Systems of polynomial equations involving sums of equal powers Given the following system of polynomial equations:
$$
\left\{\begin{array}{lclclcr}
x & + & y & + & z & = & 1
\\
x^{2} & + & y^{2} & + & z^{2} & = & 14
\\
x^{3} & + & y^{3} & + & z^{3} & = & 36
\end{array}\right.
$$
What is $x^{5} + y^{5} + z^{5}\ {\large ?}$ .

How should I approach this? Is there a general formula for this kind of system?

Let $p=x+y+z$, $q=xy+yz+zx$, $r=xyz$. So we have
$$\begin{cases}
p=1\\p^2-2q=14\\p^3-3pq+3r=36
\end{cases}$$
$$\begin{cases}
p = 1\\q = -\frac{13}{2}\\ r = \frac{31}{6}
\end{cases}$$
By consecutive eliminating the highest powers terms we get
$$x^5+y^5+z^5-(x+y+z)^5+5(xy+yz+xz)(x+y+z)^3-5(xy+yz+xz)^2(x+y+z)-5(x+y+z)^2xyz+5(xy+yz+xz)xyz=0$$
In other words, $x^5+y^5+z^5=p^5-5qp^3+5q^2p+5p^2r-5qr$ $=\frac{877}{2}$
