Kindly let me know the meaning of this sentence (taken from this article):
We went back and forth, but there was nothing we could do.
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It is actually not clear what it means. It could mean that they physically traveled between two positions several times. It could mean that they argued over what to do. It could mean that they changed their minds about what to do several times. There is not enough context to figure out what it means.
My best guess is that it means that they actively worked on obtaining permission to perform the transplant from the family, discussing every aspect, using different members of the team, and so on. But ultimately, the process just took too long and they were unable to obtain permission while there was still time.
In a general sense, "going back and forth" expresses the give and take of a negotiation or arbitration process. We suggested this and they countered that, and so we "went back and forth".
In casual speech, however, it often means simply "some events transpired, and I won't bore you with the details." As an example, "I went to the driver's license office to get my license, and after two hours of going back and forth I finally got it." This statement doesn't express that I actually negotiated an agreement to get my license, it merely expresses that I navigated the process to get it.