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Find the value of $p$ for which the following system of linear equations has infinitely many solutions :
$x + (p + 1)y = 5; (p + 1)x + 9y = 8p - 1$
$x + (p + 1)y = 5; (p + 1)x + 9y = 8p - 1$
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For infinitely many solutions, we have
$\frac{1}{p+1} = \frac{p+1}{9} = \frac{5}{8p-1}$
$\implies (p+1)^2 = 9$ and $5(p+1) = 8p - 1$
$p+1 = \pm 3 \implies p = 2, -4 \dots (i)$
Also, $5p + 5 = 8p - 1 \implies p = 2 \dots (ii)$
From $(i)$ and $(ii)$, $p = 2$
$\frac{1}{p+1} = \frac{p+1}{9} = \frac{5}{8p-1}$
$\implies (p+1)^2 = 9$ and $5(p+1) = 8p - 1$
$p+1 = \pm 3 \implies p = 2, -4 \dots (i)$
Also, $5p + 5 = 8p - 1 \implies p = 2 \dots (ii)$
From $(i)$ and $(ii)$, $p = 2$