# 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.

• Presumably, you mean a non-zero solution? – Thomas Andrews Nov 11 '14 at 22:42

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 $a(6 ,1,8,-4)$, $a\in\mathbb{R}$.