# Linear equation in one variable

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A linear equation in one variable can be written as ax+b=0 with a nonzero linear coefficient a.

mko-linear-equation · definition · v0.10.0



## 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](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/linear-equation.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-linear-equation/index.md)