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I'm confused on where the intuition came from to divide $w$ by the sum of its entries to find $q$. I don't really see the relation from the sum of its entries with "every solution being a multiple of the solution $w$".
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$\begingroup$In comments the user JMoravitz gives an answer. The solution to the question says that he wants to find the probability set of all solutions. So he finds a vector that solves the equation: $\vec{w} = (3,4)$ and then you normalize it. So the entries of $\vec{q}$.
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