elementary number theory - Linear Diophantine Equations in Three Variables

$$3x+6y+5z=7$$
The general solution to this linear Diophantine equation is as described
here (Page 7-8) is:

$$x = 5k+2l+14$$
$$y = -l$$
$$z = -7-k$$
$$k,l \in \mathbb{Z}$$

If I plug the original equation into Wolframalpha the solution is:
$$y = 5n+2x+2$$
$$z =-6n-3x-1$$
$$n \in \mathbb{Z}$$

I can rewrite this as:

$$x = l$$
$$y = 5k+2l+2$$
$$z = -6k-3l-1$$

$$k,l \in \mathbb{Z}$$

However now two equations depend on two variables ($k,l$) and one on one variable $l$.
In the first solution one equation depends on two variables and two on one variable.

Questions:

How can I come from a representation like the one from wolfram alpha for the general solution to one where all equations depend on one distinct variable except one equation.

Is there always such a representation?