Is calling a linear-equation a linear-function, misnomer or completely wrong? From my college life, I remember many professors used to call a linear-equation a linear-function, however:
A standard definition of linear function (or linear map) is:
$$f(x+y)=f(x)+f(y),$$
$$f(\alpha x)=\alpha f(x).$$
Where as linear equation is defined as:
$$f(x)=mx+b.$$
So, linear-equation is NOT a linear-function, according to the definitions defined above.
Though, for $b=0$ the linear-equation becomes a linear-function, but it is not true in general.
Question: Is it misnomer to call a linear-equation a linear-function, or it is completely wrong to say that? And linear-equation must be considered strictly as an affine-mapping. 


Is it misnomer to call a linear-equation a linear-function, or it is completely wrong to say that?

Neither. The meaning of some mathematical terms (normal, regular, smooth, etc) is context-dependent. E.g., smooth function means $C^\infty$ in some papers/books and $C^1$ in others. 
In certain contexts, linear means "additive and commutes with scalar multiplication". In other contexts, it means "a function of the form $x\mapsto ax+b$". 
