o Section 2: Solving Equations

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 1.2 Solving Equations 15 Version: Fall 2007 1.2 Solving Equations In this section, we review the equation-solving skills that are prerequisite for successful completion of the material in this text. Before we list the tools used in the equation- solving process, let’s make sure that we understand what is meant by the phrase “solve for x .” Solve for x . Using the properties that we provide, you must “isolate x ,” so that your final solution takes the form x = “Stuff,” where “Stuff” can be an expression containing numbers, constants, other variables, and mathematical operators such as addition, subtraction, multiplication, division, square root, and the like. “Stuff” can even contain other mathematical functions, such as exponentials, loga- rithms, or trigonometric functions. However, it is essential that you understand that there is one thing “Stuff” must not contain, and that is the variable you are solving for, in this case, x . So, in a sense, you want to isolate x on one side of the equation, and put all the other “Stuff” on the other side of the equation. Now, let’s provide the tools to help you with this task. Property 1. Let a and b be any numbers such that a = b . Then, if c is any number, a + c = b + c, and, a c = b c. In words, the first of these tools allows us to add the same quantity to both sides of an equation without affecting equality. The second statement tells us that we can subtract the same quantity from both sides of an equation and still have equality. Let’s look at an example. l⚏ Example 2. Solve the equation x + 5 = 7 for x . The goal is to “isolate x on one side of the equation. To that end, let’s subtract 5 from both sides of the equation, then simplify. Copyrighted material. See: 1
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16 Chapter 1 Preliminaries Version: Fall 2007 x + 5 = 7 x + 5 5 = 7 5 x = 2 . It is important to check your solution by showing that x = 2 “satisfies” the original equation. To that end, substitute x = 2 in the original equation and simplify both sides of the result. x + 5 = 7 2 + 5 = 7 7 = 7 This last statement (i.e., 7 = 7 ) is a true statement, so x = 2 is a solution of the equation x + 5 = 7 . An important concept is the idea of equivalent equations . Equivalent Equations . Two equations are said to be equivalent if and only if they have the same solution set. That is, two equations are equivalent if each of the solutions of the first equation is also a solution of the second equation, and vice-versa. Thus, in Example 2 , the equations x +5 = 7 and x = 2 are equivalent, because they both have the same solution set { 2 } . It is no coincidence that the tools in Property 1 produce equivalent equations. Whenever you add the same amount to both sides of an equation, the resulting equation is equivalent to the original equation (they have the same solution set). This is also true for subtraction. When you subtract the same amount from both sides of an equation, the resulting equation has the same solutions as the original equation.
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