---
title: "Is y(t) = Ax(t) + B linear"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:linearity-affine-ct"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/linearity-affine-ct
site: BCT Engineering
---

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

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2075 Bhadra, 4 marks
- Video: [Linear and Non-Linear Systems (Solved Problems) | Part 1](https://www.youtube.com/watch?v=QcAY1oPozhw) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/linearity-affine-ct
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Determine whether the given system is linear or not. y(t) = A x(t) + B

## Short answer

System y=Ax+B is linear only if B=0. For B≠0 it fails superposition and zero-input test, so it is affine or incrementally linear, not linear.

## Answer

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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Superposition requires a x1+b x2 give:
   A. y1+y2 only
   B. a b y1 y2
   C. a y1 + b y2
   D. y1 - y2
   Answer: C. Both additivity and homogeneity. (Key point: Superposition)

2. For y=Ax+B, the weighted sum of outputs a y1+b y2 equals:
   A. A(a x1+b x2)+B
   B. A(a x1+b x2)
   C. A(a+b)x+B
   D. A(a x1+b x2)+(a+b)B
   Answer: D. The constant B is counted a+b times. (Key point: Test)

3. With x(t)=0, y(t) = ?
   A. B, so for B≠0 the system is not linear
   B. 0
   C. A
   D. AB
   Answer: A. A linear system gives zero for zero input. (Key point: Zero input)

4. y=Ax+B with B≠0 is called:
   A. Time-variant
   B. Incrementally linear (affine)
   C. Non-causal
   D. Linear
   Answer: B. Change in output is linear in change in input. (Key point: Name)

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