106 CHAPTER 1 Equations and Inequalities Basic Terminology of Equations An equation is a statement that two expressions are equal. x + 2 = 9, 11x = 5x + 6x, x2 - 2x - 1 = 0 Equations To solve an equation means to find all numbers that make the equation a true statement. These numbers are the solutions, or roots, of the equation. A number that is a solution of an equation is said to satisfy the equation, and the solutions of an equation make up its solution set. Equations with the same solution set are equivalent equations. For example, x = 4, x + 1 = 5, and 6x + 3 = 27 are equivalent equations because they have the same solution set, 546. However, the equations x2 = 9 and x = 3 are not equivalent because the first has solution set 5-3, 36 while the solution set of the second is 536. One way to solve an equation is to rewrite it as a series of simpler equivalent equations using the addition and multiplication properties of equality. 1.1 Linear Equations ■ Basic Terminology of Equations ■ Linear Equations ■ Identities, Conditional Equations, and Contradictions ■ Solving for a Specified Variable (Literal Equations) Addition and Multiplication Properties of Equality Let a, b, and c represent real numbers. If a =b, then a +c =b +c. That is, the same number may be added to each side of an equation without changing the solution set. If a =b and c 30, then ac =bc. That is, each side of an equation may be multiplied by the same nonzero number without changing the solution set. (Multiplying each side by 0 leads to 0 = 0.) Linear Equation in One Variable A linear equation in one variable is an equation that can be written in the form ax +b =0, where a and b are real numbers and a≠0. These properties can be extended: The same number may be subtracted from each side of an equation, and each side may be divided by the same nonzero number, without changing the solution set. Linear Equations We use the properties of equality to solve linear equations.
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