General solution using Euclidean Algorithm  
I was able to come up with the integer solution that they also have in the textbook using the same method they used but I am really puzzled how they come up with a solution for all the possible integer combinations...how do they come up with that notation/equation that represents all the integer solutions ? I am talking about the very last line.

A general rule in life: When you have a linear equation(s) of the form $f(x_1,x_2,\dots, x_n)=c$, find one solution to the equation and then find a general solution to $f(x_1,\dots,x_n)=0$ and now you can obtain the general solution for the initial equation by adding the special solution you found with the general solution of the second, homogeneous, equation.
In our case the homogeneous equation is $957x+609y=0$. Divide by the gcd 87 to obtain $11x=-7y$. So the general solution for this equation is $(-7n,11n)$ for integer $n$ (you must multiply both sides by something that will give you the LCM times an integer).
