Help finding integer solutions of equation.

Question

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.

Answer 1

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 $t$.