Proving an affine set's equivalence to the solution set of $Ax=b$

I am stuck with the following equivalence about Affine Sets: “$L$ being an affine set is equivalent to $L$ being the solution set of a set of equations $Ax=b$ for some $A,b$.”

In a more mathematical statement:

$L$ is an affine set $\iff$ $L=\left\{x|Ax=b\right\}$ where $x \in \mathbb{R}^n$, for some matrix $A$ and $b$.

Proving that $L=\left\{x|Ax=b\right\}$ implies an affine set is easy, I pick $x,y \in L$. Then using the definition of affinity, such that $\alpha x + \beta y \in L$ with $\alpha + \beta = 1$, it is $A(\alpha x + \beta y)=\alpha A x + \beta A y=b$.

Proving the other direction of the equivalence is where I run into trouble. I have the definition of $L$ with $(\alpha x + \beta y) \in L$ where $\alpha + \beta =1$ and $x,y \in L$. But I don’t see how I can use this to reach $L=\left\{x|Ax=b\right\}$ to begin with. How should I go on from here?

Solutions Collecting From Web of "Proving an affine set's equivalence to the solution set of $Ax=b$"