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Question Number 156061 by mathlove last updated on 07/Oct/21

	Answered by TheSupreme last updated on 07/Oct/21

	$2 x y = 3 x + 2 y$ $x z = - 20 x + 12 z$ $2 y z = - 12 y + 15 z$ $y (2 z + 12) = 15 z$ $y = \frac{15 z}{2 z + 12}$ $x (z + 20) = 15 z$ $x = \frac{15 z}{z + 20}$ $2 (\frac{15 z}{z + 20}) (\frac{15 z}{2 z + 12}) = 15 z (\frac{3}{2 z + 12} + \frac{2}{z + 20})$ $30 z = 3 (z + 20) + 2 (2 z + 12)$ $30 z = 3 z + 60 + 4 z + 24$ $23 z = 84$ $z = \frac{84}{23} . . .$
	Answered by cortano last updated on 07/Oct/21
	${ ((((3x+2y)/(xy))=2⇒(3/y)+(2/x)=2)),((((5x−3z)/(xz))=−(1/4)⇒(5/z)−(3/x)=−(1/4))),((((4y−5z)/(yz))=−(2/3)⇒(4/z)−(5/y)=−(2/3))) :} { (((1/x)=a, (1/y)=b ,(1/z)=c)),((2a+3b=2 ; −3a+5c=−(1/4) ; −5b+4c=−(2/3))) :} ⇒ ((a),(b),(c) ) = ((( 2 3 0)),((−3 0 5)),(( 0 −5 4)) )^(−1) ((( 2)),((−(1/4))),((−(2/3))) )$
	${\begin{cases} \frac{3 x + 2 y}{xy} = 2 \Rightarrow \frac{3}{y} + \frac{2}{x} = 2 \\ \frac{5 x - 3 z}{xz} = - \frac{1}{4} \Rightarrow \frac{5}{z} - \frac{3}{x} = - \frac{1}{4} \\ \frac{4 y - 5 z}{yz} = - \frac{2}{3} \Rightarrow \frac{4}{z} - \frac{5}{y} = - \frac{2}{3} \end{cases}$ ${\begin{cases} \frac{1}{x} = a, \frac{1}{y} = b, \frac{1}{z} = c \\ 2 a + 3 b = 2; - 3 a + 5 c = - \frac{1}{4}; - 5 b + 4 c = - \frac{2}{3} \end{cases}$ $\Rightarrow (\begin{matrix} a \\ b \\ c \end{matrix}) = {(\begin{matrix} 2 3 0 \\ - 3 0 5 \\ 0 - 5 4 \end{matrix})}^{- 1} (\begin{matrix} 2 \\ - \frac{1}{4} \\ - \frac{2}{3} \end{matrix})$
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