Solving $n$ unknowns with $n$ independent equations Is it always possible to solve $n$ independent equations with $n$ unknowns?
Or is it possible to solve the following 3 equations with 3 unknowns?
$$x + y + z = a$$
$$x^2 + y^2 + z^2 = b$$
$$x^3 + y^3 + z^3 = c$$

Consider $x$ , $y$ and $z$ as the roots of the equation :
$$(X-x)(X-y)(X-z)=X^3+AX^2+BX+C=0$$
$A=-(x+y+z)=-a$
$B=xy+yz+zx=\frac{1}{2}\left((x+y+z)^2-(x^2+y^2+z^2)\right)=\frac{1}{2}(a^2-b)$
$C=-xyz=-\frac{1}{6}\left( (x+y+z)^3-3(x+y+z)(x^2+y^2+z^2)+2(x^3+y^3+z^3) \right)=-\frac{1}{6}(a^3-3ab+2c)$
$x$ , $y$ , $z$ are the roots of the equation :
$$X^3-aX^2+\frac{1}{2}(a^2-b)X-\frac{1}{6}(a^3-3ab+2c)=0$$
