1. **State the problem:** Solve the system of linear equations for $x_1$, $x_2$, and $x_3$: $$x_1 + x_2 - 5x_3 = -8$$ $$5x_1 + 4x_2 - 5x_3 = -8$$ and express the solution in the form: $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} ? \\ ? \\ ? \end{bmatrix} + s \begin{bmatrix} ? \\ ? \\ ? \end{bmatrix}$$ where $s$ is a parameter. 2. **Rewrite the system:** Equation 1: $x_1 + x_2 - 5x_3 = -8$ Equation 2: $5x_1 + 4x_2 - 5x_3 = -8$ 3. **Express $x_1$ and $x_2$ in terms of $x_3$:** Let $x_3 = s$ (parameter). From Equation 1: $$x_1 = -8 - x_2 + 5s$$ Substitute $x_1$ into Equation 2: $$5(-8 - x_2 + 5s) + 4x_2 - 5s = -8$$ Simplify: $$-40 - 5x_2 + 25s + 4x_2 - 5s = -8$$ $$-40 - x_2 + 20s = -8$$ 4. **Solve for $x_2$:** $$- x_2 = -8 + 40 - 20s$$ $$- x_2 = 32 - 20s$$ $$x_2 = -32 + 20s$$ 5. **Substitute $x_2$ back to find $x_1$:** $$x_1 = -8 - (-32 + 20s) + 5s = -8 + 32 - 20s + 5s = 24 - 15s$$ 6. **Write the solution vector:** $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 24 \\ -32 \\ 0 \end{bmatrix} + s \begin{bmatrix} -15 \\ 20 \\ 1 \end{bmatrix}$$ This means the solution set is all vectors of this form for any real number $s$. **Final answer:** $$\boxed{\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 24 \\ -32 \\ 0 \end{bmatrix} + s \begin{bmatrix} -15 \\ 20 \\ 1 \end{bmatrix}}$$