# Is the system uniquely solvable?

• MHB
• mathmari
In summary, The system of equations given is not uniquely solvable for $y = y(x)$ and $z = z(x)$ in a neighbourhood of the point $(1,-1,1)$. To check this, the implicit function theorem can be used.
mathmari
Gold Member
MHB
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)

mathmari said:
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 $(-1,1,1)$?

Opalg said:
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)

mathmari said:
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.

## 1. What does it mean for a system to be uniquely solvable?

For a system to be uniquely solvable, it means that there is only one possible solution that satisfies all of the equations or constraints within the system. This means that there are no other combinations of values that could also satisfy the system.

## 2. How can I determine if a system is uniquely solvable?

To determine if a system is uniquely solvable, you can use various methods such as substitution, elimination, or graphing. If after using these methods, you are able to arrive at a single solution that satisfies all of the equations or constraints, then the system is uniquely solvable.

## 3. Can a system have more than one solution and still be uniquely solvable?

No, if a system has more than one solution, then it is not uniquely solvable. A system can only be considered uniquely solvable if there is only one possible solution that satisfies all of the equations or constraints within the system.

## 4. Is it possible for a system to have no solution and still be uniquely solvable?

No, if a system has no solution, then it is not uniquely solvable. For a system to be uniquely solvable, there must be at least one possible solution that satisfies all of the equations or constraints within the system.

## 5. Can a system with an infinite number of solutions be considered uniquely solvable?

No, a system with an infinite number of solutions is not considered uniquely solvable. For a system to be uniquely solvable, there must be only one possible solution that satisfies all of the equations or constraints within the system.

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