In this video, the two variable equation $(1 - \alpha - \frac{\gamma}{2})\vec A + (\frac{\alpha}{2} - \frac{\gamma}{2})\vec B + (-1 + \frac{\alpha}{2} + \gamma)\vec C = \vec 0$ is solved by introducing four additional variables, $a$, $b$, $c$, and $\beta$. This seems like overkill, and I'm confused about why it even helps. The idea is that each coefficient is proportional with constant $\beta$ to $a$, $b$, $c$, respectively, but this move feels like renaming the coefficient expressions without adding new information.
It is noted in the video that if we were to make the false assumption that each coefficient must equal $0$ in order for the entire linear combination to equal $0$, then we would have three equations and two variables, but would get "lucky" and find the solution $(\frac{2}{3}, \frac{2}{3})$, which is in fact the answer that needs to be proven.
Is this "luckiness" of finding a solution to the overdetermined system secretly the reason that an answer is obtainable by the professor's ultimate solution method, or does introducing $a$, $b$, $c$, and $\beta$ actually add something needed for the proof?
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