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Properties of Equality, Identity, and Operations

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Commutative Property a + b = b + a (a)(b) = (b)(a) The Commutative Property states that the order of the numbers may change and the sum/product will remain the same. This property applies to both addition and multiplication. 2 + 3 = 3 + 2 (2)(3) = (3)(2)

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Associative Property (a + b) + c = a + (b + c) (a · b) · c = a · (b · c) The Associative Property states that the grouping of numbers can change and the sum/product will remain the same. This property applies to both addition and multiplication. (2 + 4) + 5 = 2 + (4 + 5) (2 · 4) · 5 = 2 · (4 · 5)

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Distributive Property of Multiplication a (b + c) = a(b) + a(c) a (b – c) = a(b) – a(c) The Distributive Property takes a number and multiplies it by everything inside the parentheses. This property works over addition and subtraction. 2(3 + 4) = 2(3) + 2(4) 2 (5 – 2) = 2(5) – 2(2)

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Substitution Property Solve: y = 2(x) + 4 if x = 5 This property allows you to simplify algebraic expressions for different values. You substitute the given value of the variable into the equation and solve. y = 2(5) + 4 y = 10 + 4 y = 14

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Identity Properties n · 1 = n n + 0 = n This property shows how a given number is itself when multiplied by 1 or added to 0. These are important concepts to understand when solving single and multi-step equations. The one and zero act like mirrors. 4 · 1 = 4 5 + 0 = 5

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Zero Property of Multiplication n · 0 = 0 Simply stated, any number times zero equals zero.

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Multiplicative Inverse Property ½ (2) = 1 This property is helpful when solving equations where there is a fraction “attached” to a variable by multiplication. The normal inverse operation for multiplication is division, but in this case, you will multiply both sides of the equation by the reciprocal of the fraction. ½ n – 3 = 4 ½ n -3 + 3 = 4 + 3 ½ n = 7 ½ n (2) = 7(2) n = 14

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Transitive Property If a = b and b = c, then a = c If one quantity equals a second quantity and the second quantity equals a third quantity, then the first equals the third. If 1000 mm = 100 cm and 100 cm = 1 m, Then 1000 mm = 1m

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Symmetric Property If a + b = c then c = a + b If one quantity equals a second quantity, then the second quantity equals the first. If 10 = 4 + 6, then 4 + 6 = 10

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Reflexive Property a = a a + b = a + b Any quantity is equal to itself. 7 = 7 2 + 3 = 2 + 3

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