Equation

Mathematical statement

Given two expressions L(x) and R(x) in a common context, the solution set of L(x)=R(x) consists of all allowed x for which both sides are defined and equal.

Representative formula (exact TeX)

x+1=3

Assumptions

• The allowed variable domain, expression types, and operation rules are explicitly stated.

Symbol scope

• x — variable to be solved for or checked
• = — condition that the two sides have equal values

Intuition, definition, and misconceptions

An equation is not a single expression. Each side of the equality sign is an expression, and both must be interpreted over the same variable domain.

To check a candidate solution, substitute it into the original equation and evaluate both sides separately. The candidate belongs to the solution set only when both sides are defined and equal.

Adding or subtracting the same quantity on both sides, or multiplying both sides by the same quantity, maps solutions of the original equality to the new one. Preserving exactly the same solution set under multiplication requires a nonzero factor. Division also requires proving the divisor nonzero; rules for linear equations do not automatically extend to all equations.

Common mistakes include simplifying only one side, claiming the solution set is unchanged after multiplying by zero, dividing by an expression that may be zero, treating an expression as an equation, or failing to substitute a candidate back into the original equation.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Variables and algebraic expressions — An equation consists of two expressions interpreted in one shared context.

Verification boundary

One-variable linear equations with integer coefficients support exact finite recomputation, but that does not prove completeness or correctness for every equation-solving algorithm.

not_formalized

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

Next obligation

Formalize equality, solution sets, and equivalent transformations over explicit domains while separating premises such as a nonzero divisor.

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.