## Solve a given system of linear equations over QQ

Find the nonzero solutions of the following system over $\mathbb{Q}$.

$\left\{ \begin{array}{rcrcrcrcl} x_1 & + & 2x_2 & + & 3x_3 & + & 4x_4 & = & 0 \\ 5x_1 & & & + & x_3 & + & 8x_4 & = & 0 \\ 2x_1 & + & 3x_2 & + & 7x_3 & & & = & 0 \end{array} \right.$

Let $A = \left[ \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 & 0 & 1 & 8 \\ 2 & 3 & 7 & 0 \end{array} \right]$.

Using Gauss-Jordan elimination, we see that the reduced row echelon form of $A$ is $B = \left[ \begin{array}{cccc} 1 & 0 & 0 & 13/6 \\ 0 & 1 & 0 & 31/6 \\ 0 & 0 & 1 & -17/6 \end{array} \right]$. In particular, note that $AX = 0$ and $BX = 0$ have the same solution set. So $x_4$ may be chosen arbitrarily, and then $x_1 = \frac{-13}{6}x_4$, $x_2 = \frac{-31}{6}x_4$, and $x_3 = \frac{17}{6}x_4$.