Algebra 1 Chapter 2: Linear Equations, Literal Equations, Ratios, and Inequalities

Vocabulary practice flashcards covering fundamental definitions, algebraic properties of equality, literal equations, ratios, rates, proportions, linear modeling, and inequalities from Chapter 2.

Equation

An open sentence stating that two algebraic expressions are equal.

Linear Equation

An equation that has variables with an exponent or degree of $1$.

Solution (Root)

The value of the variable that makes an equation true.

Conditional Equation

An equation that is true for only some variable values (for example, $8x = 16$, which is true when $x = 2$).

Identity

An equation that is true for all defined values (for example, $4x + 2x = 6x$).

Contradiction

An equation that produces a false statement, such as $1 = 0$, meaning there is no solution.

Addition Property of Equality

The property stating that if $a = b$, then $a + c = b + c$.

Subtraction Property of Equality

The property stating that if $a = b$, then $a - c = b - c$.

Additive Inverse

The mathematical property stating that a number and its opposite add to zero: $a + (-a) = (-a) + a = 0$.

Multiplication Property of Equality

The property stating that if $a = b$, then $a \cdot c = b \cdot c$.

Division Property of Equality

The property stating that if $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

Multiplicative Inverse

The property stating that for any nonzero number $c$, multiplying by its reciprocal yields one: $c \cdot \left(\frac{1}{c}\right) = 1$.

Multiplicative Identity

The property stating that multiplying by $1$ leaves a number unchanged: $1 \cdot a = a$.

Additive Identity

The property stating that adding $0$ to a number leaves it unchanged: $a + 0 = 0 + a = a$.

Associative Property of Addition

The algebraic property stating that grouping does not change the sum: $(a + b) + c = a + (b + c)$.

Commutative Property

The property stating that the order of terms does not affect the outcome: $a + b = b + a$ and $a \cdot b = b \cdot a$.

Transitive Property of Equality

The logical property of equality stating that if $a = b$ and $b = c$, then $a = c$.

Distributive Property

The property asserting that multiplication distributes across addition or subtraction: $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$.

Substitution Principle

The mathematical principle stating that equal quantities can be substituted for each other.

Literal Equation

An equation containing two or more variables, such as geometric or physical formulas, in which inverse operations are used to isolate a specified variable.

Ratio

A mathematical comparison of two numbers or quantities by division, written in words, colon notation ($4 : 3$), or fraction notation ($\frac{4}{3}$).

Continued Ratio

A ratio that compares three or more quantities in a specific order, such as $a : b : c = 2 : 3 : 5$.

Rate

A ratio that compares quantities expressed in different units, such as miles per hour or dollars per gallon.

Unit Rate

A rate that has a denominator of $1$, calculated by dividing the first quantity by the number of units.

Unit Price

The cost for one unit of an item, such as dollars per pound or cents per ounce, used to compare item costs.

Proportion

A mathematical statement or equation stating that two ratios are equal, expressed as $\frac{a}{b} = \frac{c}{d}$ or $a : b = c : d$.

Cross Products

The diagonal products formed in a proportion that are always equal to each other ($ad = bc$ for $\frac{a}{b} = \frac{c}{d}$).

Modeling with Linear Equations

The process of translating words and quantities from a real-world problem into an algebraic expression or linear equation to solve for an unknown.

Distance-Rate-Time Relationship

The uniform motion formula defined by $\text{distance} = \text{rate} \times \text{time}$ ($d = rt$).

Inequality

A mathematical statement comparing quantities using relational symbols: $<$ (less than), $>$ (greater than), $\le$ (less than or equal to), or $\ge$ (greater than or equal to).

Inequality Sign Reversal Rule

The rule dictating that an inequality symbol must be reversed whenever multiplying or dividing both sides by a negative number.

At Least

A phrasing in mathematical modeling indicating a minimum value or limit, represented by the greater than or equal to symbol ($\ge$).

At Most

A phrasing in mathematical modeling indicating a maximum value or limit, represented by the less than or equal to symbol ($\le$).