Section 2.1 Equations and Their Solutions 93 Student Check 1 Determine if each problem is an expression or equation. a. -4y + 2 = 3y + 1 b. -4y + 2 - 3y + 1 c. -2x2 + 4x = 3x d. 5r2 - r - 4 �������������������������������������������� As we have learned, a solution, or root, of an equation is a value of the variable that makes the equation true. The process of finding the value(s) of the variable that makes the equation true is called solving an equation for the given variable. The process of solving equations is one of the most important skills in algebra. Two fundamental properties of equation solving will be presented in Section 2.2. Before we actually solve equations, we will review how to check if a number is a solution of an equation. We did this in Section 1.3 but it is worth repeating. ���������������������� Determining if a Number Is a Solution of an Equation Step 1: Evaluate each side of the equation for the given value. Step 2: Determine if the two values are equal. a. If the two values are equal, then the given number makes the equation true and is a solution of the equation. b. If the two values are not equal, then the given number makes the equation false and is not a solution of the equation. �� ������������������������ ������������������ Determine if the given numbers are solutions of the equation. 2a. -4x + 2 = 2x - 1; x = -2, 4, or 1 2 2b. x2 = x; x = -0.5, 0, 1 Solutions 2a. x -4 x + 2 = 2 x - 1 -2 -4(-2) + 2 = 2(-2)-1 8 + 2 = -4 - 1 10 ≠ -5 Since 10 ≠ -5, x = -2 is not a solution. 4 -4(4) + 2 = 2(4) - 1 -16 + 2 = 8 - 1 -14 ≠ 7 Since -14 ≠ 7, x = 4 is not a solution. 1 2 -4a1 2 b + 2 = 2a1 2 b - 1 -2 + 2 = 1 - 1 0 = 0 Since 0 = 0, x = 1 2 is a solution. ? ? ? ? ? ? Of the given values, x = 1 2 is a solution of the equation -4x + 2 = 2x - 1. 2b. x x2 = x -0.5 (-0.5)2 = -0.5 0.25 ≠ -0.5 Since 0.25 ≠ -0.5, x = -0.5 is not a solution. 0 (0)2 = 0 0 = 0 Since 0 = 0, x = 0 is a solution. 1 (1)2 = 1 1 = 1 Since 1 = 1, x = 1 is a solution. ? ? ? Of the given values, x = 0 and x = 1 are solutions of the equation x2 = x. Objective 2 ▶ Determine if a number is a solution of an equation.
hendricks_beginning_algebra_1e_ch1_3
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