I think I just solved it, actually!

I let a3 = 7*a2 - 2*a1, and made that the new condition on W.

I then let a = (a1, a2, 7*a2 - 2*a1) and similarly for b.

Then I did a + cb = (a1 + c*b1, a2 + c*b2, 7(a2 + c*b2) - 2(a1 + c*b1))

Which satisfies my new condition on W I believe...

Can anyone verify this?