# Thread: [SOLVED] Find number of distinct pairs (x,y)?(difficult)

1. ## [SOLVED] Find number of distinct pairs (x,y)?(difficult)

The number of distinct pairs (x,y) of real numbers satisfying x = x^3 + y^4 and y = 2xy is:

A)5
B)12
C)3
D)7

2. $\displaystyle y = 2xy$

$\displaystyle y - 2xy = 0$

$\displaystyle y(1 - 2x) = 0$

$\displaystyle y = 0$ or $\displaystyle x = \frac{1}{2}$

for $\displaystyle y = 0$ ...

$\displaystyle x = x^3 + y^4$

$\displaystyle x = x^3$

$\displaystyle x - x^3 = 0$

$\displaystyle x(1 - x^2) = x(1 - x)(1 + x) = 0$

$\displaystyle (0,0)$, $\displaystyle (1,0)$ and $\displaystyle (-1,0)$

for $\displaystyle x = \frac{1}{2}$ ...

$\displaystyle \frac{1}{2} = \frac{1}{8} + y^4$

$\displaystyle y^4 - \frac{3}{8} = 0$

$\displaystyle \left(y^2 + \sqrt{\frac{3}{8}}\right)\left(y^2 - \sqrt{\frac{3}{8}}\right) = 0$

$\displaystyle \left(y^2 + \sqrt{\frac{3}{8}}\right)\left(y - \sqrt[4]{\frac{3}{8}}\right)\left(y + \sqrt[4]{\frac{3}{8}}\right) = 0$

$\displaystyle \left(\frac{1}{2}, \sqrt[4]{\frac{3}{8}}\right)$ and $\displaystyle \left(\frac{1}{2}, -\sqrt[4]{\frac{3}{8}}\right)$

looks like 5 distinct solutions to me ... hope I didn't miss any.

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# the number of pairs of reals {x,y } such that x = x×x y×y and y= 2xy

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