60 Chapter 2 Linear Equations and Inequalities in One Variable Objective 5 Example Solve the linear equation 4x - 3x + 2 = 3 -(5 - x). Solution 4x - 3x + 2 = 3 - (5 - x) 4x - 3x + 2 = 3 - 5 + x Apply the distributive property. x + 2 = -2 + x Combine like terms on each side. x + 2 - x=-2 + x - x Subtract x from each side. 2=-2 Simplify. The resulting equation 2=-2 is false; it is a contradiction which means this equation has no solution. We write the solution set as . Student Check 5 Solve the linear equation 2x - 3 - x = 2 + x + 1. Linear Equations with Infinitely Many Solutions Just as there are some linear equations with no solution, there are some linear equations that have all real numbers as solutions. This is a linear equation with infinitely many solutions, and it is called an identity. An identity is an equation that is always true. An identity occurs when both sides of the equation are equal for all values substituted in place of the variable. Some simple examples of identities are 5 = 5, -2 = -2, 0 = 0, and x = x. If an equation is an identity, then every value of the variable produces a true statement when it is substituted into the equation. We write the solution set as 5xux is a real number6 (read “the set of x such that x is a real number”) or . Again, we may not be able to determine if a linear equation is an identity by examination, but we can use the addition and multiplication properties of equality to solve the equation. If a true statement results, then the linear equation is an identity and has all real numbers as its solution set. Objective 6 ▶ Identify linear equations with infinitely many solutions. Objective 6 Example Solve the linear equation 4(x + 3) + 5 = 17 + 4x. Solution 4(x + 3) + 5 = 17 + 4x 4x + 12 + 5 = 17 + 4x Apply the distributive property. 4x + 17 = 17 + 4x Combine like terms on the left side. 4x + 17 - 4x = 17 + 4x - 4x Subtract 4x from each side. 17 = 17 Simplify. The equation 17 = 17 is true; it is an identity, which means there are infinitely many solutions. We write the solution set as 5xux is a real number6 or . Student Check 6 Solve the linear equation 6(2 - x) + 4 - 3x = 12 - (9x - 4). Objective 7 ▶ Troubleshoot common errors. Troubleshooting Common Errors Some common errors associated with solving linear equations using the addition and multiplication properties are shown.
hendricks_intermediate_algebra_1e_ch1_3
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