<s>[INST] Help finding integer solutions of equation. There are infinitely many integer solutions for the equation $4x + 6y = 8 $
My work:
$2x+3y=4$ and $2x=4-3y$ so $x=2-(\frac 32)y$
Similarly $y=\frac 43 - \frac 23 x$ are integer solutions of equation.
But correct answer is $x = −4 + 3t, y = 4 − 2t$ for all integers $t$. Help me understand where I have gone wrong.
 [/INST] There are infinitely many solutions to the equation $4x+6y=8$ since $(4,6) =2$ and $2\mid8$. We use Euclidean Algorithm to determine $m,n$ such that $4m+6n=2$. Here $m=-1,n=1$ and $8=2.4$.
Thus $x_0 = 4(-1)$ and $y_0=4.1=4$ is a particular solution. 
The solutions are given by $x=−4+3t,y=4−2t$ for all integers $t$.
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