Suppose that bar{p}, bar{q} and bar{r} are th | vedclass

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(C) Given that $\bar{s} = 4\bar{p} + 3\bar{q} + 5\bar{r}$.Also,$\bar{s} = x(-\bar{p} + \bar{q} + \bar{r}) + y(\bar{p} - \bar{q} + \bar{r}) + z(-\bar{p

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$9$

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(C) Given that $\bar{s} = 4\bar{p} + 3\bar{q} + 5\bar{r}$.Also,$\bar{s} = x(-\bar{p} + \bar{q} + \bar{r}) + y(\bar{p} - \bar{q} + \bar{r}) + z(-\bar{p} - \bar{q} + \bar{r})$.Expanding the right side,we get $\bar{s} = (-x + y - z)\bar{p} + (x - y - z)\bar{q} + (x + y + z)\bar{r}$.Comparing the coefficients of $\bar{p}, \bar{q}, \bar{r}$ with the given expression for $\bar{s}$,we have:$-x + y - z = 4$ $(i)$$x - y - z = 3$ $(ii)$$x + y + z = 5$ $(iii)$Adding $(ii)$ and $(iii)$,we get $2x = 8 \implies x = 4$.Adding $(i)$ and $(ii)$,we get $-2z = 7 \implies z = -\frac{7}{2}$.Substituting $x=4$ and $z=-\frac{7}{2}$ into $(iii)$,we get $4 + y - \frac{7}{2} = 5 \implies y = 1 + \frac{7}{2} = \frac{9}{2}$.Now,calculate $2x + y + z = 2(4) + \frac{9}{2} - \frac{7}{2} = 8 + \frac{2}{2} = 8 + 1 = 9$.

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