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}
Best Answer
Just set $z=0$, say. With a bit of luck, you'll be able to solve the resulting system: $$ \eqalign{ 3x+5y&=8\cr x+2y&=3 } $$
The solution of the above system is $y=1 , x=1 $; so, a solution to the original equation is $(1, 1 , 0)$.
For your second question, do a similar thing. Set $x_2=0$. Then you can conclude $x_1=11/4$ and $x_3=5/4$.