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The first systematic book on algebra was written in Baghdad, Iraq. However, algebraic methods were also used in China, India, Egypt and Greece long before that.
The first comprehensive book on algebraic methods was published by the Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, in Baghdad. However, algebraic methods were used for centuries before that in various countries.
-- Take the statement of the problem that you're given. -- Make sure you understand exactly what the question is, and what you're expected to find. -- Use the methods and procedures you've learned in your algebra studies to manipulate the given problem and answer the question. If you've been a student in an algebra class, then you've received several hours of explanation and practice in "doing" them.
The father of algebra is typically considered to be the Persian mathematician Muhammad al-Khwarizmi. He introduced the concept of algebra and its fundamental operations in his book "Kitab al-Jabr wa al-Muqabalah" (The Compendious Book on Calculation by Completion and Balancing). In this book, al-Khwarizmi laid the groundwork for solving equations and developing algebraic methods.
Most recently, Ngô Báo Châ for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebra-geometric methods, for which he won a Fields Medal. Also, from a historical perspective, and Egyptian named Ahmes produced a manuscript called the 'Rhind Papyrus' which showed the first simple algebraic methods. There are many lists of mathematicians which can be found using a simple internet search.