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Given that $p=\left(\begin{array}{c}5 \\ 3\end{array}\right), q=\left(\begin{array}{c}-1 \\ 2\end{array}\right)$ and $r=\left(\begin{array}{c}17 \\ 5\end{array}\right)$ and $r=\alpha r+\beta q$, where $\alpha$ and $\beta$ are scalars, express $q$ in terms of $r$ and $p$.
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$p=\left(\begin{array}{l}5 \\ 3\end{array}\right) ; q=\left(\begin{array}{c}-1 \\ 2\end{array}\right) ; r=\left(\begin{array}{c}17 \\ 5\end{array}\right)$
$r=\alpha p+\beta q$
$\left(\begin{array}{c}17 \\ 5\end{array}\right)=\alpha\left(\left(\begin{array}{l}5 \\ 3\end{array}\right)\right)+\beta\left(\left(\begin{array}{c}-1 \\ 2\end{array}\right)\right.$
$17=5 \alpha-\beta \ldots(1) ; 5=3 \alpha+2 \beta \ldots(2)$
$(1) \times 2: 34=10 \alpha-2 \beta \ldots(3)$
$(3)+(2): 39=13 \alpha \Longrightarrow \alpha=3$
$5=3(3)+2 \beta \Longrightarrow 2 \beta=5-9=-4$
$\beta=-2$
$r=3 p-2 q \Longrightarrow 2 q=3 p-r$
$q=\frac{1}{2}(3 p-r)$
by Diamond (89,036 points)

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