[tex]\displaystyle \\
\frac{x}{2} = \frac{y}{3} = \frac{z}{4} \\ \\
z = 6 \\ \\
\frac{x}{2} = \frac{y}{3} = \frac{6}{4} \\ \\
\frac{x}{2} = \frac{y}{3} = \frac{3}{2} \\ \\
\frac{x}{2} =\frac{3}{2} ~~\Longrightarrow ~~ x = \frac{2\times 3}{2} = \boxed{3} \\ \\
\frac{y}{3} = \frac{3}{2} ~~\Longrightarrow ~~ y = \frac{3\times 3}{2} = \boxed{ \frac{9}{2} } \\ \\
\Longrightarrow ~~ x+y = 3+ \frac{9}{2} = \frac{6+9}{2} =\boxed{\boxed{\frac{15}{2} }}[/tex]
