104 Chapter 2 Linear Equations and Inequalities Multiplication and Division Properties of Equality Let a, b, and c represent algebraic expressions, . 1. Multiplication property of equality: If then 2. *Division property of equality: If then a b, ac bc a b *The division property of equality follows directly from the multiplication property because division is defined as multiplication by the reciprocal. Figure 2-2 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 3 10 lb 3 10 lb 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 10 lb 2 10 lb 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 c 0 10 lb 10 lb 10 lb 10 lb 10 lb 10 lb 10 lb 10 lb Figure 2-3 10 lb 10 lb 5 lb 5 lb TIP: The product of a number and its reciprocal is always 1. For example: 1 5 152 1 7 2 a 2 7 b 1
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