viniterranova
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
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\[x ^{y}=1\div y ^{2} and y ^{x}=1/\sqrt{x} \] u mean??
estudier
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From the second one, can I say that y^2 = 1/sqrt2...
viniterranova
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ok. Keep going.
viniterranova
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It´s right.
estudier
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So that gives you y and you can sub in the first to get x...
viniterranova
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I solved this question before, but i can´t remember all the steps.
viniterranova
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May you show me, how?
ash2326
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@viniterranova Let's continue, would you mind showing me your work?
ujjwal
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From the second i can say that y=1
viniterranova
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I mad x^y = y.
viniterranova
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So, if x^y=y I got the following expression. y= 1/y^2.
ujjwal
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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!!
ash2326
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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
viniterranova
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Thanks ash2326.
viniterranova
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If I mad x^y=y, and rewriting the expressoin, i can got y-1/y^2=0. What do they think?
ash2326
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How could you use the same variable "y"?
it's defined differently. I have something. Let me show you
viniterranova
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ok.
ash2326
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\[x^y=\frac{1}{y^2}\]
\[y^x=\frac{1}{\sqrt x}\]
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
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@viniterranova do you get this?
mukushla
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\[y\times ( -\log y)=-2\log y\]gives y=1 and y=2
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ash2326
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Yes @mukushla I missed that
then y=1 and x=1
mukushla
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neat sol
viniterranova
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Great.
viniterranova
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It´s rigth.
viniterranova
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I got y=1 and x= 1 in another way.
mark_o.
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i think x=1/2 and y=2
from xy=1
if x=1/2 then
y/2=1
y=2 ans
mark_o.
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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 ....
viniterranova
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Thanks everyone for the help.
mark_o.
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ok yw, have fun....