How many real solutions does the system of equations -x- - 3x - y-y- - 3y - z-z- - 3z - x -have-

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Condition for a Conic to Be a Pair of Straight Lines

How many real...

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How many real solutions does the system of equations x³ - 3x = y y³ - 3y = z z³ - 3z = x have?

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If x + y + z = xyz. Then

$\frac{3 x - x^{3}}{1 - 3 x^{2}}$ + $\frac{3 y - y^{3}}{1 - 3 y^{2}}$ + $\frac{3 z - z^{3}}{1 - 3 z^{3}}$ =

x + y + z = x y z

\frac{3 x - x^{3}}{1 - 3 x^{2}} + \frac{3 y - y^{3}}{1 - 3 y^{2}} + \frac{3 z - z^{3}}{1 - 3 z^{2}} = \frac{3 x - x^{3}}{1 - 3 x^{2}} \cdot \frac{3 y - y^{3}}{1 - 3 y^{2}} \cdot \frac{3 z - z^{3}}{1 - 3 z^{2}}

α

x + 3 y + 5 z = α x

5 x + y + 3 z = α y

3 x + 5 y + z = α z

x^{2} + y^{2} = 25

x + y + z = 5

x + 2 y + 3 z = 9

x + 3 y + α z = β

β - α

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