The word 'algebra' comes from the Arabic "al jabr," which translates roughly as "the reunion of broken parts." An Arabian mathematician wrote a book 1200 years ago which used the term, the title of which translated something like "the science of restoring what is missing and equating like with like." This expression comes down to us as "solving for X."
Our current culture is the beneficiary of science and technology, all of which is explainable mathematically. Although few of us use that level of math on a regular daily basis, algebra, as one branch of mathematics, is important for the resolution of scientific and technical issues. For non-scientists and technicians, the benefit of its study comes from the development of the logic and reason the student must encounter in finding solutions. A student may never do another problem after finishing an Algebra course, but reasoning skills learned are important for anyone considering him or herself a thinking individual.
American Heritage Dictionary
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