Value

Just subtract the given equations!
$(x + y + y) - (y - z) = 18 - 2\\ x + y + z = \color{#69047E}{\boxed{16}}$

$\small y-z=2$ $\iff y=z+2$ $\mathrm{So}$ $\small x+y+y=18$ $\iff x+y+(z+2)=18$ $\large \iff x+y+z+2=18$ $\Large \iff x+y+z=18-2$ $\LARGE \iff x+y+z=18-2= \huge \boxed{16}$

Solve for $y$ in terms of $z$ in the second equation then substitute to the first equation.

$y-z=2$ $y=2+z$

Substitute:

$x+y+2+z=18$

$x+y+z=\boxed{16}$