124 Chapter 2 Linear Equations and Inequalities in One Variable �� ���������������� �������� More on Solving Linear Equations Sections 2.2 and 2.3 laid the groundwork for us to solve linear equations. In this section, we will solve linear equations that require a few more steps than what we have encountered so far and we will also explore equations that have special solution sets. Linear Equations Containing Fractions We have solved equations with fractions in earlier sections in which we simply worked with the fractions to obtain the solution. Working with fractions can often be cumbersome, so we can use the multiplication property of equality to first clear fractions from the equation and then solve it. Consider the expression 3 4 x - 1 6 . The least common denominator (LCD) of the fractions in this expression is 12. If we multiply the expression by 12, we get 12a3 4 x - 1 6 b = 12 1 a3 4 xb - 12 1 a1 6 b = 36 4 x - 12 6 = 9x - 2 When the expression is multiplied by its LCD, the fractions are cleared and the expression completely changes. But if we perform this operation to both sides of an equation with fractions, the new equation is equivalent to the original equation. So, if we solve an equation with fractions, we can clear fractions before we apply the steps to solve the equation. These steps are presented next in a general strategy for solving linear equations in one variable. These steps can be applied to solving any linear equation in one variable. ���������������������� Solving Linear Equations in One Variable—A General Strategy Step 1: Write an equivalent equation that doesn’t contain fractions or decimals, if necessary. a. Multiply both sides of the equation by the LCD to clear fractions in the equation if they occur. b. If decimal numbers appear in the equation, multiply both sides of the equation by the power of 10 that eliminates the number with the most decimal places. Step 2: Apply the distributive property to remove any parentheses if they occur. Step 3: Use the addition property of equality to get all the variable terms on one side of the equation and all constants on the other side of the equation. Step 4: Isolate the variable by applying the multiplication property of equality. Step 5: Check the proposed solution by substituting it into the original equation. �� ������������������������ ������������������ Solve each equation by first clearing fractions. Check each answer. 1a. 3x 4 - 5 = x 4 1b. a 6 - 1 8 = a 12 1c. - 2 3 (x - 6) = 4 5 (x + 10) 1d. y - 5 4 - 2y 9 = 1 6 ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Solve linear equations with fractions. 2. Solve linear equations with decimals. 3. Solve linear equations with no solution. 4. Solve linear equations with infinitely many solutions. 5. Troubleshoot common errors. Objective 1 ▶ Solve linear equations with fractions.
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