Paraphrasing, chapter 2.4 (Linear Dependence and Span), page 35 of Deep Learning by Goodfellow, Bengio, and Courville, claims something like the following:
If $\mathbf{x}$ and $\mathbf{y}$ are solutions to the system of equations $A\mathbf{x} = \mathbf{b}$, then $\mathbf{z} = \alpha \mathbf{x} + (1 - \alpha)\mathbf{y}$ is also a solution for any real $\alpha$.
I don't recall ever seeing this claim when studying linear algebra, and I would appreciate it if people could please take the time to inform me of the theorem and proof, so that I may understand it.
It's just that\begin{align}A\mathbf z&=A\bigl(\alpha\mathbf x+(1-\alpha)\mathbf y\bigr)\\&=\alpha A\mathbf x+(1-\alpha)A\mathbf y\\&=\bigl(\alpha+(1-\alpha)\bigr)\mathbf b\\&=\mathbf b.\end{align}