Check if the system is linear The system:
 $$ T(x[n]) = ax[n] + bx[n-3] $$

For me it seems that the system is linear:

$$
\begin{align} 
T(\alpha_1x_1[n]+\alpha_2x_2[n]) &amp; = a(\alpha_1x_1[n]+\alpha_2x_2[n]) + b(\alpha_1x_1[n-3]+\alpha_2x_2[n-3]) \\
&amp; = \alpha_1(ax_1[n] + bx_1[n-3]) + \alpha_2(ax_2[n] + bx_2[n-3]) \\
&amp; = \alpha_1T(x_1[n])+\alpha_1T(x_2[n])
\end{align}$$

Thus it's linear, however in the presentation I got it says it's not linear. (without reasoning) Where I'm making the mistake (or maybe there is a mistake in the presentation)
I believe there's either a mistake in the presentation or the presentation is using a different definition of linear.

For example, the system is linear in $x$ from a system perspective, but it's affine in $x[n]$ (and, therefore not linear) because of the $bx[n-3]$ offset.

On this site, we tend to go with the system definition rather than split hairs about linearity versus affine-ness.