Solve simple nonlinear equations in the form [A]x=b

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I have a simple set of nonlinear equations

1) 3x = 30

2) x+2y = 20

3) x + y*z = 15

Clearly the solution to this is (10,5,1) but I want to find a robust way to solve this type of problem [A]x=b (where [A] is a simple function of x) which doesn't involve numerically solving using Newtons method.

{{3,0,0},{1,2,0},{1,z,0}} {{x},{y},{z}} = {{30}, {20}, {15}}

Picture in matrix form