c 0. a b, ac bc a b 3 10 lb 3 10 lb 10 lb 10 lb 10 lb 10 lb 10 lb 10 lb 10 lb 10 lb Figure 2-2 124 Chapter 2 Linear Equations and Inequalities Figure 2-3 10 lb 10 lb 10 lb 2 10 lb 2 5 lb 5 lb Multiplication and Division Properties of Equality Let a, b, and c represent algebraic expressions, where 1. Multiplication property of equality: If then 2. *Division property of equality: If then *The division property of equality follows directly from the multiplication property because division is defined as multiplication by the reciprocal. If then, To understand the multiplication property of equality, suppose we start with a true equation such as If both sides of the equation are multiplied by a constant such as 3, the result is also a true statement (Figure 2-2). 10 10 3 10 3 10 30 30 Similarly, if both sides of the equation are divided by a nonzero real number such as 2, the result is also a true statement (Figure 2-3). 10 10 10 2 To solve an equation in the variable x, the goal is to write the equation in the 10 2 form x number. In particular, notice that we desire the coefficient of x to be 1. That is, we want to write the equation as 1x number.Therefore, to solve an equation such as we can multiply both sides of the equation by the reciprocal of the x-term coefficient. In this case, multiply both sides by the reciprocal of 5, which is Multiply by The coefficient of the x-term is now 1. 5x 15 1x 3 x 3 1 5 . 1 5 15x2 1 5 1152 15 . 5x 15, 5 5 10 10. a c b c a 1 c b 1 c a c b c Concept Connections 11. The division property of equality states that if then provided that c 0. Why is the condition necessary? TIP: The product of a number and its reciprocal is always 1. For example: 1 5 152 1 7 2 a Answer 11. c 0 because division by zero is undefined. 2 7b 1 c 0 a c b c a b,
miller_introductory_algebra_3e_ch1_3
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