Kinds of trees

Relevant wiki: Setting Up Equations

Multiply $\color{#D61F06}(2)$ by $-2$ then add to $\color{#D61F06}(1)$ ,

$[-4x-6y-8z=-52]$ $+$ $[4x+2y+z=22]$ $\implies$ $-4y-7z=-30$ $\color{#D61F06}(4)$

Multiply $\color{#D61F06}(2)$ by $3$ and multiply $\color{#D61F06}(3)$ by $-2$ then add the results,

$[6x+9y+12z=78]$ $+$ $[-6x-2y-6z=-38]$ $\implies$ $7y+6z=40$ $\color{#D61F06}(5)$

Multiply $\color{#D61F06}(4)$ by $7$ and multiply $\color{#D61F06}(5)$ by $4$ then add the results to solve for $z$ ,

$[-28y-49z=-210]$ $+$ $[28y+24z=160]$ $\implies$ $-25z=-50$ $\implies$ $z=2$

It follows that,

$y=4$ and $x=3$

Finally,

$x+y+x+z+y+z=2x+2y+2z=6+8+4=$ $\color{#20A900}\boxed{18}$ $\boxed{\text{answer}}$