1. **State the problem:** Find the fundamental solutions of the system of non-homogeneous linear equations: $$\begin{cases} 2x_1 + x_2 + 4x_3 + x_4 = 5 \\ x_1 - 3x_2 + x_3 + x_4 = 7 \end{cases}$$ 2. **Understand the goal:** Fundamental solutions refer to the general solution of the system, which is the sum of a particular solution to the non-homogeneous system plus the general solution to the associated homogeneous system. 3. **Write the associated homogeneous system:** $$\begin{cases} 2x_1 + x_2 + 4x_3 + x_4 = 0 \\ x_1 - 3x_2 + x_3 + x_4 = 0 \end{cases}$$ 4. **Solve the homogeneous system:** From the second equation: $$x_1 = 3x_2 - x_3 - x_4$$ Substitute into the first equation: $$2(3x_2 - x_3 - x_4) + x_2 + 4x_3 + x_4 = 0$$ Simplify: $$6x_2 - 2x_3 - 2x_4 + x_2 + 4x_3 + x_4 = 0$$ $$7x_2 + 2x_3 - x_4 = 0$$ Express $x_2$ in terms of $x_3$ and $x_4$: $$x_2 = \frac{-2x_3 + x_4}{7}$$ Recall: $$x_1 = 3x_2 - x_3 - x_4 = 3\left(\frac{-2x_3 + x_4}{7}\right) - x_3 - x_4 = \frac{-6x_3 + 3x_4}{7} - x_3 - x_4 = \frac{-6x_3 + 3x_4 - 7x_3 - 7x_4}{7} = \frac{-13x_3 - 4x_4}{7}$$ 5. **Write the general solution of the homogeneous system:** Let $x_3 = s$, $x_4 = t$ be free parameters: $$\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = s \begin{pmatrix} -\frac{13}{7} \\ -\frac{2}{7} \\ 1 \\ 0 \end{pmatrix} + t \begin{pmatrix} -\frac{4}{7} \\ \frac{1}{7} \\ 0 \\ 1 \end{pmatrix}$$ 6. **Find a particular solution to the non-homogeneous system:** Use the original system: $$\begin{cases} 2x_1 + x_2 + 4x_3 + x_4 = 5 \\ x_1 - 3x_2 + x_3 + x_4 = 7 \end{cases}$$ Try $x_3 = 0$, $x_4 = 0$ for simplicity. From second equation: $$x_1 - 3x_2 = 7$$ From first equation: $$2x_1 + x_2 = 5$$ Solve the system: Multiply second equation by 3: $$6x_1 + 3x_2 = 15$$ Add to first equation multiplied by 1: $$x_1 - 3x_2 = 7$$ Add: $$7x_1 = 22 \Rightarrow x_1 = \frac{22}{7}$$ Substitute back: $$\frac{22}{7} - 3x_2 = 7 \Rightarrow -3x_2 = 7 - \frac{22}{7} = \frac{49 - 22}{7} = \frac{27}{7}$$ $$x_2 = -\frac{9}{7}$$ 7. **Write the particular solution:** $$\begin{pmatrix} \frac{22}{7} \\ -\frac{9}{7} \\ 0 \\ 0 \end{pmatrix}$$ 8. **Write the general solution of the non-homogeneous system:** $$\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} \frac{22}{7} \\ -\frac{9}{7} \\ 0 \\ 0 \end{pmatrix} + s \begin{pmatrix} -\frac{13}{7} \\ -\frac{2}{7} \\ 1 \\ 0 \end{pmatrix} + t \begin{pmatrix} -\frac{4}{7} \\ \frac{1}{7} \\ 0 \\ 1 \end{pmatrix}$$ This is the fundamental solution set of the system.