102 Chapter 2 Linear Equations and Inequalities in One Variable Student Check 1 Determine if the equation is a linear equation in one variable. a. 2y - 3 = 7 b. a - 1 2 = 3 2 c. y4 + 8y2 = 9 The Addition Property of Equality We have learned that a solution of an equation is the value of the variable that makes a true statement. It would be too tedious to use evaluating an equation as a method for solving it. The goal of this section is to determine the solution of a linear equation by performing a series of steps that yield simpler equations that are equivalent to the original equation. Equivalent equations are equations that have the same solution set. The process of producing equivalent equations will enable us to isolate the variable to one side of the equation. These steps will ultimately produce an equation of the form x = some number or some number = x Since two sides of an equation are equal, or equivalent, to each other, we must perform the same operations to each side of the equation to keep the equation balanced. • It is helpful to think of a balance scale to see this relationship. • If each side of the equation represents some weight, these two weights must be the same since they are set equal to one another. • If we add something to one side of the equation or “scale,” we must add it to the other side to keep the scales balanced. • If we subtract something from one side of the equation or “scale,” we must subtract it from the other side to keep the scales balanced. Consider the equation, x + 2 = 5. The following diagram shows each side of the equation on the balance scale. Note the scales are balanced since the two sides are equal. x + 2 = 5 1 1 1 1 1 11x To isolate the variable x on the left side of the scale, we must take 2 units away from the left side of the scale. To maintain balance, we must also take 2 units away from the right side of the scale. x = 3 1 1 1 x This leaves us with x on the left side of the scale and 3 on the right side of the scale. This action leads us to the solution of the equation, x = 3. The equations x + 2 = 5 and x = 3 are called equivalent equations since they have the same solution set. We obtained the second equation by subtracting two from each side of the original equation, that is, x + 2 - 2 = 5 - 2 is equivalent to x = 3. This leads to the addition property of equality. �������������������� Addition Property of Equality If a = c, then a + b = c + b and a - b = c - b. Objective 2 ▶ Use the addition property of equality to solve a linear equation.
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