3.8 Proof in Algebra

Vocabulary flashcards covering the properties and definitions used in 3.8 Proof in Algebra algebraic proofs.

Distributive Property

The algebraic property stating that multiplying a sum by a value multiplies each addend individually, justifying steps such as $-a(b + c) = (-a)b + (-a)c$ and $(a + b) \times \frac{1}{b} = a \times \frac{1}{b} + b \times \frac{1}{b}$.

Property of Opposites

The algebraic property used to simplify products involving negative signs, justifying the step $(-a)b + (-a)c = -ab + (-ac)$.

Definition of Subtraction

The mathematical definition stating that adding a negative quantity is equivalent to subtracting that quantity, justifying the transition from $-ab + (-ac)$ to $-ab - ac$.

Definition of Division

The algebraic definition stating that dividing by a non-zero number is equivalent to multiplying by its reciprocal, justifying steps such as $(ab) \times \frac{1}{b}$ for $(ab) \times b^{-1}$ and rewrites like $\frac{a + b}{b} = (a + b) \times \frac{1}{b}$.

Associative Property of Multiplication

The property stating that grouping factors differently does not alter their product, used in proofs to regroup $(ab) \times \frac{1}{b}$ into $a \times \frac{b \times 1}{b}$.

Property of Reciprocal

The property stating that the product of any non-zero real number and its reciprocal is equal to $1$, justifying $b \times \frac{1}{b} = 1$.

Identity Property of Multiplication

The property stating that multiplying any real number by $1$ leaves the value unchanged, justifying the simplification $a \times 1 = a$.

Commutative Property of Addition

The property stating that the order in which two terms are added does not affect their sum, justifying the final rearrangement $\frac{a}{b} + 1 = 1 + \frac{a}{b}$.

Algebraic Proof of $-a(b + c) = -ab - ac$

A three-step proof justified sequentially by: (1) distributive property, (2) property of opposites, and (3) definition of subtraction.

Algebraic Proof of $(ab) \times b^{-1} = a$ (where $b \neq 0$)

A four-step proof showing $(ab) \div b = a$ justified sequentially by: (1) definition of division, (2) associative property of multiplication, (3) property of reciprocal, and (4) identity property of multiplication.

Algebraic Proof of $\frac{a + b}{b} = 1 + \frac{a}{b}$ (where $b \neq 0$)

A five-step proof justified sequentially by: (1) definition of division, (2) distributive property, (3) property of reciprocal, (4) definition of division, and (5) commutative property of addition.