(((a − b))/((c − d))) = 3 (((a − c))/((b − d))) = 4 (((a − d))/((b − c))) = ?


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Question Number 28554 by naka3546 last updated on 27/Jan/18

$\frac{(a - b)}{(c - d)} = 3$ $\frac{(a - c)}{(b - d)} = 4$ $\frac{(a - d)}{(b - c)} = ?$
	Answered by ajfour last updated on 27/Jan/18

	$a - b = 3 (c - d) . . . (i)$ $a - c = 4 (b - d) . . . (i i)$ $(i i) - (i) :$ $\Rightarrow b - c = 3 (b - c) + b - d$ $o r 2 (b - c) = d - b . . . (I)$ $(i) - 3 (i i) :$ $- 2 a - b + 3 c = 3 c - 3 d - 12 b + 12 d$ $\Rightarrow - 2 (a - d) = - 11 (b - d) . . . (I I)$ $(I I) \div (I) :$ $\frac{a - d}{b - c} = - 11 .$
	Answered by mrW2 last updated on 27/Jan/18

	$a - b = 3 c - 3 d . . . (i)$ $a - c = 4 b - 4 d$ $a - 4 b = c - 4 d . . . (i i)$ $(i) - (i i)$ $3 b = 2 c + d$ $\Rightarrow b = \frac{2 c + d}{3}$ $\Rightarrow a = b + 3 c - 3 d = \frac{2 c + d + 9 c - 9 d}{3} = \frac{11 c - 8 d}{3}$ $a - d = \frac{11 c - 8 d - 3 d}{3} = \frac{11 (c - d)}{3}$ $b - c = \frac{2 c + d - 3 c}{3} = - \frac{(c - d)}{3}$ $\Rightarrow \frac{a - d}{b - c} = \frac{\frac{11 (c - d)}{3}}{- \frac{(c - d)}{3}} = - 11$
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