Is y[n] = ax[n] + b linear

A system is linear if it obeys superposition: $a_1x_1[n] + a_2x_2[n] \rightarrow a_1y_1[n] + a_2y_2[n]$.

Given: $y[n] = a\,x[n] + b$

Individual outputs: $y_1[n] = a\,x_1[n] + b$ and $y_2[n] = a\,x_2[n] + b$.

Weighted sum of outputs:

$$a_1y_1[n] + a_2y_2[n] = a\left(a_1x_1[n] + a_2x_2[n]\right) + (a_1 + a_2)\,b$$

Output for the combined input $x_3[n] = a_1x_1[n] + a_2x_2[n]$:

$$y_3[n] = a\left(a_1x_1[n] + a_2x_2[n]\right) + b$$

These are equal only if $(a_1 + a_2)b = b$ for all $a_1, a_2$, i.e. only if $b = 0$.

Zero-input check: $x[n] = 0$ gives $y[n] = b$. A linear system must give zero output for zero input.

Conclusion:

• For $b \ne 0$ the system is non-linear. It is incrementally linear: the difference between two outputs, $y_1 - y_2 = a(x_1 - x_2)$, is linear in the difference of inputs. It can be seen as a linear system $a\,x[n]$ plus a fixed zero-input response $b$.
• For $b = 0$, $y[n] = a\,x[n]$ is linear.