Aches and pains, especially in the back, are something that many people deal with on a daily basis. The idea of working out completely pain-free is a bit of a fantasy -- there always seems to be a sore muscle somewhere.
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Performing pushups with back pain is OK as long as you don't have from a serious back injury. The exercise may even help relieve some of the pain you're experiencing. It may be worth a shot to alter your pushup form or try a similar exercise until your back pain subsides.
The standard pushup form places a lot of stress on the back; altering your technique may offer just enough relief to complete your workout. Dropping to your knees or elevating your upper body during pushups will put less pressure on your back.
If you find it uncomfortable to perform pushups but still want to work your arms and chest, give bench presses a go. This exercise allows you to lie on an exercise bench, offering complete back support, while you press weights vertically to target your triceps and pectoral muscles.