1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x_1 - 3x_2 = 5 \\ -x_1 + x_2 + 5x_3 = 2 \\ x_2 + x_3 = 0 \end{cases}$$ 2. **Use substitution or elimination:** From the third equation, express $x_3$ in terms of $x_2$: $$x_3 = -x_2$$ 3. **Substitute $x_3$ into the second equation:** $$-x_1 + x_2 + 5(-x_2) = 2$$ $$-x_1 + x_2 - 5x_2 = 2$$ $$-x_1 - 4x_2 = 2$$ 4. **Rewrite the system with two equations:** $$\begin{cases} x_1 - 3x_2 = 5 \\ -x_1 - 4x_2 = 2 \end{cases}$$ 5. **Add the two equations to eliminate $x_1$:** $$(x_1 - 3x_2) + (-x_1 - 4x_2) = 5 + 2$$ $$-7x_2 = 7$$ 6. **Solve for $x_2$:** $$x_2 = \frac{7}{-7} = -1$$ 7. **Find $x_1$ using the first equation:** $$x_1 - 3(-1) = 5$$ $$x_1 + 3 = 5$$ $$x_1 = 2$$ 8. **Find $x_3$ using $x_3 = -x_2$:** $$x_3 = -(-1) = 1$$ **Final solution:** $$\boxed{(x_1, x_2, x_3) = (2, -1, 1)}$$