Section 2.1 Solving Linear Equations 59 4d. 0.05(x - 2) + 0.10x = 3.65 1000.05(x - 2) + 0.10x = 100(3.65) Multiply each side by 100. 1000.05(x - 2) + 100(0.10x) = 365 Apply the distributive property. 5(x - 2) + 10x = 365 Simplify. 5x - 10 + 10x = 365 Apply the distributive property. 15x - 10 = 365 Combine like terms. 15x - 10 + 10 = 365 + 10 Add 10 to each side. 15x = 375 Simplify. 15x 375 = 15 15 Divide each side by 15. x = 25 Simplify. It is important to note that clearing decimals is not required in this example. 0.05(x - 2) + 0.10x = 3.65 0.05x - 0.1 + 0.10x = 3.65 Apply the distributive property. 0.15x - 0.1 = 3.65 Combine like terms. 0.15x - 0.1 + 0.1 = 3.65 + 0.1 Add 0.1 to each side. 0.15x = 3.75 Simplify. 0.15x 3.75 = 0.15 0.15 Divide each side by 0.15. x = 25 Simplify. The value 25 makes the equation true, so the solution set is {25}. The check is left for the reader. Student Check 4 Solve each linear equation. a. 9x - 3 + 2x = 7x - 2 -13 b. 4(3 - 2x) = 6 - (3x - 1) c. 3y 4 - y 3 = 1 12 d. 0.25(x + 5) + 0.05x = 4.85 Linear Equations with No Solution Thus far, all of the linear equations we have solved had one real number as a solution. These equations are examples of conditional equations. A conditional equation is an equation that is true for some values of the variable and not true for other values. Not every linear equation, however, has a solution. A linear equation with no solution is called a contradiction. A contradiction is an equation that is never true. Some simple examples of contradictions are 3 = 5, -19 = 9, and 0 = 1. If an equation is a contradiction, there is no value of the variable that produces a true statement when it is substituted into the equation. When an equation has no solution, we write the solution set as the empty set, , or 5 6 . Although we may not be able to determine if a linear equation is a contradiction by examination, we can use the addition and multiplication properties of equality to go through the process of solving the equation. If a false equation results, then the linear equation is a contradiction and has no solution. Objective 5 ▶ Identify linear equations with no solution.
hendricks_intermediate_algebra_1e_ch1_3
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