2.3-2.11 Principles | Advanced Algebra 8

RPE — Reflexive Principle for Equality

x = x

SPE — Symmetric Principle for Equality

If x = y, then y = x

TPE — Transitivity Principle for Equality

If x = y and y = z, then x = z

CPA — Commutative Principle for Addition

x + y = y + x

CPM — Commutative Principle for Multiplication

x • y = y • x or xy = yx

APA — Associative Principle for Addition

(x + y) + z = x + (y + z)

APM — Associative Principle for Multiplication

(x • y) • z = x • (y • z) or (xy)z = x(yz)

DPS — Definition Principle of Subtraction

x − y = x + (−y)

DPD — Definition Principle of Division

x ÷ y = x • 1/y, where y ≠ 0

DOS — Definition Principle for Squaring

x • x = x² or xx = x²

DZP — Definition of Zero Power

x⁰ = 1, where x ≠ 0

DPMA — Distributive Principle for Multiplication Over Addition

x(y + z) = (xy) + (xz) OR (x + y)z = (xz) + (yz)

DPMS — Distributive Principle for Multiplication Over Subtraction

x(y − z) = (xy) − (xz) OR (x − y)z = (xz) − (yz)

DPOA — Distributive Principle for Oppositing Over Addition

−(x + y) = (−x) + (−y)

DPDA — Distributive Principle for Division Over Addition

(x + y) ÷ z = (x ÷ z) + (y ÷ z), where z ≠ 0

DPDS — Distributive Principle for Division Over Subtraction

(x − y) ÷ z = (x ÷ z) − (y ÷ z), where z ≠ 0

IPO — Introduction Principle for Oppositing

x + (−x) = 0

IPR — Introduction Principle to Reciprocals

x • 1/x = 1, where x ≠ 0

PM+1 — Principle for Multiplying by +1

x • (+1) = x

PM1 — Principle for Multiplying by −1

x • (−1) = −x

PM0 — Principle for Multiplying by Zero

x • 0 = 0

PA0 — Principle for Adding Zero

x + 0 = x

OOP — Opposite of an Opposite Principle

−(−x) = x

ZPP — Zero Product Property

If xy = 0, then x = 0 or y = 0

TPAE — Transformation Principle of Addition for Equality

If x = y, then x + z = y + z OR z + x = z + y

TPSE — Transformation Principle of Subtraction for Equality

If x = y, then x − z = y − z

TPME — Transformation Principle of Multiplication for Equality

If x = y, then xz = yz OR zx = zy

TPDE — Transformation Principle of Division for Equality

If x = y, then x ÷ z = y ÷ z, where z ≠ 0