[tex]4x=6y=8z = k \\ x= \frac{k}{4} \\ \\ y= \frac{k}{6}\\ \\ z= \frac{k}{8} \\ x+y+z=52 \\ \frac{k}{4}+ \frac{k}{6}+\frac{k}{8}=52 \\ \\ \frac{6k}{24}+ \frac{4k}{24}+\frac{3k}{24}=52 \\ \\ \frac{6k+4k+3k}{24}=52 \\ \\ \frac{13k}{24}=52 \\ 13k=52*24 \\ k= \frac{52*24}{13} =4*24=96[/tex]
[tex]x= \frac{k}{4} = \frac{96}{4}= \boxed{24} \\ \\ y= \frac{k}{6} =\frac{96}{6} = \boxed{16} \\ \\ z= \frac{k}{8} =\frac{96}{8}= \boxed{12} [/tex]
