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Mathematical Properties Algebra I

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Associative Property of Addition and Multiplication The associative property means that you will get the same result no matter how you group the numbers that you add or multiply. Ex) (a + b) + c = a + (b + c) (3 + 4) + 8 = 3 + (4 + 8) (ab)c = a(bc) (3 · 4)8 = 3(4 · 8) Change groups/ order stays the same.

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Commutative Property of Addition and Multiplication The commutative property means that you can switch the order of the numbers that you add or multiply and still get the same result. Ex) x + y = y + x 7 + 4 = 4 + 7 xy = yx (7)(4) = (4)(7) Change the order.

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Distributive Property The distributive property distributes the number you multiply by to the other numbers that are grouped in parentheses. Ex) a(b + c) = ab + ac (b + c)a = ba + ca 2(5 + 6) = 2(5) + 2(6) (5 + 6)2 = (5)(2) + (6)(2) Multiply when distributing.

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Identity Property of Addition and Multiplication The identity property means that if you add 0 to a number or multiply a number by 1, the result is the same number. Ex) x + 0 = x and x · 1 = x 3 + 0 = 3 and 3 · 1 = 3 The identity or value of the number stays the same.

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Inverse Property of Addition The sum of an integer and its additive inverse is equal to zero. Ex) 3 + (-3) = 0 b + (-b) = 0 Add the opposite.

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Inverse Property of Multiplication Two numbers whose product is 1 are multiplicative inverses of each other. Ex) 4. 5 = 1 and X. Y = 1 5 4 Y X Multiply by the reciprocal.

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Multiplicative Property of Zero Any number multiplied by zero will take on the value of zero. Ex) 3 · 0 = 0 and x · 0 = 0 Anything times zero is zero.

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

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Symmetric Property If one quantity equals a second quantity, then the second quantity equals the first. Ex) If a = b; then b = a If 9 = 6 + 3; then 6 + 3 = 9 If, then

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Transitive Property If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity. Ex) If x = y and y = z; then x = z If 5 + 7 = 8 + 4 and 8 + 4= 12; then 5 + 7 = 12 If, and, then

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Substitution Property A quantity may be substituted for its equal in any expression. Ex) If a = b, then a may be replaced for b in any expression. If n = 15, then 3n = 3 · 15 Used to solve many problems

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Properties step by step 2 (3 · 2 – 5) + 3 · 1/3

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Properties step by step 2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6

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Properties step by step 2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1

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Properties step by step 2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1 2 + 3 · 1/3 mult. identity 2 · 1 = 2

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Properties step by step 2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1 2 + 3 · 1/3 mult. identity 2 · 1 = 2 2 + 1 mult. inverse 3 · 1/3 = 1

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Properties step by step 2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1 2 + 3 · 1/3 mult. identity 2 · 1 = 2 2 + 1 mult. inverse 3 · 1/3 = 1 3 substitution 2 + 1 = 3

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