A Diophantine equation with coefficients

Divide by $2$. $3x+5y=7$.

Particular solution. $x=4$, $y=-1$: $12-5=7$. So $(4,-1)$ works.

General solution. Homogeneous: $3x+5y=0$, so $x=5k$, $y=-3k$. Thus
$$(x,y)=(4+5k,\,-1-3k),\qquad k\in\mathbb{Z}.$$

Conclusion. All solutions: $(4+5k,-1-3k)$ for $k\in\mathbb{Z}$.