1.17 One-Step Equation Definitions

One-Step Equation Definitions establishes the vocabulary for the simplest class of solvable algebraic equations, those requiring only a single application of a property of equality to isolate the variable, along with the two structural forms such equations most commonly take and the process of separating the variable from the rest of the equation.

One-Step Equation

A one-step equation is an equation in which the variable can be fully isolated by applying exactly one property of equality—one addition, subtraction, multiplication, or division applied to both sides—after which the variable stands alone on one side with its value revealed on the other. The equation x + 7 = 12 is a one-step equation, since subtracting 7 from both sides immediately isolates x.

Variable Isolation

Variable isolation is the goal and process of rewriting an equation, through one or more equivalence steps, until the variable appears alone on one side of the equation with a numerical value on the other side. In a one-step equation, isolation is achieved in a single equivalence step; more complex equations require additional steps, but the underlying goal of isolation remains the same throughout elementary algebra.

Additive Equation Form

The additive equation form is a one-step equation in which the variable is combined with a constant through addition or subtraction, such as x + 7 = 12 or x − 4 = 9. Isolating the variable in an additive equation form requires applying the addition or subtraction property of equality—performing the opposite operation on both sides to remove the added or subtracted constant from the variable's side.

Multiplicative Equation Form

The multiplicative equation form is a one-step equation in which the variable is combined with a constant through multiplication or division, such as 5x = 30 or x/3 = 6. Isolating the variable in a multiplicative equation form requires applying the multiplication or division property of equality—performing the opposite operation on both sides to remove the multiplied or divided constant from the variable's side.

Together, these definitions distinguish the two basic structural forms a one-step equation can take—additive and multiplicative—and describe variable isolation as the single equivalence step that resolves each form directly, forming the simplest and most foundational case of the broader equation-solving process used throughout elementary algebra.