Let's determine the ratio $x: y: z$ if

$$
(5 x+4 y-6 z):(4 x-5 y+7 z):(6 x+5 y-4 z)=1: 27: 18
$$
From the given ratio, it follows that

$$
5 x+4 y-6 z=a, \quad 4 x-5 y+7 z=27 a, \quad 6 x+5 y-4 z=18 a
$$

Solving this system of equations, we get:

$$
x=3 a, \quad y=4 a, \quad z=5 a
$$

and thus

$$
x: y: z=3: 4: 5
$$

(Alfréd Haar, Budapest.)

The problem was also solved by: Déri Zs., Deutsch I., Harsányi Z., Kürti I., Ragány B., Simonyi Gy.