\(5x+y⋮19\\ 7\left(5x+y\right)⋮19\\ 35x+7y⋮19\\ 16x-12y+19x+19y⋮19\\ 4\left(4x-3y\right)+19\left(x+y\right)⋮19\\ \left\{{}\begin{matrix}35x+17⋮19\\19\left(x+y\right)⋮19\\4\left(4x-3y\right)+19\left(x+y\right)=35x+17y\end{matrix}\right.\Rightarrow4\left(4x-3y\right)⋮19\\ \left(4,19\right)=1\Rightarrow4x-3y⋮19\)