How do I solve this simultaneous $$a + b + c = 2\to (1)$$ $$ax + by + cz = 0\to (2)$$ $$ax^2 + by^2 + cz^2 = \frac{2}{3}\to (3)$$ $$ax^3 + by^3 + cz^3 = 0\to (4)$$ $$ax^4 + by^4 + cz^4 = \frac{2}{5}\to (5)$$ $$ax^5 + by^5 + cz^5 = 0\to (6)$$ This simultaneous has 6 equations, 6 unknown and some variables are raised to the power of 5. I have been battling with this question for days now... The best is could come up with
Multiplying equation (2) by $x^2$
$$ax^3 + byx^2 + czx^2 = 0\to (7)$$
Then (7) - (4) becomes
$$by(x^2 - y^2) + cz(x^2 - z^2) = 0\to (8)$$ Multiplying equation (2) by x^4 then subtracting (6) I have: $$by(x^4 - y^4) + cz(x^4 - z^4) = 0$$
Which then leads to:
$$y^2 = z^2$$ $$Conclusively, y = \pm z$$ Aside from this, every other thing I have tried is looking way off. Is there any better way of doing this? Maybe a matrix or something. Thanks in advance.