Is the system uniquely solvable?

[mathmari]

Hey!

Is the system $$x^3+y^3+z^3=1 \\ x\cdot y\cdot z=-1$$ in a region of the point $(1; 1; 1)$ uniquely solvable for $y = y (x) $ and $z = z (x)$ ?

How can we check that? Could you give me a hint? (Wondering)

 

[Opalg]

Hey!

Is the system $$x^3+y^3+z^3=1 \\ x\cdot y\cdot z=-1$$ in a region of the point $(1; 1; 1)$ uniquely solvable for $y = y (x) $ and $z = z (x)$ ?

How can we check that? Could you give me a hint? (Wondering)

The point $(1,1,1)$ does not lie on either of those surfaces. Did you mean the point $(-1,1,1)$?

 

[mathmari]

The point $(1,1,1)$ does not lie on either of those surfaces. Did you mean the point $(-1,1,1)$?

Oh I meant $(1;-1;1)$. (Tmi)

 

[S.G. Janssens]

Hey!

Is the system $$x^3+y^3+z^3=1 \\ x\cdot y\cdot z=-1$$ in a region of the point $(1; 1; 1)$ uniquely solvable for $y = y (x) $ and $z = z (x)$ ?

How can we check that? Could you give me a hint? (Wondering)

If by "region" you mean "neighbourhood", then my hint would be to use the implicit function theorem.