$\dfrac{x}{3}=\dfrac{y}{4}=\dfrac{z}{2}=k$ ($k\ne 0$)
$\Rightarrow x=3k; y=4k; z=2k$
$x^3-y^3+z^3=-27$
$\Rightarrow (3k)^3-(4k)^3+(2k)^3=-27$
$\Leftrightarrow -29k^3=-27$
$\Leftrightarrow k=\sqrt[3]{\dfrac{27}{29}}$
$\to x=3\sqrt[3]{\dfrac{27}{29}}$, $y=4\sqrt[3]{\dfrac{27}{29}}$, $z=2\sqrt[3]{\dfrac{27}{29}}$