in this question you have not mentioned which area of your back is hurting, is it upper or lower back.
PE can mimic back ache (in the upper part of spine), but the pain is more on the front than on the back, there is cough with sputum
and often blood can also accompany. The embolus usually arises from the lower limb vessels and after a period of immobilization or major surgery. also your age dosent fit into the risk factors for PE.
But other than that there are several other causes of back ache with cough. you should consult a doctor and get your tests done. It can be disc disease
or acute spasm also. do not worry.