converting between two cordinate systems

I'm trying to make a conversion between two coordinate systems, pixel x,y and latitude, longitude. Say we have two points, the first being 53,12 which represents latitude 42 longitude -119.5, the second 42,56 representing latitude 39 longitude -119.3. How would I find the pixel distance between the two longitudes and two latitudes?

It may seem like a silly request giving all the datums and map projection math readily available on wikipedia but this particular map im using does not follow any standard projection. I'm mathematically representing a mesh of several xy points mapping to latitudes and longitudes to properly calibrate plotting cities in a state.