How to find the minimum value of $ax+by+cz$ given $x,y,z$ are non negative integers that satisfy $x+2y+3z=d$? how to find the minimum value of the given expression $ax+by+cz$ if the constraints are $x,y,z$ are non negative integers that satisfy $x+2y+3z=d$, where $a,b,c,d$ all are positive integers and known beforehand.
Or if I substitute one variable from the equation the question can be also asked as
How to find minimum non negative value of $Ax+By+C$ given that $x,y$ are non negative integers and A,B are integers?
 A: Note that
$$
\left\{ \matrix{
  ax + by + cz = u \hfill \cr 
  x + 2y + 3z = d \hfill \cr}  \right.
$$
represents a line in 3D.
You want to find the minimum value of $u$ for which the line contains non-negative integer points.
A vector ($\bf t$) parallel  to the line is given by the cross product of the vectors normal to the planes
$$
{\bf t} = \left( {3b - 2c,\;c - 3a,\,\,2a - b} \right)
$$
and we can write the parametric equation of the line as
$$
\left\{ \matrix{
  x = d - 2{{u - ad} \over {b - 2a}} + \left( {3b - 2c} \right)s \hfill \cr 
  y = {{u - ad} \over {b - 2a}} + \left( {c - 3a} \right)s \hfill \cr 
  z = \left( {2a - b} \right)s \hfill \cr}  \right.
$$
i.e.
$$
\left\{ \matrix{
  s = {n \over {2a - b}}\quad \left| {\,n \in \mathbb N} \right. \hfill \cr 
  x = d + 2{{u - ad + \left( {3b - 2c} \right)n} \over {2a - b}} \hfill \cr 
  y = {{ - u + ad + \left( {c - 3a} \right)n} \over {2a - b}} \hfill \cr 
  z = n \hfill \cr}  \right.
$$
Since $a,b,c,d$ are positive integers, and assuming $0<2a-b$
$$
\left\{ \matrix{
  \left( {2a - b} \right)x = \left( {2a - b} \right)d + 2\left( {u - ad} \right) + 2\left( {3b - 2c} \right)n \hfill \cr 
  \left( {2a - b} \right)y =  - \left( {u - ad} \right) + \left( {c - 3a} \right)n \hfill \cr 
  z = n \hfill \cr}  \right.
$$
That means that, first of all, $u$ and $n$ must satisfy the congruence
relations
$$
\left\{ \matrix{
  v = u - ad \hfill \cr 
  \left( {2a - b} \right)d + 2v + 2\left( {3b - 2c} \right)n \equiv 0\quad \bmod \left( {2a - b} \right) \hfill \cr 
   - v + \left( {c - 3a} \right)n \equiv 0\quad \bmod \left( {2a - b} \right) \hfill \cr 
  z = n \hfill \cr}  \right.
$$
which upon some simplifications leads to
$$
\left\{ \matrix{
  \left( {2c - 3a} \right)n \equiv 0\quad \bmod \left( {2a - b} \right) \hfill \cr 
  v = u - ad \equiv \left( {c - 3a} \right)n \equiv 3cn\quad \bmod \left( {2a - b} \right) \hfill \cr 
  z = n \hfill \cr}  \right.
$$
Once the solutions to these are found, it will be the case to check them against the original equations
to properly take into account the signs, and verify that $x$ and $y$ are positive, and choose the minimum $u$ for that.
Example, in the case $a=b=c=d=1$, the congruences give
$$
\left\{ \matrix{
   - n \equiv 0\quad \bmod \left( 1 \right) \hfill \cr 
  v = u - 1 \equiv  - 2n \equiv 3n\quad \bmod \left( 1 \right) \hfill \cr 
  z = n \hfill \cr}  \right.\quad  \Rightarrow \quad \left\{ \matrix{
  0 \le z = n \in Z \hfill \cr 
  u \in Z \hfill \cr}  \right.
$$
and substituting back
$$
\left\{ \matrix{
  x = 2u + 2n - 1 \hfill \cr 
  y = 1 - u - 2n \hfill \cr 
  z = n \hfill \cr}  \right.\quad  \Rightarrow \quad \left\{ \matrix{
  0 \le x + y = u \hfill \cr 
  u \le 1 - 2n \hfill \cr 
  0 \le n \hfill \cr}  \right.\quad  \Rightarrow \quad \left\{ \matrix{
  u_{\,\min }  = 1 \hfill \cr 
  n = 0 \hfill \cr 
  x = 1,\;y = 0,z = 0 \hfill \cr}  \right.
$$
The other approach that you suggested is also viable
$$
\eqalign{
  & \left\{ \matrix{
  ax + by + cz = u \hfill \cr 
  x + 2y + 3z = d \hfill \cr}  \right.\quad  \Rightarrow \quad \left\{ \matrix{
  0 \le x = d - 2y - 3z \hfill \cr 
  \left( {2a - b} \right)y + \left( {3a - c} \right)z + \left( {ad - u} \right) = 0 \hfill \cr}  \right.\quad  \Rightarrow   \cr 
  &  \Rightarrow \quad \left\{ \matrix{
  0 \le x = d - 2y - 3z \hfill \cr 
  Ay + Bz + C = 0 \hfill \cr}  \right. \cr} 
$$
For the resolution of the 2D diophantine equation in the 2nd line, please refer to this related post.
The solutions to it shall then respect the inequality in the 1st line.
From the cited post you get that your problem is, in general, related to:
  - the Modular Inverse operation;
  - the Popoviciu's Theorem;
  - the Frobenius Coin Exchange problem.
and that, for some particular values of $a,b,c,d$, although non-negative, you might even have 
no solution, for whichever value of $u$.
