Bashing is for the weak, WE factor

Algebra Level 5

In the following linear system:

\(a(a+2)-4b(b+3)=8, \\ 8(b-2c)(b^2+2bc+4c^2)-48bc(b-2c)=(b+2c)(b^2-2bc+4c^2)+6bc(b+2c), \\ a^4+81c^4=12a^3c-54a^2c^2+108ac^3\)

there are two ordered triples \( (u, v, w) \) and \( (x, y, z) \) which satisfy the system. If \(u+v+w+x+y+z\) can be represented as \(-\frac{p}{q}\) where p,q are positive co-prime integers, find \(p+q\).

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