<s>[INST] What is the solution set of the given homogenous system? Write the solution set of the given homogenous system in parametric vector form:
\begin{align} 2x_{1}+2x_{2}+4x_{3} &= 0\\
-4x_{1}-4x_{2}-8x_{3} &= 0\\
0x_{1}-3x_{2}-3x_{3} &= 0\\
\end{align}
My attempt:
\begin{align} 2x_{1}+2x_{2}+4x_{3} &= 0\\
-4x_{1}-4x_{2}-8x_{3} &= 0\\
0x_{1}-3x_{2}-3x_{3} &= 0\\
\end{align}
Divided the second row by $-4$ and the third row by $-3$
\begin{align} x_{1}+x_{2}+2x_{3} &= 0\\
x_{1}+x_{2}+2x_{3} &= 0\\
0x_{1}+x_{2}+x_{3} &= 0\\
\end{align}
Row $2$ and $3$ plus the first row multiplied by $-1$
\begin{align} x_{1}+x_{2}+2x_{3} &= 0\\
0x_{1}+0x_{2}+0x_{3} &= 0\\
-x_{1}+0x_{2}-x_{3} &= 0\\
\end{align}
Thanks in advance.
 [/INST] That looks fine so far except that the bottom left $0$ in the very last equation should be a $-1$. The middle equation tells you nothing, so you can eliminate it. You'll end up with a whole line of solutions (all multiples of a single vector). 
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