confused in linear property?

Question

I have a system

$$y(t)=3x(t)+2\cos(\pi t/3)$$

I am confused if this function/system is linear or not? As if only we had $y=3x$, it would be definitely linear but now due to cos term, scenario complicates?

Answer 1

A linear system is one in which two things happen

1) Scaling the input by $a$, scales the output by $a$

2) The the output of the sum of two inputs to the system is the same as adding the outputs of each input individually

This is not the same as saying the graph is a line. For example, $y=ax+b$ (where $x$ is the input) is not a linear system (more specifically it is affine). Doubling the input does not double the output (i.e. $2y = 2ax + 2b \ne a(2x) + b$). In your system, doubling the input also does not double the output, so this is not a linear system

Answer 2

It would be linear as a function of $x$, but not as a function of $t$. I.e.: if you assume $x$ is the variable and $t$ a constant $y(x)$ is linear, because is of the form $y(x)=ax+b$, where $a=3$ and $b=2\cos(\pi t/3)$. However, $y(t)$ with respect to $t$ doesn't satisfy this property.