There are several reasons why your back might hurt, and why you have chest pain.
When these two symptoms occur together, it can be worrisome.
There are several explanations for chest pain that appears with back pain.

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This article covers causes of chest and back pain.
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What Could Cause Chest and Back Pain?
Your back is also susceptible to muscle strain.
Immediate medical care is crucial to surviving a heart attack.
For muscle pain, stretching, physical therapy, and over-the-counter (OTC) pain relievers may help.
Despite this, the combination of chest and back pain is not always an emergency.
Pain can be either chronic or short term.
Harvard Health Publishing.Other conditions may be causes of chest pain.
American Heart Association.Angina (chest pain).
June 2023;14:1104817. doi:10.3389/fneur.2023.1104817
National Institutes of Health.Back pain.
National Institute of Neurological Disorders and Stroke.Low back pain fact sheet.
British Heart Foundation.Vasospastic angina.
Centers for Disease Control and Prevention.About heart attack symptoms, risk, and recovery.
American Heart Association.What causes chest pain?
American Heart Association.Stable angina.