112 Chapter 2 Linear Equations and Inequalities in One Variable �� ���������������� �������� The Multiplication Property of Equality ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Use the multiplication property of equality to solve linear equations. 2. Apply both the addition and multiplication properties of equality to solve linear equations. 3. Solve application problems. 4. Solve consecutive integer problems. 5. Troubleshoot common errors. To rent a moving truck for one day, a company charges $44.95 plus $0.45 per mile. If a family budgets $300 for a moving truck, how many miles will they be able to drive the truck? To answer this question, we have to solve the equation 44.95 + 0.45x = 300, where x is the number of miles driven. In this section, we will learn how to find the solution of this type of equation where the variable term has a coefficient other than one. The Multiplication Property of Equality The addition property of equality, presented in Section 2.2, only enables us to solve for the variable when its coefficient is 1. We need another property to solve equations which have coefficients other than 1 on the variable. Some examples of equations of this form are 3x - 4 = 5 2 3 x = 6 Consider how we can solve the equation 2x = 6 using our reasoning skills. 2x = 6 → This equation means 2 times some number is 6. We know that 2 times 3 is 6, so the solution of this equation is x = 3. The equations 2x = 6 and x = 3 are equivalent equations. To derive the first equation from the second equation, we must divide each side of the equation by 2. 2x = 6 2x = 2 6 2 x = 3 Note dividing by 2 is the same as multiplying by 1 2 , so we could have also multiplied each side by 1 2 to obtain the same result. That is, 2x = 6 1 2 (2x) = 1 2 (6) x = 3 This leads us to another property of solving equations. �������������������� Multiplication Property of Equality If a = c, and b ≠ 0, then ab = cb and a b = c b This property tells us that if two expressions are equal or equivalent, we can • multiply both expressions by the same nonzero number and the expressions will remain equal. • divide both expressions by the same nonzero number and the expressions will remain equal. Objective 1 ▶ Use the multiplication property of equality to solve linear equations.
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