Is the general equation for a straight line not considered a linear function in linear algebra? Is the general equation for a straight line, which we called a linear function in highschool, i.e.
$$f(x)=mx+c \tag{1}$$
not considered to be a linear function according to the linear algebra definition that it need to satisfy
$$f(\alpha x_{1} + \beta x_{2}) = \alpha f(x_{1}) + \beta f(x_{2}) \tag{2}$$
since the two sides of equation $(2)$ would differ by the constant term if using equation $(1)$ as the definition of $f$? As far as I understand the first equation I posted would be considered an affine function but not a linear function in linear algebra (right?).
Are there just (annoyingly) two completely different definitions of what a linear function is or is there a reason that these two definitions have the same name (i.e. are the concepts related)?

You are correct, the term linear is sligtly misused in high school, when affine functions from $\mathbb R$ to $\mathbb R$ are also called linear.
That said, two points that make this a little less annoying:


*

*"linear" in this term simply means "forming a line", and you can't argue with the fact that $f(x)=kx+n$ forms a line in $\mathbb R^2$...

*Technically, you could say that functions are always mappings from some set to the field $\mathbb R$, and you could then maybe get away with it by saying that we have one definition for linear functions (that of $y=kx+n$), and another for linear mappings (the one with additivity). It's stretching it a bit, I know.

