<s>[INST] Does this system has a solution? Solve the system of equations
$$\begin{cases}2x_1+x_3+5x_4=0 \\x_1+2x_2-x_3=0\\x_1+x_2+2x_4=0\end{cases}$$
I created the matrix for this system and I found that it has many solution but I'm not sure if that true of not.
 [/INST] As you said, it has many solutions.
Working with the matrix we get
$ \left( \begin{array}{cccc}
2 & 0 & 1 & 5\\
1 & 2 & -1 & 0 \\
1 & 2 & 0 &2 \end{array} \right) \to$ $ \left( \begin{array}{cccc}
1 & 0 & 1/2 & 5/2\\
0 & 2 & -3/2 & -5/2 \\
0 & 2 & -1/2 & -1/2 \end{array} \right) \to$ $ \left( \begin{array}{cccc}
1 & 0 & 1/2 & 5/2\\
0 & 1 & -3/4 & -5/4 \\
0 & 0 & 1 &2 \end{array} \right)\\$ $\to\left( \begin{array}{cccc}
1 & 0 & 0 & 3/2\\
0 & 1 & 0 & 1/4 \\
0 & 0 & 1 &2 \end{array} \right)$
Follows that the solutions are of the form $a(6 ,1,8,-4)$, $a\in\mathbb{R}$.
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