---
title: "Is y[n] = 2x[n] + 5 linear"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:linearity-2x-plus-5"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/linearity-2x-plus-5
site: BCT Engineering
---

# Is y[n] = 2x[n] + 5 linear

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2076 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-2x-plus-5
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Check whether the discrete-time system described by the following input-output relation is linear or non-linear. y[n] = 2 × x[n] + 5

## Short answer

The system y[n] = 2x[n] + 5 is non‑linear because zero input yields output 5, and superposition does not hold; it is incrementally linear.

## Answer

A system is **linear** if it satisfies superposition (additivity and homogeneity):

$$
a_1x_1[n]+a_2x_2[n]\ \rightarrow\ a_1y_1[n]+a_2y_2[n]
$$

#### Test
Let $x_1[n]\rightarrow y_1[n]=2x_1[n]+5$ and $x_2[n]\rightarrow y_2[n]=2x_2[n]+5$.

Response to the combined input $x_3=a_1x_1+a_2x_2$:

$$
y_3[n]=2\big(a_1x_1[n]+a_2x_2[n]\big)+5=2a_1x_1[n]+2a_2x_2[n]+5
$$

Weighted sum of the individual outputs:

$$
a_1y_1[n]+a_2y_2[n]=2a_1x_1[n]+2a_2x_2[n]+5(a_1+a_2)
$$

These are equal only if $a_1+a_2=1$, not for all constants. So $y_3[n]\ne a_1y_1[n]+a_2y_2[n]$ in general.

**Quick check (zero-input test):** for a linear system, zero input gives zero output. Here $x[n]=0$ gives $y[n]=5\ne0$.

Numerical example: $x[n]=1\rightarrow y=7$; doubling the input, $x[n]=2\rightarrow y=9\ne 14$.

**Answer: the system $y[n]=2x[n]+5$ is non-linear.** It is called **incrementally linear**: it is a linear system ($2x[n]$) plus a constant offset (5), so differences of outputs do respond linearly to differences of inputs. (It is memoryless, causal, time-invariant and stable.)

## Check yourself

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

1. Which condition must hold for a system to be linear?
   A. a1x1+a2x2 gives a1y1+a2y2 for all inputs and constants
   B. Shifting the input shifts the output by the same amount
   C. Bounded input always gives bounded output
   D. Output depends only on present and past inputs
   Answer: A. Linearity means additivity plus homogeneity, i.e. superposition. (Key point: Superposition)

2. What does x[n]=0 give for y[n]=2x[n]+5, and what does that show?
   A. y=0, so the system is linear
   B. y=5, which is not 0, so the system is not linear
   C. y=5, which shows the system is time-varying
   D. y=2, which shows the system has memory
   Answer: B. A linear system must give zero output for zero input; here the output is 5. (Key point: Zero-input test)

3. In the superposition test the weighted sum of outputs contains 5(a1+a2) while the response to the combined input contains only 5. When do these agree?
   A. Always, for any a1 and a2
   B. Only when a1=a2=0
   C. Only when a1+a2=1, not for all constants
   D. Only when x1[n]=x2[n]
   Answer: C. The constant offset gets scaled by (a1+a2) in one case and not in the other, so they match only if a1+a2=1. (Key point: Why superposition fails)

4. For y[n]=2x[n]+5, x=1 gives y=7. What does x=2 give, and what does this show?
   A. y=14, so homogeneity holds
   B. y=9, so the system is time-varying
   C. y=12, so the system is incrementally linear
   D. y=9, not 14, so homogeneity fails
   Answer: D. Doubling the input should double the output (14) for a linear system, but the output is 9. (Key point: Numerical check)

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