Linear equation in one variable

Mathematical statement

In an explicitly stated rational- or real-number context, ax+b=0 with a≠0 is a linear equation in one variable and has the unique solution x=-b/a. When a=0, the statement is a degenerate constant equation and is outside this definition.

Representative formula (exact TeX)

x+1=0

Assumptions

• The variable x and coefficients a and b lie in an explicitly stated rational- or real-number context.
• The linear coefficient a is nonzero.

Symbol scope

• x — the single unknown variable
• a — the nonzero linear coefficient
• b — the constant term

Intuition, definition, and misconceptions

“Linear” means that the unknown occurs only in a first-degree term. The representative form is ax+b=0 with a≠0.

Over the rational or real numbers, ax+b=0 gives ax=-b. Dividing by the explicitly nonzero coefficient a yields the unique solution x=-b/a, which can then be substituted into the original equation for verification.

If a=0, the equation degenerates to b=0. When b=0 every permitted value of x satisfies it; when b≠0 there is no solution. Neither case may be presented as a linear equation with a unique solution.

Common mistakes include allowing a=0, confusing a line equation with any expression containing x, dividing by a coefficient not proved nonzero, or transforming an equation without substituting the result back into the original equation.

Prerequisites

• Equation — A linear equation in one variable is an equation with a specific expression shape and solution-set condition.

Verification boundary

Small integer coefficients and exact rational solutions support finite recomputation, but do not formally prove the theory of all linear systems, linear functions, or general equation solving.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind exact statements for a≠0, ax+b=0, the unique solution -b/a, and the degenerate a=0 classification.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.