Find all solutions of the system

s(x) + s(y) = x

x + y + s(z) = z

s(x) + s(y) + s(z) = y − 4,

where x, y and z are positive integers, and each of s(x), s(y) and s(z) denotes the sum of digits of
x, y and z, respectively, in decimal notation.

My solution, done before Ady's hint, matches Charlie's.

I just have not written it up yet.

I observed that y must be a multiple of 9 (from the 1st equation).

(Also, then, x must be a multiple of 9 from the second equation, although this inference was not used).

Substituting equations 1 and 2 into 3 yields

z = 2y - 4

From equation 2, x = z - s(z) - y

So, it is easy to build a table of z and x for y = 9, 18, 27, 36, etc.

x and y cannot be bigger than 100, so the table is qyuite manageable.

Testing those entries using equation 1, yields just two solutions