Section 2.4 More on Solving Linear Equations 131 So, x + 1 = x + 1 is an example of an equation with infinitely many solutions. It is true for every value of the variable. This type of equation is called an identity. ������������������������ An identity is an equation that is true for all values of the variable. An identity has infinitely many solutions. The solution set is all real numbers, denoted . Again, this type of equation looks no different from a conditional equation. Through the equation-solving process, we obtain a true statement (e.g., 5 = 5 or x = x). If we solve an equation and a true statement remains, then the equation is an identity and has infinitely many solutions of the equation. The solution set is the set of all real numbers, denoted by . �� ������������������������ ������������������ Solve each equation. 4a. 2x + 3 = 2(x - 2) + 7 4b. 4(3x + 5) - (x - 3) = 5(3x + 5) - (4x + 2) Solutions 4a. 2x + 3 = 2(x - 2) + 7 2x + 3 = 2x - 4 + 7 Apply the distributive property. 2x + 3 = 2x + 3 Combine like terms. 2x + 3 - 2x = 2x + 3 - 2x Subtract 2x from each side. 3 =3 Simplify. The resulting equation, 3 = 3, is a true statement, so this equation is an identity. The solution set is all real numbers, or . 4b. 4(3x + 5) - (x - 3) = 5(3x + 5) - (4x + 2) 12x + 20 - x + 3 = 15x + 25 - 4x - 2 Apply the distributive property. 11x + 23 = 11x + 23 Combine like terms. 11x + 23 - 23 = 11x + 23 - 23 Subtract 23 from each side. 11x = 11x Simplify. This resulting equation, 11x = 11x, is always true, so this equation is an identity. The solution set is all real numbers, or . Student Check 4 Solve each equation. a. 2(6x + 4) = 4(3x + 2) b. 7x + 3(2x - 1) = 4(3x - 1) + x + 1 ������������ Once an equation is simplified on each side, we know we have an identity when the variable terms on each side of the equation are the same and the constant terms on each side of the equation are the same. Troubleshooting Common Errors Some common errors associated with solving linear equations are shown next. Objective 5 ▶ Troubleshoot common errors.
hendricks_beginning_algebra_1e_ch1_3
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