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?