (-38)] = [0, -13, 5, -21]$$ - Multiply row 2 by 2 and add to row 3: $$R_3 \to R_3 + 2R_2: [-6 + 6, 3 + (-8), -6 + 8, 30 + (-38)] = [0, -5, 2, -8]$$ Matrix now: $$\left[\begin{array}{ccc|c}0 & -13 & 5 & -21 \\ 3 & -4 & 4 & -19 \\ 0 & -5 & 2 & -8 \end{array}\right]$$ - Swap rows 1 and 2 to get a leading 1 in the top-left: $$\left[\begin{array}{ccc|c}3 & -4 & 4 & -19 \\ 0 & -13 & 5 & -21 \\ 0 & -5 & 2 & -8 \end{array}\right]$$ - Divide row 1 by 3: $$R_1 \to \frac{1}{3}R_1: [1, -\frac{4}{3}, \frac{4}{3}, -\frac{19}{3}]$$ - Eliminate $y$ in row 3 using row 2: Multiply row 2 by $\frac{5}{13}$ and add to row 3: $$R_3 \to R_3 + \frac{5}{13}R_2: [0, -5 + (-5), 2 + \frac{10}{13}, -8 + \frac{-105}{13}] = [0, 0, \frac{36}{13}, -\frac{209}{13}]$$ 4. **Back substitution:** - From row 3: $$\frac{36}{13}z = -\frac{209}{13} \implies z = -\frac{209}{13} \times \frac{13}{36} = -\frac{209}{36}$$ - From row 2: $$-13y + 5z = -21 \implies -13y = -21 - 5z = -21 - 5 \times \left(-\frac{209}{36}\right) = -21 + \frac{1045}{36} = \frac{-756 + 1045}{36} = \frac{289}{36}$$ $$y = -\frac{289}{36} \times \frac{1}{13} = -\frac{289}{468} = -\frac{17}{26}$$ - From row 1: $$x - \frac{4}{3}y + \frac{4}{3}z = -\frac{19}{3}$$ Substitute $y$ and $z$: $$x = -\frac{19}{3} + \frac{4}{3}y - \frac{4}{3}z = -\frac{19}{3} + \frac{4}{3} \times \left(-\frac{17}{26}\right) - \frac{4}{3} \times \left(-\frac{209}{36}\right)$$ Calculate: $$x = -\frac{19}{3} - \frac{68}{78} + \frac{836}{108} = -\frac{19}{3} - \frac{34}{39} + \frac{209}{27}$$ Find common denominator 351: $$-\frac{19}{3} = -\frac{2223}{351}, \quad -\frac{34}{39} = -\frac{306}{351}, \quad \frac{209}{27} = \frac{2717}{351}$$ Sum: $$x = \frac{-2223 - 306 + 2717}{351} = \frac{188}{351} = \frac{8}{15}$$ 5. **Final solution:** $$x = \frac{8}{15}, \quad y = -\frac{17}{26}, \quad z = -\frac{209}{36}$$