Simultaneously fitting two straight lines (with a constrain) to observed data

I am trying to solve the following problem with Bayesian statistics.

I have two data arrays (say y1 and y2) that are measured/obtained at the same x-values. However, at each x-value, the one data array should always be equal or larger than the other, more specifically y1 >= y2. Given these two datasets, I want to simultaneously fit 2 straight lines which satisfy the above mentioned constrain to the observed data by using Bayesian statistics.

I have tried to solve this as follows:
I simplified the constrain to be y1 = y2 + k, where k is some constant which is equal or larger than 0.
I use this equation as the model to simultaneously fit the two straight lines.
From this fit I retrieve the values for k and y1 (through Bayesian statistics).
To obtain the values for y2, I simply take y2 = y1 - k.

Is this is a valid approach to solve this problem? If not could you give me any suggestions to solve this problem? Any help and advise will be appreciated.