How to construct a solution to a homogeneous system of linear equations?

Question

given the system of linear equations

$\begin{array}{rrrrr}x_{1} & + & x_{2} & + & 2 x_{3} & = & 2 \\ 2 x_{1} & + & 5 x_{2} & - & x_{3} & = & 2 \\ 3 x_{1} & + & 6 x_{2} & + & x_{3} & = & 4\end{array}$

I found the solutions

$x=\left(\begin{array}{c}\frac{8}{3}-\frac{11}{3} t \\ -\frac{2}{3}+\frac{5}{3} t \\ t\end{array}\right)$ ,

$y=\left(\begin{array}{ccc}\frac{6}{5} -\frac{11s}{5} \\ s\\ \frac{2}{5}+\frac{3s}{5}\end{array}\right)$

Now I have to construct a solution to the following homogenous system out of x and y:

$\begin{array}{rrrrr}x_{1} & + & x_{2} & + & 2 x_{3} & = & 0 \\ 2 x_{1} & + & 5 x_{2} & - & x_{3} & = & 0 \\ 3 x_{1} & + & 6 x_{2} & + & x_{3} & = & 0\end{array}$

How can this be done?

Answer 1

Since both $x$ and $y$ are solutions of the original system, $x-y$ is a solution of the homogenous system.