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Mathematics LibreTexts

3: The Fundamentals of Algebra

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    In this section we’ll learn how to manipulate symbols in order to combine like terms and restore and balance equations.

    • 3.0: Prelude to the Fundamentals of Algebra
      Abu Jafr Muhammad ibn Musa al-Khwarizmi was one of the greatest Arab mathematicians of his time. While living in Baghdad during the ninth century AD he became the Chief Librarian at the House of Wisdom, a library and major center of intellectual study. In 820 CE, al-Khwarizmi wrote Al-Kitab al-mukhtasar ti Hisab al-jabr w’al-muqabala, translated to, The Compendious Book on Calculation by Restoration and Reduction, the first book to generalize solving equations using the principles of equality.
    • 3.1: Mathematical Expressions
      In this section we turn our attention to translating word phrases into mathematical expressions. We begin with phrases that translate into sums. There is a wide variety of word phrases that translate into sums. When we combine numbers and variables in a valid way, using operations such as addition, subtraction, multiplication, division, exponentiation, and other operations and functions as yet unlearned, the resulting combination of mathematical symbols is called a mathematical expression.
    • 3.2: Evaluating Algebraic Expressions
      In this section we will evaluate algebraic expressions for given values of the variables contained in the expressions.
    • 3.3: Simplifying Algebraic Expressions
      The commutative property allows us to change the order of multiplication without affecting the product or answer. The associative property allows us to regroup without affecting the product or answer.
    • 3.4: Combining Like Terms
      A term is a single number or variable, or it can be the product of a number (called its coefficient) and one or more variables (called its variable part). The terms in an algebraic expression are separated by addition symbols.
    • 3.5: Solving Equations Involving Integers II
      We return to solving equations involving integers, only this time the equations will be a bit more advanced, requiring the use of the distributive property and skill at combining like terms.
    • 3.6: Applications
      Because we’ve increased our fundamental ability to simplify algebraic expressions, we’re now able to tackle a number of more advanced applications.

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