54 Chapter 2 Linear Equations and Inequalities in One Variable The Addition Property of Equality Now that we know what a linear equation is and how to determine if a value is a solution of a linear equation, we will learn the process for solving a linear equation. When we solve a linear equation in one variable, our goal is to find the solution set of the equation by isolating the variable to one side of the equation. We do this by producing a series of equivalent equations until we reach an equation of the form x = some number or some number = x Equivalent equations are equations with the same solution set. Since two sides of an equation are equal, whatever we do to one side of an equation must be done to the other side to maintain this equality. For instance, we can add the same number to each side of an equation or subtract the same number from each side of an equation. Consider the equation x - 3 = 2. When we add 3 to each side of the equation, we get x - 3 = 2 x - 3 + 3 = 2 + 3 x = 5 Adding 3 to each side of the equation x - 3 = 2 produces an equivalent equation of the form x = 5, which provides the solution of the equation. This operation is an example of the addition property of equality. Property: Addition Property of Equality For real numbers a, b, and c, a = b is equivalent to a + c = b + c. When the same value is added to two equal quantities (or each side of an equation), the two resulting expressions are also equal. Because we define subtraction in terms of addition, this property also guarantees that when we subtract the same number from each side of an equation, an equivalent equation is obtained. Consider the equation x + 5=-1. We can either add -5 to each side of the equation or subtract 5 from each side of the equation to obtain an equivalent equation in which x is isolated on one side. x + 5=-1 x + 5=-1 x + 5 + (-5)=-1 + (-5) x + 5 - 5=-1 - 5 x=-6 x=-6 Note that in either case, the same solution is obtained. Procedure: Using the Addition Property of Equality to Solve an Equation Step 1: Determine the operation that will isolate the variable on one side of the equation. Perform this operation on each side of the equation. Remember the inverse property for addition: a + (-a)=-a + a = a - a = 0. Step 2: Simplify each side of the equation, as necessary. The result should be of the form x = some number or some number = x. Step 3: Check the solution by substituting the value into the original equation. Step 4: Write the solution in set notation. Objective 2 Examples Use the addition property of equality to solve each linear equation and check the solution. 2a. 12.5 = d - 2.5 2b. 12 + 2g = 3g + 15.4 Objective 2 ▶ Use the addition property of equality.
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