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Properties of Equality, Identity, and Operations 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) 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) 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) 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 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 Zero Property of Multiplication n·0=0 Simply stated, any number times zero equals zero. 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. ½n–3=4 ½ n -3 + 3 = 4 + 3 ½n=7 ½ n (2) = 7(2) n = 14 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 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 Reflexive Property a=a a+b=a+b Any quantity is equal to itself. 7=7 2+3=2+3