<s>[INST] 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?
$$x + y + z = a$$
$$x^2 + y^2 + z^2 = b$$
$$x^3 + y^3 + z^3 = c$$
 [/INST] This problem can be solved via finding roots of polynomial of $n$-th degree.
$2$ links are helpful here:


*

*Newton's Identities;

*Vieta's formulas.  


Denote (according to $1$st link) 
$$
p_1 = x+y+z=a;\\
p_2 = x^2+y^2+z^2=b;\\
p_3 = x^3+y^3+z^3-c;
$$
and
$$
e_1 = x+y+z;\\
e_2 = xy+xz+yz;\\
e_3 = xyz.
$$
Then
$$
e_1=p_1=a;\\
2e_2 = e_1p_1-p_2=a^2-b;\\
3e_3 = e_2p_1-e_1p_2+p_3 = \dfrac{a^2-b}{2}a-ab+c;
$$
so
$$
e_1 = a;\\
e_2 = \dfrac{a^2-b}{2};\\
e_3 = \dfrac{a^3-3ab+2c}{6}.
$$
Then (according to $2$nd link) $x,y,z$ are three roots of polynomial
$$
p(x) = x^3-e_1x^2+e_2x-e_3.
$$

Example ($a=18, b=134, c=1152$):
$$
x+y+z=18;\\
x^2+y^2+z^2=134;\\
x^3+y^3+z^3=1152;
$$
then
$$
e_1=18, \quad e_2=95, \quad e_3=150;
$$
so $x,y,z$ are the roots of polynomial 
$$
p(x) = x^3-18x^2+95x-150.
$$
$(x=3,y=5,z=10)$ and their permutations.
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