Shifting subset sum solution by constant positive integer
Your error is that you add your constant to the left-hand side of the equality multiple times, and only once to the right side. For example, $1+1=2$ clearly holds. If you add $8$ to both sides of it, you get $1+1+8=10$ which also holds. Erroneously you might add it twice to the left hand side and get $9+9=10$, which does not hold.

In your case, $0$ should not become $8$, but instead $8 \cdot 5 = 40$. Now a solution exists again.