How to raise a wooden floor to same height as adjoining concrete floor before carpetting?
Plywood to build up the wood floor height is an appropriate way to address this problem. For use under carpet some regions may allow use of other products for this purpose such as particle board but plywood is still the best.

Installation of the plywood also gives an opportunity to firm up the floor and work at eliminating any possible squeaking in the old floor. It is best to screw down the plywood into the original floor joists. You can probably find where these are by looking where the old boards were nailed. It can be handy to apply some masking tape to the bottom of the adjacent walls and marking the location of the joist center lines so that after you lay down the plywood it is easy to locate the joints and snap chalk lines across the plywood for a screwing guide. 

You may still need to do some special work at the plywood to concrete transition zone because there may still be some minor height variation after the plywood is installed. This can be handled by troweling in a layer of an appropriate floor leveling compound and feathering it out away from the transition zone.