<s>[INST] System of equations modulo $n$ Suppose we have the following system of equations module $n$
\begin{align*}
2x-y & \equiv 1\mod{n}\\
4x+3y & \equiv 2 \mod{n}
\end{align*}
Determine the integers $n$ for which the system has solutions.
I have posted my solution as an answer below.
 [/INST] Let $2x-y=1+an$ and $4x+3y=2+bn$ where $a,b$ are integers
$2+bn=4x+3(2x-1-an)\iff5(2x-1)=(3a+2b)n$
As $(3,2)=1, $ any integer can be expressed in terms of $3a+2b$
Let us find $a,b$ such that $3a+2b=5c\iff3(a-c)=2(c-b)$
$\dfrac{2(c-b)}3=a-c$ which is an integer
$\implies3|2(c-b)\iff3|(c-b),c-b=3d$ (say for some integer $d$) 
$\implies b=c-3d,a=c+2d$
$\implies2x-1=cn$ 
As the the left hand side is odd, the above equation will have a solution iff $n$ is odd
Please check for $y$
</s>