I am new here and I just want to ask if the following system has a general solution:

If a, b and c are given such that:

$$
x + y + z = a
$$
$$
x^2 + y^2 + z^2 = b
$$
$$
x^8 + y^8 + z^8 = c
$$
Is there a way to find out if there exists a general solution to find x, y and z?