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May 3rd, 2019, 11:03 PM   #1
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solve for x and y

$\displaystyle x^{3}y =1$ and $\displaystyle xy^{3}=1$

What are the solution sets?
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May 4th, 2019, 02:24 AM   #2
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Assuming that $x$ is real, $(x,\,y) = (1,\,1)$ or $(-1,\,-1)$.

What did you mean by "solution sets"?
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May 4th, 2019, 07:34 AM   #3
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Originally Posted by MATHEMATICIAN View Post
$\displaystyle x^{3}y =1$ 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
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May 5th, 2019, 03:30 AM   #4
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What might be the other solution beside those real solutuons?
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May 5th, 2019, 04:07 AM   #5
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$(x,\,y) = (x,\,1/x^3)$, where $x = \pm i$ or $\pm\sqrt{\pm i}$.
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