Discrete Math Understanding a proof involving the definition of divisibility In this first course on discrete mathematics, the instructor provided this following solution to a question. The question was asked us to prove the following (the solution is provided as well):
 
My question is where did the following expressions come from. It seems to be substitution, but I am not sure from where:
$a=2(2a+b)-(3a+2b)$
and
$b=2(3a+2b)-3(2a+b)$
Note: I have an understanding that $a|b$ can be written in the form $b=qa$, where $a,b,q$ are integers.
Edit: Please consider this is a first year course on discrete math, and I have no prior knowledge of linear algebra, etc.

Maybe this interpretation of the calculation will help. We know that $d$ divides $3a+2b$. Thus
$$3a+2b=ds\tag{1}$$
for some integer $s$. Similarly, 
$$2a+b=dt\tag{2}$$ 
for some integer $t$.  We have two equations in $a$ and $b$. Eliminate $b$ by multiplying the second equation through by $2$, and "subtracting" the first equation. We get
$$a=(2)(2a+b)-(3a+2b)=2dt-ds=d(2t-s),$$
and now it is clear that $d\mid a$.
