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 $x\mapsto ax+b$".