Help me understand linear function. today in my high school lesson I learned for linear function. I know that each linear function has the form of $f(x)=kx+n$ where $k,x,n ∈ \mathbb R$. Now let's say be have those two functions:
$f_1(x)=kx+n_1\\
f_2(x)=kx+ n_2$
Since $k$ is same in both functions, we know that it represents those function whose graph will be parallel, but why and how to prove that those functions will be parallel?

At a high school level, I would say that a fair definition for parallel would be saying that the distance (i.e., the difference) between the lines remains constant. This is straightforward to prove:
$f_1(x)-f_2(x)=(kx + n_1) - (kx + n_2) = (k-k)x + (n_1-n_2) = n_1-n_2$,
therefore the difference $f_1(x)-f_2(x)$ does not depend on $x$, i.e. it is the same for all $x \in \mathbb{R}$.
Comment 1: Precisely speaking, a linear function is a function of the form $f(x)=kx$. Functions of the form you defined ($f(x)=kx + b$) are better called affine functions. This is because linearity is a well defined term in mathematics which implies (among other things) that $f(0)=0$, i.e. that the function passes through the origin.
Comment 2: You used the letter $n$ to denote a real number, which is not wrong, but it's very unusual. Usually, the letter $n$ is reserved to denote integer numbers or even natural numbers only (i.e. non-negative integer numbers). Unless you cannot really think of any other symbol to denote a real number, do not use the letter $n$, to avoid inducing confusion on the reader.
