Determine whether the system is linear?
In the following, I suggest that, before using the generic $T(\alpha_1 x_1+\alpha_2 x_2)$ versus  $\alpha_1 T( x_1)+\alpha_2T( x_2)$, it can be more informative to try with simpler partial tests, or try counterexamples, based on your intuition.

I don't really understand the motivation behind the second group of equations, or why $b$ gets multiplied by $\alpha_i$.

Here, you are using the generic version of the linearity test, which is good, but can raise some doubts, as apparent from your question. You can, in complement, use other simpler tests, that can show that the system is not linear, as a double check. Those are counter-examples. For instance:


is the output of the zero signal zero? For a linear system ,  $T(\vec{0})=0$. 
is a single  output  with a zero coefficient  zero? This tests if $T(0.\vec{x})=0.T(\vec{x}) = 0$ 
is a single  output linear? This tests if $T(\alpha.\vec{x})=\alpha.T(\vec{x})$ 
is a simple addition   linear? This tests if $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$ 


Those, if not passed,  prove that the system is non-linear. And instead of using the generic version, they can show to your (clever) professor that you have some intuition about what is going on, and the risk of errors is reduced. I personally  appreciate a lot when students use minimal arguments: they go straight to the point, spend less time on such questions, to focus on more involved ones.

Of course, if the system is linear, more is required. 

Here, your system is non-linear... unless $b=0$.