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 2 years ago
Find the value of x and y in this system of equations. x^y=1/y^2 and y^x=1/sqrt(x).
 2 years ago
Find the value of x and y in this system of equations. x^y=1/y^2 and y^x=1/sqrt(x).

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sriramkumar
 2 years ago
Best ResponseYou've already chosen the best response.0\[x ^{y}=1\div y ^{2} and y ^{x}=1/\sqrt{x} \] u mean??

estudier
 2 years ago
Best ResponseYou've already chosen the best response.0From the second one, can I say that y^2 = 1/sqrt2...

estudier
 2 years ago
Best ResponseYou've already chosen the best response.0So that gives you y and you can sub in the first to get x...

viniterranova
 2 years ago
Best ResponseYou've already chosen the best response.0I solved this question before, but i can´t remember all the steps.

viniterranova
 2 years ago
Best ResponseYou've already chosen the best response.0May you show me, how?

ash2326
 2 years ago
Best ResponseYou've already chosen the best response.2@viniterranova Let's continue, would you mind showing me your work?

ujjwal
 2 years ago
Best ResponseYou've already chosen the best response.0From the second i can say that y=1

viniterranova
 2 years ago
Best ResponseYou've already chosen the best response.0So, if x^y=y I got the following expression. y= 1/y^2.

ujjwal
 2 years ago
Best ResponseYou've already chosen the best response.0I don't know how you got these.. But i am sure they (your above expressions) will still give correct answer since i get y=1 from 2nd expression!!

ash2326
 2 years ago
Best ResponseYou've already chosen the best response.2I know you are substituting but don't use the same variable \[x^y=y\] This will be troublesome, use another variable, if you want to but I don't see how it'd help @viniterranova

viniterranova
 2 years ago
Best ResponseYou've already chosen the best response.0If I mad x^y=y, and rewriting the expressoin, i can got y1/y^2=0. What do they think?

ash2326
 2 years ago
Best ResponseYou've already chosen the best response.2How could you use the same variable "y"? it's defined differently. I have something. Let me show you

ash2326
 2 years ago
Best ResponseYou've already chosen the best response.2\[x^y=\frac{1}{y^2}\] \[y^x=\frac{1}{\sqrt x}\] Take log both sides, for both the equations \[y\log x=2\log y\] \[x\log y=\frac 1 2\log x\] multiply both the equations, we would get \[xy\log x \times \log y=\log x \times\log y\] or \[xy=1\] We have the equation \[y\log x=2\log y\] put x=1/y \[y\log{\frac 1y}=2\log y\] we get \[y\times ( \log y)=2\log y\] \[y=2\] so \[x=\frac 12\]

ash2326
 2 years ago
Best ResponseYou've already chosen the best response.2@viniterranova do you get this?

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.1\[y\times ( \log y)=2\log y\]gives y=1 and y=2 

ash2326
 2 years ago
Best ResponseYou've already chosen the best response.2Yes @mukushla I missed that then y=1 and x=1

viniterranova
 2 years ago
Best ResponseYou've already chosen the best response.0I got y=1 and x= 1 in another way.

mark_o.
 2 years ago
Best ResponseYou've already chosen the best response.0i think x=1/2 and y=2 from xy=1 if x=1/2 then y/2=1 y=2 ans

mark_o.
 2 years ago
Best ResponseYou've already chosen the best response.0from y log x =2log y sub x=1/y y log(1/y)=2log y y(log y)=2log y dividing both sides by log y gives y=2log y/(log y) y=2 ans..... and from xy=1 2x=1 then x=1/2 ans ....

viniterranova
 2 years ago
Best ResponseYou've already chosen the best response.0Thanks everyone for the help.
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