$$Ax=b$$

It is not possible to have more than one but less than infinitely many solutions for a particular $b$; if both $x$ and $y$ are solutions then:

$$z=\alpha x+(1-\alpha)y$$

is also a solution for any real $\alpha$.

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Can anyone explain this? How does the right side simplify to $z$? And intuitively, why is the $(1-\alpha)$ needed as it seems arbitrary to me? It's not like an inverse or anything.