Is y(t) = Ax(t) + B linear

A system is linear if it obeys superposition: for inputs $x_1, x_2$ and constants $a, b$, the input $a x_1(t) + b x_2(t)$ must give the output $a y_1(t) + b y_2(t)$.

Given: $y(t) = A\,x(t) + B$

Outputs for two separate inputs:

$$\begin{aligned} y_1(t) &= A x_1(t) + B \\ y_2(t) &= A x_2(t) + B \end{aligned}$$

Weighted sum of the outputs:

$$a y_1(t) + b y_2(t) = A\left[a x_1(t) + b x_2(t)\right] + (a+b)B$$

Output for the combined input $x_3(t) = a x_1(t) + b x_2(t)$:

$$y_3(t) = A\left[a x_1(t) + b x_2(t)\right] + B$$

These two are equal only if $(a+b)B = B$ for all $a, b$, i.e. only when $B = 0$.

Zero-input test: for $x(t) = 0$, $y(t) = B \ne 0$. A linear system must give zero output for zero input, so the test fails.

Conclusion:

• If $B \ne 0$: the system is not linear. It is called an incrementally linear (affine) system: the change in output is linear in the change in input.
• If $B = 0$: $y(t) = A x(t)$ is linear.