TITLE: The form of a solution in a linear system
QUESTION [5 upvotes]: 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)?

REPLY [3 votes]: just conveniently look at $z$ and you see 
$$z=\alpha=11t+7$$
If you substitute this for $\alpha$ in the first form, you get the second form. You may also invert this relationship between $t,\alpha$ to get
$$t = \frac{\alpha-7}{11} $$
If you substitute to the second form, you get the first form.