Well it really depends...
I'm assuming that by body system you mean organ.
If a vital organ such as your kidneys failed, of course you would die.
But, if an organ like your appendix needs to be removed, it can also be fatal even though scientists are not sure what it does.
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A body system is a group of structures and organs that provides functions for the organism as a whole. For example, the circulatory system includes the heart and all blood vessels, while the nervous system includes the brain, spinal cord, sense organs, and nerves. The appendix, discussed by the previous contributor, is not a system but is a nonessential part of the digestive system (which IS essential to life).
A complex organism, like a human being, relies upon the synchronized working of several physiologic systems to sustain life. The FAILURE of ANY of these SYSTEMS, with one exception, will ultimately lead to the progressive failure of the other systems and DEATH. Think about it: each and every system -- nervous, circulatory, respiratory, urinary, digestive, connective/musculoskeletal, and endocrine -- is ESSENTIAL to human life.
The exception? The reproductive system. An individual human can live without a reproductive system, but of course cannot reproduce himself. The reproductive system is needed to ensure the continuity of the species as a whole, but a single instance of the organism can live without it. The reproductive system is, however, often grouped with the urinary (genito-urinary) or endocrine systems. When grouped in these manners, the reproductive is part of an ESSENTIAL system.
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First off, there are eleven systems. Second, if any of them failed entirely, the person would die.