I'm solving this equation in integers: $6 \, x + 10 \, y + 15 \, z = 1$. I can find many solutions, by combining two two-variable equations into $x = u \, p$, $y = v \, p$, $z = q$, where $u = 2 + 5 \, t$, $v = -1 - 3 \, t$, and $p = 8 + 15 \, s$, $q = -1 - 2 \, s$, with $s$ and $t$ integers. (see ) But I'm obviously missing some solutions. How does Mathematica arrive at this elegant general solution?
$$ 3x+6y+5z=7 $$ The general solution to this linear Diophantine equation is as described    $$ x = 5k+2l+14 $$ $$ y = -l $$ $$ z = -7-k $$ $$ k,l \in \mathbb{Z} $$  If I plug the original equation into  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?
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