Solve system in $a,b,c$ What's the best way to solve the system of equations ?: 
$$
\left\{\begin{array}{rcccccr}
a     & + & b     & + & c     & = & 1
\\
a^{2} & + & b^{2} & + & c^{2} & = & 7
\\
a^{3} & + & b^{3} & + & c^{3} & = & 13 
\end{array}\right.
$$
If I square the first one I get $2\left(\,ab + bc + ac\,\right)=-6$, and if I multiply the second by $a + b + c$ I get some other expression that equals $-6$ but I'm not sure how to proceed. 

By using Newton's identities, we can find
that 
$$s_2:=ab+bc+ca=\frac{1}{2}(a+b+c)^2-\frac{1}{2}(a^2+b^2+c^2)=-3$$ 
and 
$$s_3:=abc=\frac{1}{3}(a^3+b^3+c^3)-\frac{1}{3}(a+b+c)^3+(ab+bc+ca)(a+b+c)=1.$$ 
Then we obtain $(a,b,c)$ by solving the polynomial equation
$$x^3-s_1x^2+s_2x-s_3=x^3-x^2-3x-1=(x+1)(x^2-2x-1)\\=(x+1)(x-(1+\sqrt{2}))(x-(1-\sqrt{2}))=0$$
where $s_1=a+b+c$.
