An affine set is defined as the set $\{\theta x+(1-\theta)y:\theta\in\Bbb R\}$ where $x,y\in\Bbb R^n$ are fixed. There is a claim that an affine set is the solution of some inhomogenous linear system $\bf{Ax}=b$.
I could not figure out why this is true. I searched the internet and was also not able to find an easy material to read since I am new to this concept. I would be grateful if someone could help.
Just an example
Consider in $\mathbb{R}^2$ $y=(1,0)$ and $x=(1,1).$ Then
$$\{\theta x+(1-\theta)y:\theta\in\Bbb R\}=\{y+\theta (x-y):\theta\in\Bbb R\}=\{(1,\theta):\theta\in\Bbb R\}.$$ This is just the line $x=1.$ It is just the solution of the system
$$\pmatrix{1& 0\\ 0 & 0}\pmatrix{x\\ y}=\pmatrix{1\\ 0}.$$