Four equations with five unknown and the Wolframalpha can solve them

Question

$$a+b+c=1\tag 1$$ $$a*k_1+b*k_2+c=1\tag2$$ $$2a*k_1^2+2b*k_2^2+2c=1\tag3$$ $$3a*k_1^3+3b*k_2^3+3c=1\tag4$$ the Wolframalpha gives the solution as follow:

How did this happen?????

Answer 1

I don't know what WA is doing, but that's certainly not the only solution. The general solution can be written as $$ \eqalign{a&=-\,{\frac {3}{ 2\left( k_{{1}}-1 \right) \left( 6\,k_{{1 }}-1 \right) }}\cr b &=-\,{\frac {9}{2 \left( 3\,k_{{1}}+2 \right) \left( 6\,k_{{1}}-1 \right) }}\cr c&={\frac {6\,{k_{{1}}}^{2}-2\,k_{ {1}}-1}{2 \left( k_{{1}}-1 \right) \left( 3\,k_{{1}}+2 \right) }}\cr k_{{2}}&=-k_{{1}}+\frac13} $$ with $k_1$ arbitrary (but not equal to $1$, $1/6$ or $-2/3$).