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Homogeneous system

$$\left(\begin{array}{lll|l}{1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0}\end{array}\right)$$

$$\left(\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right) =\left(\begin{array}{l}{0} \\ {0} \\ {0}\end{array}\right)$$

trivial sol. 

$$\left(\begin{array}{lll|l}{1} & {a} & {b} & {0} \\ {0} & {1} & {c} & {0} \\ {0} & {0} & {1} & {0}\end{array}\right)$$

$$\left(\begin{array}{lll|l}{1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right)$$

$$\left(\begin{array}{lll|l}{1} & {0} & {1} & {0} \\ {0} & {1} & {2} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right)$$

Let $$x_{3}=s$$

$$x_{1}=-s$$
$$x_{2}=-2s$$

$$\left(\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right)=s\left(\begin{array}{c}{-1} \\ {-2} \\ {1}\end{array}\right)$$

$$\left(\begin{array}{lllll|l}{1} & {0} & {0} & {1} & {4} & {0} \\ {0} & {1} & {0} & {2} & {5} & {0} \\ {0} & {0} & {1} & {3} & {6} & {0}\end{array}\right)$$

$$x_{4}=r$$
$$x_{5}=t$$

$$x_{1}=-r-4t$$
$$x_{2}=-2 r-5 t$$
$$x_{3}=-3r-6 t$$

$$\left[\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}} \\ {x_{4}} \\ {x_{5}}\end{array}\right]=r\left[\begin{array}{c}{-1} \\ {-2} \\ {-3} \\ {1} \\ {0}\end{array}\right]+t\left[\begin{array}{c}{-4} \\ {-5} \\ {-6} \\ {0} \\ {1}\end{array}\right]$$

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