[tex]\displaystyle\\
\{x,y,z\}~d.p~\{2,5,9\}\\\\
\frac{x}{2}= \frac{y}{5} = \frac{z}{9} =k\\\\
x=2k \\
y=5k\\
z=9k\\\\
2x+ 3y+ 4z = 110\\
2\cdot 2k+ 3\cdot5k+ 4\cdot 9k = 110\\
4k+15k+36k=110\\
55k=110\\\\
k= \frac{110}{55}=2\\\\
x=2k =2\cdot 2=\boxed{\bf 4}\\
y=5\cdot 2= \boxed{\bf 10}\\
z=9\cdot 2= \boxed{\bf 18}[/tex]