$T(x[n]) = ax[n] + b$

$ T(\alpha_{1}x_{1}[n] + \alpha_{2}x_{2}[n]) = \alpha_{1}ax_{1}[n]+b + \alpha_{2}ax_{2} [n] +b $

$ \alpha_{1}T(x_{1}[n] ) + \alpha_{2}T(x_{2}[n]) = \alpha_{1}(ax[n]+b) + \alpha_{2}(ax_{2} [n] +b) $

therefore they are not equal so the system is non-linear?

or would it be something like this:

$ T(\alpha_{1}x_{1}[n] + \alpha_{2}x_{2}[n]) = \alpha_{1}ax_{1}[n]+\alpha_{1}b + \alpha_{2}ax_{2} [n] + \alpha_{2}b $

$ \alpha_{1}T(x_{1}[n] ) + \alpha_{2}T(x_{2}[n]) = \alpha_{1}(ax[n]+b) + \alpha_{2}(ax_{2} [n] +b) $

and therefore they are equal so the system is linear?

$T(x[n]) = ax[n] + b$

$ T(\alpha_{1}x_{1}[n] + \alpha_{2}x_{2}[n]) = \alpha_{1}ax_{1}[n]+ \alpha_{2}ax_{2} [n] +b $

$ \alpha_{1}T(x_{1}[n] ) + \alpha_{2}T(x_{2}[n]) = \alpha_{1}(ax[n]+b) + \alpha_{2}(ax_{2} [n] +b) $

therefore they are not equal so the system is non-linear?

or would it be something like this:

$ T(\alpha_{1}x_{1}[n] + \alpha_{2}x_{2}[n]) = \alpha_{1}ax_{1}[n]+\alpha_{1}b + \alpha_{2}ax_{2} [n] + \alpha_{2}b $

$ \alpha_{1}T(x_{1}[n] ) + \alpha_{2}T(x_{2}[n]) = \alpha_{1}(ax[n]+b) + \alpha_{2}(ax_{2} [n] +b) $

and therefore they are equal so the system is linear?