Solutions of System of Equations

[anemone]

Find all solutions of the system of equations

$s(x)+s(y)=x\\ x+y+s(z)=z\\ s(x)+s(y)+s(z)=y-4$

where $x,\,y$ and $z$ are positive integers, and $s(x),\,s(y)$ and $s(z)$ are the numbers of digits in the decimal representations of $x,\,y$ and $z$ respectively.

 

[jvicens]

After solving the system of equations, I have found that there are no solutions that satisfy all three equations simultaneously. This is because the first two equations imply that $s(x)$ and $s(y)$ are equal to $x$ and $y$ respectively, which means that $x$ and $y$ must have the same number of digits. However, the third equation states that the sum of $s(x)$, $s(y)$, and $s(z)$ is equal to $y-4$, which is not possible since $s(x)$ and $s(y)$ already add up to $x+y$, which is greater than $y-4$. Therefore, there are no positive integer solutions for this system of equations.