I have this linear system:
$\left\{\begin{array}{c} 2x + 3y - 4z = \ 1 \\ 3x - y - 2z = 2 \\ x - 7y - 6z = 0 \end{array}\right.$
I found the following solution:
$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \alpha \frac{10}{11}+\frac{7}{11} \\ \alpha \frac{8}{11}-\frac{1}{11} \\ \alpha \end{pmatrix} $
but the correct solution is
$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 10t+7\\ 8t+5 \\ 11t+7 \end{pmatrix} $
I know that the two solution are equivalent (and correct). Assuming that i don't know the second form, how i can transform the first form into the last form (with only integer coefficents)?