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 May 3rd, 2019, 11:03 PM #1 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 894 Thanks: 61 Math Focus: सामान्य गणित solve for x and y $\displaystyle x^{3}y =1$ and $\displaystyle xy^{3}=1$ What are the solution sets?
 May 4th, 2019, 02:24 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,746 Thanks: 2133 Assuming that $x$ is real, $(x,\,y) = (1,\,1)$ or $(-1,\,-1)$. What did you mean by "solution sets"?
May 4th, 2019, 07:34 AM   #3
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Quote:
 Originally Posted by MATHEMATICIAN $\displaystyle x^{3}y =1$ and $\displaystyle xy^{3}=1$ What are the solution sets?
Solving for y the first equation says
$\displaystyle y = \dfrac{1}{x^3}$

Putting this into the second equation:
$\displaystyle x \left ( \dfrac{1}{x^3} \right )^3 = 1$

$\displaystyle \dfrac{1}{x^8} = 1$

Solving this gives the two real solutions 1 and -1.

-Dan

 May 5th, 2019, 03:30 AM #4 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 894 Thanks: 61 Math Focus: सामान्य गणित What might be the other solution beside those real solutuons?
 May 5th, 2019, 04:07 AM #5 Global Moderator   Joined: Dec 2006 Posts: 20,746 Thanks: 2133 $(x,\,y) = (x,\,1/x^3)$, where $x = \pm i$ or $\pm\sqrt{\pm i}$. Thanks from topsquark

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