6.1 Order of Operations and Solving Linear Equations 295 Alia works in a surf shop. She earns a weekly salary plus a commission on all items that she sells. In this section, we will learn that Alia’s weekly earnings can be represented with a linear equation. Furthermore, we will learn that many real-world applications can be represented with linear equations. Order of Operations and Solving Linear Equations SECTION 6.1 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand algebraic concepts. 7 Use the order of operations. 7 Solve linear equations. 7 Solve problems involving financial literacy. Why This Is Important Representing problems with equations and then solving these equations is a powerful problem-solving tool that can be used in many areas of our lives. In addition to representing weekly earnings, linear equations can also be used to calculate the price we pay, including sales tax, when we purchase items. Furthermore, price markups and discounts that retailers apply to prices of items can also be represented with linear equations. Having an understanding of how linear equations can be used to solve these and other everyday financial problems can lead to us making better financial decisions. English philosopher Alfred North Whitehead explained the power of algebra when he stated, “By relieving the brain of unnecessary work, a good notation sets the mind free to concentrate on more advanced problems.” Algebraic Concepts Algebra is a generalized form of arithmetic. The word algebra is derived from the Arabic word al-jabr (meaning “reunion of broken parts”), which was the title of a book written by the mathematician Muhammed ibn-Musa al Khwarizmi in about a.d. 825. Algebra uses letters of the alphabet called variables to represent numbers. Often the letters x and y are used to represent variables. However, any letter may be used as a variable. A symbol that represents a specific quantity is called a constant . Multiplication of numbers and variables may be represented in several different ways in algebra. Because the “times” sign might be confused with the variable x, a dot between two numbers or variables indicates multiplication. Thus, ⋅ 3 4 means 3 times 4, and ⋅ x y means x times y. Placing two letters or a number and a letter next to one another, with or without parentheses, also indicates multiplication. Thus, x3 means x xy 3 times , means x y times , and x y ( )( ) means x y times . An algebraic expression (or simply an expression ) is a collection of variables, numbers, grouping symbols, and operation symbols. Some examples of algebraic expressions are + + + − + + x x x x x x x , 2, 3(2 3), 3 1 2 3 , and 7 3 2 The parts that are added or subtracted in an algebraic expression are called terms . The expression + x 0.51 40.1 contains two terms: x 0.51 and + 40.1. The expression − − x y 4 3 5 contains three terms: − x y 4 , 3 , and −5. The + and − signs that break the expression into terms are part of the terms. When listing the terms of an expression, however, it is not necessary to include the + sign at the beginning of the term. The numerical part of a term is called its numerical coefficient or, simply, its coefficient . In the term x4 , the 4 is the numerical coefficient. In the term y4 , − the 4− is the numerical coefficient. Like terms are terms that have the same variables with the same exponents on the variables. Unlike terms have different variables or different exponents on the variables. Did You Know? Foundations of Algebra Greek philosopher Diophantus of Alexandria ( A.D . 250), who invented notations for powers of a number and for multiplication and division of simple quantities, is thought to have made the first attempts at algebra. Not until the sixteenth century, however, did French mathematician François Viète (1540–1603) use symbols to represent numbers, the foundation of symbolic algebra. The work of René Descartes (1596–1660), though, is considered to be the starting point of modern-day algebra. In 1707, Sir Isaac Newton (1643–1727) gave symbolic mathematics the name universal arithmetic . Ohrim/Shutterstock
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