Machinery's Handbook, 31st Edition
24
ALGEBRA ALGEBRA
In engineering, manufacturing, and industrial applications, physical laws govern the behavior of all quantities. Algebraic formulas (equations) are the models for these laws. They usually consist of algebraic expressions, the most common being polynomials, ra- tional expressions, and radicals. Most of the formulas used in this Handbook contain one or more of these. This section gives a foundation for understanding the algebra indispens- able to solving equations. Definitions.— The vocabulary of algebra extends to all mathematics. The essential defi - nitions are given here. Operation: Addition, subtraction, multiplication, division, root-taking, raising to a power, taking a logarithm. Inverse Operation: An operation that reverses another operation. Addition and subtrac- tion are inverse operations, as are multiplication and division. Taking the n th root is the inverse of raising a number to a power. Finding an antilogarithm is the inverse of finding a logarithm. Constant: A known quantity, either a number standing alone or a letter that is assumed to be given or known in an application. In 5 x + 14, 14 is the constant. Usually, the letters a , b , and c are used to represent constants, as in the linear equation ax + by = c . Note: e and π are commonly seen constants. Variable: An unknown quantity, represented by a letter such as n , x , y , x , t . Note: e and π are not variables. Exponent: The power to which a variable or number is raised. Monomial: A single variable or number or a product of numbers and variables. Exam- ples: 5, x , – 4 y 2 , 12 xy 2 z 3 . Exponents in monomials are whole numbers, 0, 1, 2, 3, . . . , so x –1 = 1/ x and x 1/2 = x are not monomials. Coefficient: The numerical factor in a term. Examples (in bold): 5 x , 16 n , – 2 r . The coeffi - cient of a variable standing alone is understood to be 1, for example, x = 1 x ; the coefficient of – x is – 1. Term: Monomials are terms, but so are expressions that are not monomials: 1/ x , x , 8 x 1/3 , log x , and so on. Like Terms: This usually refers to monomials with the same variable and exponent, and having any real number coefficients, such as x and 7 x ; 2 n 2 and n 2 /4; 2 rst /5 and 14 rst , and so on. Any constant a can be written as ax 0 , so all constants are like terms. But x 1/2 and 4 x 1/2 also are like terms. Expression: Numbers and variables with operators (addition: +, multiplication: × or ⋅ , etc.). Equation: Two expressions set equal to one another with the equal sign = . Examples: 5 x = x 2 – 6; 3 14 x = . Solving equations for the unknown is the foundation of algebra. Inequality: Two expressions set against one another by >, <, ≥, ≤, or ≠ . Evaluating Algebraic Expressions.—An expression is evaluated by substituting given values of the variable. For example, x 2 – 2 x + 7 evaluated at x = –3 is (–3) 2 – 2(– 3) + 7 = 9 + 6 + 7 = 22. Another example, 1 2 − x evaluated at x = − − = = . Combining Like Terms.— Like terms are added and subtracted by combining their co- efficients and leaving the rest of the term as is. For example: 3 7 3 2 2 14 1 3 1 3 ( ) 2 1 9 8 9 1 1 is 2 2 2 2 5 x x x x + = – – – 2 n n + − = n n + − = rst rst rst 11 – n – + = rst 2 + − = n 2 n n 2 14 rst – + = 2 2 2 n 2 5 n – 2 14 rst 2 2 2 5 rst rst
4 n
4 n –
3 x x x x 7 3 = – = –
+ 7 3 x x x –
rst rst 11 rst
–
+ =
1
4
4
4
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