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αx+βy+ζz=p[itex]_{1}[/itex]

Where α,β,ζ are constants x,y,z are variables, and p is a prime, how would I use Galois theory and/or number theory to find the number of solutions if the other equations could all be written in the form

α[itex]_{i}[/itex]x[itex]_{i}[/itex]+β[itex]_{i}[/itex]y[itex]_{i}[/itex]+ζ[itex]_{i}[/itex]z[itex]_{i}[/itex]=p[itex]_{i}[/itex] where, once again, each α,β,ζ are constants, each p is a distinct prime, and x,y,z are all variables?

Thanks in advance.