1. **State the problem:** Solve the system of equations using matrix elimination: $$\begin{cases} x_1 - 3x_2 = 5 \\ -x_1 + x_2 + 5x_3 = 2 \\ x_2 + x_3 = 0 \end{cases}$$ 2. **Write the augmented matrix:** $$\left[\begin{array}{ccc|c} 1 & -3 & 0 & 5 \\ -1 & 1 & 5 & 2 \\ 0 & 1 & 1 & 0 \end{array}\right]$$ 3. **Perform row operations to get upper triangular form:** - Add row 1 to row 2: $$R_2 = R_2 + R_1 \Rightarrow \left[\begin{array}{ccc|c} 1 & -3 & 0 & 5 \\ 0 & -2 & 5 & 7 \\ 0 & 1 & 1 & 0 \end{array}\right]$$ 4. **Eliminate the entry below the pivot in column 2:** - Add half of row 2 to row 3: $$R_3 = R_3 + \frac{1}{2}R_2 \Rightarrow \left[\begin{array}{ccc|c} 1 & -3 & 0 & 5 \\ 0 & -2 & 5 & 7 \\ 0 & 0 & 3.5 & 3.5 \end{array}\right]$$ 5. **Back substitution:** - From row 3: $$3.5x_3 = 3.5 \Rightarrow x_3 = 1$$ - From row 2: $$-2x_2 + 5(1) = 7 \Rightarrow -2x_2 = 2 \Rightarrow x_2 = -1$$ - From row 1: $$x_1 - 3(-1) = 5 \Rightarrow x_1 + 3 = 5 \Rightarrow x_1 = 2$$ **Final solution:** $$\boxed{(x_1, x_2, x_3) = (2, -1, 1)}$$