# Solve an equation with two specific domains?

I asked how to write a code in Mathematica with one specific domain for one of its variables. Now I have a questions:

If two of the variables are from specific domain for which they have been taken correspondingly?

for example:

Solve[{z == 2 x + y && x > 0 && y > 0}, {x, y, z}, Integers] // Column


where $x\in\{1,2,5,6,9\}$, $y\in\{2,7,11,15,18\}$.

1) for the first case I mean it takes for {x,y}: {1,2}, {2,7}, {5,11}, {6,15}, {9,18}

2) for the second case it takes all $5^2$ cases, where 5 is the cardinal of the set.

P.S. for the second case I wrote the following

Clear[x, y, z]
set1 = {1,2,5,6,9};
set2 = {2,7,11,15,18};
Solve[{z == 2 x + y && x > 0 && y > 0 && Or @@ (Equal[x, #] & /@ set1) &&
Or @@ (Equal[y, #] & /@ set2)}, {x, y, z}, Integers] // Column


but for the first case?

thanks!

For this problem, Solve is not needed.
(sol1 = {#[[1]], #[[2]], 2 #[[1]] + #[[2]]} & /@

(sol2 = (Outer[{#1, #2, 2 #1 + #2} &, {1, 2, 5, 6, 9}, {2, 7, 11, 15, 18}] //