<s>[INST] How should I go about solving this system of equations using Gauss Elimination? $${ x }_{ 1 }+2{ x }_{ 2 }-{ x }_{ 3 }+2{ x }_{ 4 }=0\\ { x }_{ 2 }+{ x }_{ 3 }-2{ x }_{ 4 }+2{ x }_{ 5 }=0\\ 2{ x }_{ 1 }+{ x }_{ 2 }-5{ x }_{ 3 }-4{ x }_{ 5 }=0$$
Steps I took:
$$\left[\begin{array}{rrrrr|r}
    1 & 2 & -1 & 2 & 0 & 0 \\
    0 & 1 & 1 & -2 & 2 & 0 \\
    2 & 1 & -5 & 0 & -4 & 0
  \end{array}\right]$$
$$\Rightarrow { R }_{ 3 }={ R }_{ 3 }-2{ R }_{ 1 } =$$ 
$$\left[\begin{array}{rrrrr|r}
    1 & 2 & -1 & 2 & 0 & 0 \\
    0 & 1 & 1 & -2 & 2 & 0 \\
    0 & -3 & -2 & -6 & -4 & 0
  \end{array}\right]$$
$$\Rightarrow { R }_{ 2 }={ R }_{ 3 }-2{ R }_{ 2 } =$$ 
$$\left[\begin{array}{rrrrr|r}
    1 & 2 & -1 & 2 & 0 & 0 \\
    0 & -1 & 0 & 10 & 0 & 0 \\
    0 & -3 & -2 & -6 & -4 & 0
  \end{array}\right]$$
$$\Rightarrow { R }_{ 3 }={ R }_{ 3 }-3{ R }_{ 2 } =$$ 
$$\left[\begin{array}{rrrrr|r}
    1 & 2 & -1 & 2 & 0 & 0 \\
    0 & -1 & 0 & -10 & 0 & 0 \\
    0 & 0 & -2 & 24 & -4 & 0
  \end{array}\right]$$
Where do I go from here in order to find the solutions for this system? I can already tell that I ma going to have an infinite amount of solutions because I have more unknown than constraints. I just don't know how to (properly) apply Gauss Elimination to this. A hint in the right direction would help a lot.
 [/INST] multiplying the first equation by $-2$ and adding to the third equation we get
$$x_1+2x_2-x_3+3x_4=0$$
$$x_2+x_3+2x_4+2x_5=0$$
$$-3x_2-2x_3-x_4-4x_5=0$$
multiplying the second equation by $3$ and adding to the third we obtain
$$x_2+2x_5=0$$
</s>