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Question Number 100976 by  M±th+et+s last updated on 29/Jun/20

find all possible values of x,y,z in terms of a,b,c gor a triplet (x,y,z) that satisfy x+(1/y)=a y+(1/z)=b z+(1/x)=c

findallpossiblevaluesofx,y,zintermsofa,b,cgoratriplet(x,y,z)thatsatisfyx+1y=ay+1z=bz+1x=c

Commented by john santu last updated on 29/Jun/20

⇒x = ((ay−1)/y) ∧ y = ((bz−1)/z) ⇒ z+ (y/(ay−1)) = c ∧ y = ((bz−1)/z) ⇒ z + (((bz−1)/z)/(a(((bz−1)/z))−1)) = c z + ((bz−1)/(abz−a−z)) = c z + ((bz−1)/(z(ab−1)−a)) = c z(z(ab−1)−a)+bz−1 = c(z(ab−1)−a) (ab−1)z^2 +(b−a)z−1=(abc−c)z−ac (ab−1)z^2 +(b−a+c−abc)z+ac−1=0 z = ((abc+a−b−c ± (√((b−a+c−abc)^2 −4(ac−1)(ab−1))))/(2(ab−1)))

⇒x=ay−1y∧y=bz−1z⇒z+yay−1=c∧y=bz−1z⇒z+bz−1za(bz−1z)−1=cz+bz−1abz−a−z=cz+bz−1z(ab−1)−a=cz(z(ab−1)−a)+bz−1=c(z(ab−1)−a)(ab−1)z2+(b−a)z−1=(abc−c)z−ac(ab−1)z2+(b−a+c−abc)z+ac−1=0z=abc+a−b−c±(b−a+c−abc)2−4(ac−1)(ab−1)2(ab−1)

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