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Algebraic properties in DragonBox Algebra 12+ Rule 1: Additive Identity Property If we add 0 to any number, we will end up with the same number. π+0=π π₯+0=π₯ Rule 2: Additive Inverse Property If we add a number by the opposite of itself, we will end up with 0. π + (βπ) = 0 π₯ + (βπ₯) = 0 Rule 3: Properties of Equality (I) If π = π, π‘βππ π + π = π + π Add c to each side Rule 4: Multiplicative Inverse Property If we multiply a number by its reciprocal, we will end up with 1. 1 =1 π 1 π₯β =1 π₯ πβ Rule 5: Multiplicative Identity Property If we multiply 1 to any number, we will end up with the same number. πβ1=π π₯β1=π₯ Rule 6: Properties of Equality (II) ππ ππ = ππ and π β 0, then π = π Divide both sides by c Rule 7: Properties of Equality (III) ππ π = π, π‘βππ ππ = ππ Multiply both sides by c Rule 8: Shortcut ππ π₯ + π = π, π‘βππ π₯ = π + (β1) β π Rule 9: Properties of Negation β1 π = βπ β1 1 = β1 β βπ = π β β2 = 2 βπ π = β ππ = π βπ β2 π₯ = β 2π₯ = 2(βπ₯) βπ βπ = ππ β2 βπ₯ = 2π₯ β π + π = βπ + βπ β π₯ + 2 = βπ₯ + β2 = βπ₯ β 2 Rule 10: Rules of Signs for fractions π βπ π βπ π β = = πππ = π π βπ βπ π the negative can go anywhere in the fraction and two negatives equal a positive Rule 11: Distributive Property (I) When we are adding and multiplying with a parenthesis, we can distribute the multiplication through the addition. π π + π = ππ + ππ π₯ 2 + 3π¦ = 2π₯ + 3π₯π¦ Rule 12: Distributive Property (II) When we are adding and dividing with a parenthesis, we can distribute the division through the addition. π+π π π = + π π π 2 + 3π¦ 2 3π¦ = + π₯ π₯ π₯ Rule 13: Factoring property (I) ππ + ππ = π π + π 2π₯ + 3π₯π¦ = π₯ 2 + 3π¦ Rule 14: Factoring property (II) π π π+π + = π π π 2 3π¦ 2 + 3π¦ + = π₯ π₯ π₯ Properties and Operations of Fractions Let a, b, c and d be real numbers, variables, or algebraic expressions such that b and d do not equal 0. Rule 15: Generate Equivalent Fractions π ππ = π β 0 π ππ multiplying the top and bottom by the same thing keeps the fraction the same value Rule 16: Add/Subtract with Like Denominators π π πβπ β = π π π if the denominators are the same, add or subtract the top of the fraction Rule 17: Add/Subtract with Unlike Denominators π π ππ ± ππ β = π π ππ find a common denominator Rule 18: Multiply Fractions π π ππ β = π π ππ top times the top and bottom times the bottom Rule 19: Create a parameter x β a + b = c is equivalent to x β y = c and y = (a + b) Rule 20: Commutative Property of Addition We can add numbers in any order. π+π =π+π π₯+2=2+π₯ Rule 21: Commutative Property of Multiplication We can also multiply numbers in any order. ππ = ππ π₯ β 2 = 2π₯ Rule 22: Associative Property of Addition We can group numbers in a sum any way we want and get the same answer. π+π +π =π+ π+π π₯ + 2 β 3π¦ = π₯ + (2 β 3π¦) Rule 23: Associative Property of Multiplication We can group numbers in a product any way we want and get the same answer. ππ π = π ππ π₯ β 2 3π¦ = π₯(2 β 3π¦) Rule 24: Properties of Zero π β 0 = π 0 added or subtracted to anything equals itself π β 0 = 0 0 multiplied by anything equals 0 0 = 0 π β 0 π 0 divided by anything equals 0 Donβt forget that you can never divide by 0 π ππ π’ππππππππ 0 We cannot divide by 0