Consider a system that is described by the linear equation: $y = 2x + 1$. Is this system linear or non-linear? From what I can tell, although the system is described by a linear equation is not linear, because the additivity property of the superposition principle does not stand.
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See also: https://math.stackexchange.com/q/2054117/9464 – Jun 18 '17 at 20:48
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It's an inhomogenous linear equation. Maybe that helps you.
But you're right in the sense, that $x\mapsto2x+1$ is not a linear map.
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A linear system is of the form $A{\bf x}=b$ where $A$ is some matrix and $b$ is the known vector. The linearity is from $A$. Note that $y=2x+1$ is the same as $2x-y=-1$.