## viniterranova 3 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).

1. sriramkumar

$x ^{y}=1\div y ^{2} and y ^{x}=1/\sqrt{x}$ u mean??

2. estudier

From the second one, can I say that y^2 = 1/sqrt2...

3. viniterranova

ok. Keep going.

4. viniterranova

It´s right.

5. estudier

So that gives you y and you can sub in the first to get x...

6. viniterranova

I solved this question before, but i can´t remember all the steps.

7. viniterranova

May you show me, how?

8. ash2326

@viniterranova Let's continue, would you mind showing me your work?

9. ujjwal

From the second i can say that y=1

10. viniterranova

11. viniterranova

So, if x^y=y I got the following expression. y= 1/y^2.

12. ujjwal

I 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!!

13. ash2326

I 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

14. viniterranova

Thanks ash2326.

15. viniterranova

If I mad x^y=y, and rewriting the expressoin, i can got y-1/y^2=0. What do they think?

16. ash2326

How could you use the same variable "y"? it's defined differently. I have something. Let me show you

17. viniterranova

ok.

18. ash2326

$x^y=\frac{1}{y^2}$ $y^x=\frac{1}{\sqrt x}$ 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 $x=\frac 12$

19. ash2326

@viniterranova do you get this?

20. mukushla

$y\times ( -\log y)=-2\log y$gives y=1 and y=2 ------

21. ash2326

Yes @mukushla I missed that then y=1 and x=1

22. mukushla

neat sol

23. viniterranova

Great.

24. viniterranova

It´s rigth.

25. viniterranova

I got y=1 and x= 1 in another way.

26. mark_o.

i think x=1/2 and y=2 from xy=1 if x=1/2 then y/2=1 y=2 ans

27. mark_o.

from 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 ....

28. viniterranova

Thanks everyone for the help.

29. mark_o.

ok yw, have fun....