What exactly do we call a linear equation? I know there's a difference between the linear algebra "linear function" and the "affine function", 
but it confused me about what we consider a "linear equation" -precisely how many variables allowed, what order could a variable carry?
 A: Here are various comments on linear functions versus linear equations.
First there is the idea of a linear combination. Essentially, linear combination means a "sum of multiples" of something. For example
$$3x-2y+4z$$
is a linear combination of the variables $x,y$ and $z$, that is, a sum of multiples of $x,y$ and $z$.
If you set this expression equal to a constant value such as
$$3x-2y+4z=7$$
then you have a linear equation.
A linear function, on the other hand, is a function with the property that the function of a linear combination of two or more variables equals the same linear combination of the function of the two variables. This definition is difficult to understand without an example.
Let $f(x)=3x$. Then find $f(4a+7b)$
Solution:
$$f(4a+7b)=3(4a+7b)=12a+21b$$
Now look at the same linear combination of $f(a)$ and $f(b)$.
$$4f(a)+7f(b)=4(3a)+7(3b)=12a+21b$$
It is the same. That is
$$f(4a+7b)=4f(a)+7f(b)$$
This notion of linearity occurs in many areas of mathematics.
Notice, for example, that definite integrals are linear. For example
$$\int_0^15x-6x^2\,dx=5\int_0^1x\,dx-6\int_0^1x^2\,dx$$
Differentiation is also linear. For example
$$ \frac{d}{dx}\left(3\sin x+5e^x\right)=3\frac{d}{dx}\sin x+5\frac{d}{dx}e^x $$
Should, at some point, you study Laplace transforms you will find that they are linear transforms.
$$ \mathcal{L}\{a\cdot f(t)+b\cdot g(t)\}=a\cdot\mathcal{L}\{f(t)\}+b\cdot\mathcal{L}\{g(t)\} $$
