If one asked to solve the set of equation below with the associated homogenous system, I'd know how to do it.
$S \leftrightarrow \begin{cases} 3x + 5y + z = 8\\\ x + 2y - 2z = 3 \end{cases}$
S' \leftrightarrow \begin{cases} 3x + 4y + z = 0\\\ x + 2y - 2z = 0 \end{cases}
You'd find the solution of the homogeneous system S' to be: \begin{equation} (x, y, z) = \{ k\cdot (-12, 7, 1) | k \in \mathbb{R} \} \end{equation}
With the particular solution of $S$... \begin{equation} (x, y, z) = (1, 1, 0) \end{equation}
You can count them up and you'd find: \begin{equation} (x, y, z) = \{(1 - 12k, 1+ 7k, k)|k \in \mathbb{R}\} \end{equation}
And your original system of equations $S$ is solved.
Now I've got one question: how do you find such a particular solution to a non-homogeneous system of equations. How do you find $(1, 1, 0)$ in this case?
Another example:
How do I find one particular solution to this non homogeneous system? \begin{cases} x_1 + x_2 +x_3 =4\\ 2x_1 + 5x_2 - 2x_3 = 3 \end{cases}