Pre-determined points in Shamir's secret
Based on your question, it is not entirely clear exactly how the $x_i$ and $y_i$ values are coupled. 

As I understand it, you want to make a situation where when someone has all $x_i$, they can recover all $y_i$. A simple way to do this would be to encrypt all the $y_i$ with a strong symmetric algorithm, e.g. AES-256-GCM, using a key $k$ and make the encrypted blob public. Then use Shamir's with the $x_i$ to protect the key $k$. Then, when all $x_i$ are assembled, the key $k$ can be recovered and the encrypted blob decrypted, rendering the $y_i$ values.

This may or may not be what you actually want, depending on your use case.