ALGEBRA II PROPERTIES PROVABLE FROM AXIOMS

Review Axioms and Properties handout. Commutative Property for Addition (CPA) : = Commutative Property for Multiplication (CPM) : 7 4 = 4 7 Associative Property for Addition (APA) : (7 + 5) + 9 = 7 + (5 + 9) Associative Property for Multiplication (APM) : (7 5) 9 = 7 (5 9) Distributive Property for Multiplication Over Addition (DPMA) : 4(x + 5) = 4 x Additive Identity : x + 0 = x Multiplicative Identity : x 1 = x Additive Inverse : 7 + (-7) = 0 Multiplicative Inverse : Property for Multiplying by -1 (PM-1) : 4 -1 = -4, = 6 Property for Multiplying by Zero (PMZ) : x 0 = 0, 0 x = 0 Definition of Division : Definition of Subtraction : 7 + (-4) = 7 – 4 Reflexive Property : 14 = 14 Symmetric Property : If = 12, then 12 = Transitive Property : If a = b and b = c, then a = c Addition Property of Equality (APE) : If x = y, then x + z = y + z. You can add the same number to both sides of an equation and not affect the solution. Multiplication Property of Equality (MPE) : If x = y, then xz = yz, z ≠ 0. You can multiply both sides of an equation by the same non-zero number. Converse : A statement is true “both ways” you read it. For example : If a figure is a triangle, then the sum of the angles is 180º. The converse reads : If the sum of the angles is 180º, then the figure is a triangle. Substitution : If a = b + c, then (usually later in the proof), d = a, then d = b + c. Trichotomy (Comparison Property) : Given any two real numbers a and b, exactly one of the following is true : a > b, a < b, or a = b.

1) Prove : If x + z = y + z, then x = y. STATEMENTREASON a) x + z = y + z a) Given b) x + z + (-z) = y + z + (-z) b) APE c) x + [z + (-z)] = y + [z + (-z)] c) APA d) x + 0 = y + 0 d) Inverse e) x = y e) Identity We just proved the converse of APE.

2) Prove : If x + b = a, then x = a + (-b) STATEMENTREASON a) x + b = aa) Given b) x + b + (-b) = a + (-b) b) APE c) x + 0 = x + (-b) c) Inverse d) x = a + (-b) d) Identity Usually, identity follows inverse.

3) Prove : If ab = b and b ≠ 0, then a = 1. STATEMENTREASON a) ab =b and b ≠ 0a) Given b)b) MPE c) c) APM d) a 1 = 1d) Inverse e) a = 1 e) Identity

Prove : If ax + b = c and a ≠ 0, then STATEMENTREASON a) ax + b = c, a ≠ 0a) Given b) ax + b + (-b) = c + (-b)b) APE c) ax + [b + (-b)] = c + (-b)c) APA d) ax + 0 = c + (-b)d) Inverse e) ax = c + (-b)e) Identity f) f) MPE g)g) APM h) h) Inverse i)i) Identity