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77 Sect 3.3 - Multiplication and Division Properties of Equality Objective 1: Multiplication and Division Properties of Equality. Multiplication and division properties of equality work in a similar fashion to the addition and subtraction properties of equality. Consider the following: Solve the following: Ex. 1 3x = 27 Ex. 2 w 5 = 11 Solution: Solution: Since 3•9 = 27, then x = 9. Since 55 ÷ 5 = 11, then Or, 27 ÷ 3 = 9. € w = 55. Or, 11•5 = 55. In example #1, the second way we solved the problem was by dividing 27 by 3. On the left side, if we divide 3x by 3, we get x. So, by dividing both sides of 3x = 27, we get the equation x = 9. In example #2, the second way we solved the problem was by multiplying 1.1 by 5. On the left side. if we multiply w 5 by 5, we get w. So, by multiplying both sides of w 5 = 11 by 5, we get w = 55. This means we can multiply or divide both sides of an equation by any non-zero quantity without changing the answer. These are known as the Multiplication and Division Properties of Equality. € € Multiplication and Division Properties of Equality: If a, b, and c are algebraic expressions with c ≠ 0, and if a = b, then 1. ac = bc is equivalent to a = b. Multiplication Property 2. a c = b c is equivalent to a = b. Division Property There are several comments that need to be made about these properties. First, we cannot multiply both sides of an equation by zero. If we do, we will get an equation that has a different solution than the original equation and hence it will not be an equivalent equation. We also cannot divide both sides by zero since division by zero is undefined. Solve the following: Ex. 3 – 25x = – 475 Ex. 4 – 17x = 51 € 78 Solution: Divide both sides by – 25 to solve for x: – 25x = – 475 – 25 – 25 x = 19 Check: – 25(19) = – 475 – 475 = – 475 True So, x = 19. x −11 Ex. 5 Solution: Divide both sides by – 17 to solve for x: – 17x = 51 – 17 – 17 x=–3 Check: – 17(– 3) = 51 51 = 51 True So, x = – 3. =–4 Solution: Multiply both sides by – 11 € reciprocal of x ( −11 ) = – 11(– 4) 3 44 Check: € € Ex. 8 – x = – 65 Solution: Divide by – 1: −x −65 = Check: € −1 (– 15) = – 5 – 5 = – 5 True – 32p = 0 Solution: Divide by – 32: −32p 0 = −32 −32 p=0 – (65) = – 65 – 65 = – 65 True So, x = 65. Objective 2: 1 3 So, t = – 15. x = 65 € to solve for t: t = – 15 Check: =–4 −11 € –4=–4 So, x = 44. −1 4 7 ( 31 t) = 3(– 5) x = 44 Ex.€7 =–5 Solution: Multiply both sides by the € to solve for x: – 11 1 t 3 Ex. 6 € Check: € – 32(0) = 0 0 = 0 True So, p = 0. Comparing the properties of equality In solving linear equations, it is important to identify which property needs to be used to solve an equation. We do this by examining the operation between the variable and the number next to it. Whatever that operation is, we use the reverse process to solve the equation. 79 Solve, but do not check the answer: Ex. 9 – 7x = 42 Solution: Since the operation between – 7 and x is multiplication, we need to divide both sides by – 7: – 7x = 42 –7 –7 x=–6 Ex. 11 – x 7 = 42 Solution: Since the operation between – 7 and x is division, we need €to multiply both sides by – 7: ( –7 – x 7 ) = – 7(42) x = – 294 € Ex. 13 4(y – 3) + 6 = 7 – 13 Solution: Distribute the 4: 4y – 12 + 6 = 7 – 13 Combine like terms: 4y – 6 = – 6 Add 6 to both sides: 4y – 6 = – 6 +6=+6 4y =0 Divide both sides by 4: 4y = 0 4 4 y=0 Ex. 10 – 7 + x = 42 Solution: Since the operation between – 7 and x is addition, we need to add 7 to both sides: – 7 + x = 42 +7 =+7 x = 49 Ex. 12 – 7 – x = 42 Solution: Since the operation between – 7 and x is subtraction, we need to add 7 to both sides: – 7 – x = 42 +7 =+7 – x = 49 To solve for x, divide both sides by – 1 – x = 49 –1 –1 x = – 49 Ex. 14 7y – 3(2y + 3) = – 9 Solution: Distribute the – 3: 7y – 6y – 9 = – 9 Combine like terms: y–9=–9 Add 9 to both sides: y–9=–9 +9=+9 y=0