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MTH 10905 Algebra THE MULTIPLICATION PROPERTY OF EQUALITY CHAPTER 2 SECTION 3 Identity Reciprocals Reciprocal – two numbers are reciprocals when their product equals 1. If a is a non-zero number the reciprocal is 1 a The reciprocal of a positive is positive and the reciprocal of a negative is a negative Identity Reciprocals The reciprocal of 0 does not exist. 0 1 0 0 1 0 0 first we cannot have a zero on the bottom of a fraction second zero divided by zero is zero. Exp: the reciprocal of 3 is Exp: the reciprocal of -2 is 1 3 because 1 2 3 1 3 1 1 3 3 2 1 2 because 1 1 2 2 Identity Reciprocals Exp: the reciprocal of 4 5 is 5 4 because 4 1 4 4 4 5 4 5 20 1 4 1 1 5 5 5 5 4 5 4 20 5 Exp: the reciprocal of 2 7 is 7 2 because 2 1 4 4 4 5 4 5 20 1 4 1 1 5 5 4 5 4 20 7 5 5 Multiplication Property to Solve Equation Multiplication Property of Equality if a = b then a · c = b · c for any real number a, b, and c We can multiply any non-zero number to both sides without changing the solution. We can solve equations in the form of ax = b using the multiplication property To isolate the variable we will multiply by the reciprocal of the numerical coefficient . Multiplication Property to Solve Equation Exp: Exp: 4 x 12 1 4 12 1 x 4 1 1 4 12 x 4 x3 1 x 12 3 3 1 3 x 12 1 3 1 3 1 12 3 x 1 3 1 1 x 36 Multiplication Property to Solve Equation Exp: 5b 65 1 1 5 b 65 5 5 1 5 65 1 b 5 1 1 5 65 b 5 b 13 Division is defined in the term of multiplication a means a 1 b b this allows is to divide both sides by a non-zero number r Exp: 7 4 r 4 7 4 4 4 r 7 4 1 4 r 28 Multiplication Property to Solve Equation Exp: 3 x9 4 Exp: 9x 5 4 3 4 x 9 3 4 3 1 1 9 ( x) 5 9 9 4 3 9 4 x 3 4 1 3 1 9 5 1 x 9 1 1 9 36 x 3 x 12 5 x 9 Multiplication Property to Solve Equation Exp: -24 6 w 1 6 1 24 w 6 1 6 1 24 w 6 1 -24 w -6 4w Exp: 0.25 x 1.50 1 1.50 1 0.25 ( x) 0.25 1 0.25 x 1.50 0.25 x6 Multiplication Property to Solve Equation When solving an equation in the form of ax = b: 1. for a fractions multiply both sides by the reciprocal of a 2. for whole numbers divide both sides by a Exp: 5 -4x 8 Exp: 2 -7 x 5 4 x 5 1 4 8 4 7 5 2 5 ( x) 1 2 5 2 5 x 32 35 x 2 Solve Equation in the form of –x = a Remember that x = a is the same as 1x = a Therefore, -x = a is the same as -1x = a Exp: - y 10 1y 10 1 1 y 10 Exp: - x 19 1x 19 1 1 x 19 Do some steps Mentally to Solve Equations As you become comfortable you can do some of the steps mentally Exp: 10 x 70 70 x 10 x7 Exp: 1 x 16 4 x (16)(4) x 64 HOMEWORK 2.3 Page 118 – 119 #9, 11, 19, 25, 31, 35, 49, 57