Linear Equations in One Variable and Properties of Equality

Vocabulary terms covering definitions, properties of equality, equation classifications, and solving procedures for linear equations in one variable.

Linear Equation in One Variable

An equation of the form $Ax + B = 0$, where $A$ and $B$ are real numbers.

Reflexive Property of Equality

The property stating that $a = a$, meaning a number is equal to itself.

Symmetric Property of Equality

The property stating that if $a = b$, then $b = a$, meaning the two sides of an equation can be interchanged.

Transitive Property of Equality

The property stating that if $a = b$ and $b = c$, then $a = c$, meaning quantities that are equal to the same thing are equal.

Addition Property of Equality

The property stating that if $a = b$, then $a + c = b + c$, meaning the same number can be added to each side of an equation.

Subtraction Property of Equality

The property stating that if $a = b$, then $a - c = b - c$, meaning the same number can be subtracted from each side of an equation.

Multiplication Property of Equality

The property stating that if $a = b$, then $ac = bc$, meaning the same number can be multiplied to each side of an equation.

Division Property of Equality

The property stating that if $a = b$, then $\frac{a}{c} = \frac{b}{c}$ where $c \neq 0$, meaning each side of an equation can be divided by the same nonzero number.

Guess-and-Check

The process of replacing a variable (such as $x$) with a specific real number and evaluating both sides of an equation to determine whether the equality is true.

Solution (or Root)

A number which makes an equation true when substituted in place of the variable.

Solving the Equation

The process of finding the solution or root of an equation.

Solution Set

The set of all solutions to an equation.

Equivalent Equations

Equations that share the exact same solution set.

Steps in Solving a Linear Equation in One Variable

1. Simplify each side of the equation by performing all indicated operations; 2. Apply properties of equality to isolate the variable of interest; 3. Check the solution in the original equation.

Identity

An equation whose solution set is the set of all real numbers because the equality is always true regardless of what real number replaces the variable.

Inconsistent Equation (or Contradiction)

An equation that has no solution (its solution set is the empty set) because the resulting statement is always false for any real number replacement.

Conditional Equation

A non-identity equation that has at least one solution and is true only under specific conditions rather than for all real numbers.

Clearing Fractions in an Equation

The procedure of multiplying both sides of an equation by the least common denominator ($\text{LCD}$) of the fractions using the multiplication property of equality.

Clearing Decimals in an Equation

The procedure of multiplying both sides of an equation by a power of $10$, with as many zeros as the maximum number of decimal places among the terms.