Algebra is used all the time. Many equations can be created for many things, and it's all thanks to algebra. Physics, Engineering, Computers, and most other technical professions require algebra, as well as higher mathematics. For someone who plans not going into the technical professions, basic things such as balancing your checkbook, and dealing with taxes often require atleast some knowledge of algebra.
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As by "who" i assume you mean what profession. Algebra is used a lot in banking, economics, and doctors who need formulas. But overall algebra is mostly used by college leveled advanced algebra teachers.
Algebra is used to help determine distances in time, or places
Algebra is used in lots of differant ways in everyday life because it's maths.
Yes it is.
There were no Muslims around 4000 years ago when the Babylonians began using what we now call algebra.