1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x_1 + x_2 = -5 \\ x_2 + x_3 = -2 \\ x_3 + x_4 = -1 \\ x_1 + x_4 = -4 \end{cases}$$ 2. **Write the system in matrix form:** We want to find vectors $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}$ satisfying the above. 3. **Express variables in terms of others:** From the first equation: $$x_1 = -5 - x_2$$ From the second: $$x_3 = -2 - x_2$$ From the third: $$x_4 = -1 - x_3 = -1 - (-2 - x_2) = -1 + 2 + x_2 = 1 + x_2$$ 4. **Use the fourth equation to check consistency:** $$x_1 + x_4 = (-5 - x_2) + (1 + x_2) = -5 - x_2 + 1 + x_2 = -4$$ The $x_2$ terms cancel out, and the equation holds true, so the system is consistent. 5. **Parameterize the solution:** Let $s = x_2$ be a free parameter. Then: $$x_1 = -5 - s$$ $$x_2 = s$$ $$x_3 = -2 - s$$ $$x_4 = 1 + s$$ 6. **Write the general solution as a vector sum:** $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} -5 \\ 0 \\ -2 \\ 1 \end{bmatrix} + s \begin{bmatrix} -1 \\ 1 \\ -1 \\ 1 \end{bmatrix}$$ **Final answer:** $$\boxed{\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} -5 \\ 0 \\ -2 \\ 1 \end{bmatrix} + s \begin{bmatrix} -1 \\ 1 \\ -1 \\ 1 \end{bmatrix}}$$