Solving a mixed linear‑cubic system

Solve the system
$3 x - 4 y + 7 z = 0,$
$2 x - y - 2 z = 0,$
$3 x^{3} - y^{3} + z^{3} = 18.$

~10 minAnswer checked instantlylinear equationscubic substitution

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. I. Ratio, Examples I, No. 20

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Checked against the answer in H. S. Hall & S. R. Knight, Higher Algebra (1891).