1. **State the problem:** Solve the system of equations using the augmented matrix method: $$\begin{cases} x + 2y - z = 3 \\ x + 3y + z = 5 \\ 3x + 8y + 4z = 17 \end{cases}$$ 2. **Write the augmented matrix $M$:** $$M = \left[\begin{array}{ccc|c} 1 & 2 & -1 & 3 \\ 1 & 3 & 1 & 5 \\ 3 & 8 & 4 & 17 \end{array}\right]$$ 3. **Use row operations to reduce $M$ to row echelon form:** - Subtract row 1 from row 2: $$R_2 \to R_2 - R_1: \left[\begin{array}{ccc|c} 1 & 2 & -1 & 3 \\ 0 & 1 & 2 & 2 \\ 3 & 8 & 4 & 17 \end{array}\right]$$ - Subtract 3 times row 1 from row 3: $$R_3 \to R_3 - 3R_1: \left[\begin{array}{ccc|c} 1 & 2 & -1 & 3 \\ 0 & 1 & 2 & 2 \\ 0 & 2 & 7 & 8 \end{array}\right]$$ - Subtract 2 times row 2 from row 3: $$R_3 \to R_3 - 2R_2: \left[\begin{array}{ccc|c} 1 & 2 & -1 & 3 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 3 & 4 \end{array}\right]$$ 4. **Back substitution to solve for variables:** - From row 3: $3z = 4 \Rightarrow z = \frac{4}{3}$ - From row 2: $y + 2z = 2 \Rightarrow y + 2\times\frac{4}{3} = 2 \Rightarrow y = 2 - \frac{8}{3} = -\frac{2}{3}$ - From row 1: $x + 2y - z = 3 \Rightarrow x + 2\times\left(-\frac{2}{3}\right) - \frac{4}{3} = 3$ Simplify: $$x - \frac{4}{3} - \frac{4}{3} = 3 \Rightarrow x - \frac{8}{3} = 3 \Rightarrow x = 3 + \frac{8}{3} = \frac{17}{3}$$ 5. **Final solution:** $$\boxed{\left(x, y, z\right) = \left(\frac{17}{3}, -\frac{2}{3}, \frac{4}{3}\right)}$$ This means the system has a unique solution with these values for $x$, $y$, and $z$.