linear equation
Enable an agent to recognise a linear equation, determine its solution conditions, and choose transformations or solution methods that preserve its meaning.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an agent to recognise a linear equation, determine its solution conditions, and choose transformations or solution methods that preserve its meaning.
A linear equation is an equation expressible as a₁x₁ + ⋯ + aₙxₙ = b, where the unknowns xᵢ occur only to the first power, the coefficients aᵢ and constant b are independent of the unknowns, and at least one coefficient is nonzero.
It can be Identify designated unknowns and classify linearity under explicit parameter assumptions.; Reduce the equation to a linear form while retaining inherited domain restrictions.; Isolate an unknown when its coefficient is invertible, splitting parameter cases when necessary.; Describe or parameterise the admissible solution set and verify candidate assignments.; Interpret a real-valued equation as a line, plane or hyperplane when its coefficients justify that interpretation.; Evaluate numerical residuals and assess whether a computed solution meets an explicitly chosen tolerance..
Distinguishing features
It asserts equality and constrains admissible values; an expression such as 3x + 2 alone is not an equation.
After valid simplification, designated unknowns occur only to the first power and are not multiplied together or retained inside nonlinear operations.
Linearity is relative to the designated unknowns: ax = b is linear in x when a is a parameter, but generally nonlinear when both a and x are unknowns.
It contains one scalar equality; several simultaneous equalities belong to a system, even when written as one vector equation.
A nonzero right-hand constant is allowed: x + y = 1 is a linear equation whose real solution set is affine, although the corresponding residual function is not a linear map.
Scope
+ Designation of unknowns, parameters, coefficients and the constant term
+ Original notation and a domain-preserving linear form
+ Admissible values and restrictions inherited from the original equation
+ Consistency, solution sets and parameter-dependent degeneracies
+ Equivalent transformations and verification of candidate solutions
+ Geometric and applied interpretations of a single linear constraint
- Systems of multiple equations and their collective rank or consistency
- Linear inequalities and optimisation objectives
- Linear functions and linear maps considered independently of an equation
- Differential, integral and operator equations whose unknowns are functions
- Nonlinear equations and approximate linearisation of nonlinear models
- Authorship, editions, rights and physical carriers of mathematical texts
Characteristics
- Designated unknowns
- Ordered list of symbols interpreted as scalar unknowns Determines whether an occurrence is a coefficient, parameter or nonlinear interaction.
- Scalar domain
- Real numbers, complex numbers, rational numbers, integers, a specified finite field, or another explicitly defined domain Controls allowed arithmetic and whether existence, uniqueness and geometric claims apply.
- Coefficient assignment
- Mapping from each designated unknown to its coefficient, including parameter dependencies Supports classification, solution and detection of branches where coefficients vanish.
- Constant term
- Scalar b in the recorded form Σaᵢxᵢ = b Distinguishes homogeneous constraints and determines consistency when every coefficient vanishes.
- Admissibility restrictions
- Excluded values, interval restrictions, integrality requirements or other predicates Prevents algebraic simplification from silently admitting values forbidden by the original equation or context.
- Consistency and solution multiplicity
- Undetermined, no solution, one solution, multiple finite solutions, or infinitely many solutions, conditional on assumptions Expresses the equation's current mathematical status without assuming a real-valued domain.
- Homogeneity
- Homogeneous when b = 0; nonhomogeneous when b ≠ 0; parameter-dependent when unresolved Supports valid claims about zero solutions and vector-space structure.
- Numerical representation
- Exact symbolic values, exact rational values, rounded floating-point values, or uncertain measured values Determines whether exact equality, numerical residuals or uncertainty bounds are appropriate.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 6 bundles · 11 layers · 15 findings · 25 questions.
Equation identity and linearity Establishes which equality is being modelled and relative to which unknowns it is linear.
The same notation can represent a linear equation, a nonlinear constraint or an expression depending on symbol roles and valid simplifications.
Equality and symbol roles
Records the original equality and separates unknowns from fixed or conditional inputs.
Designated unknowns determine linearity
A coefficient may depend on parameters but must be independent of the designated unknowns.
- Which symbols are unknowns to solve for, and which are coefficients or parameters? definition
- Would treating any current parameter as an additional unknown introduce products or other nonlinear dependence? boundary
Linear form and degeneracy
Tests reduction to a linear form and makes zero-coefficient outcomes explicit.
Validated linear form
The recorded form Σaᵢxᵢ = b must follow from valid simplification; all-zero coefficients require separate identity or contradiction handling.
- What domain-valid steps produce Σaᵢxᵢ = b from the original equality? action
- Can all coefficients vanish, and does that branch reduce to 0 = 0 or 0 = b with b nonzero? boundary
Domains and parameter conditions Defines the arithmetic setting and the conditions under which coefficients and unknowns are interpreted.
An equation such as 2x = 1 has different solution behaviour over the reals, integers and fields of characteristic two.
Scalar domain and admissibility
Separates the ambient scalar domain from additional restrictions on assignments.
Domain controls solvability
Division, solution counts and admissibility must be assessed in the specified domain rather than presumed from real arithmetic.
- In which scalar domain are the equation's operations and equality interpreted? definition
- Which otherwise algebraic solutions are excluded by integrality, bounds or restrictions inherited from the original expression? boundary
Parameter branches
Tracks coefficient conditions that change available operations or solution behaviour.
Coefficient invertibility is conditional
Isolating an unknown by division requires an invertible coefficient; parameter values may invalidate that step.
- Under which parameter conditions is the coefficient selected for division invertible? boundary
- What separate solution analysis is required when that coefficient is zero or otherwise noninvertible? action
Transformations and verification Controls algebraic manipulation and checks whether a proposed solution satisfies the original constraint.
Unrecorded division, cancellation or numerical tolerance can change the solution set or misclassify a candidate.
Solution-preserving transformations
Records transformation conditions and distinguishes equivalence from one-way implication.
Equivalence requires operation conditions
Adding a defined expression to both sides preserves equality; multiplying or dividing requires attention to invertibility and retained restrictions.
- Does each transformation preserve exactly the admissible solution set, or only imply the next equation? action
- Which nonzero assumptions or excluded values must accompany cancellation or denominator clearing? boundary
Candidate and residual checks
Verifies assignments against the original equation using exact or explicitly approximate criteria.
Verification matches number representation
Exact candidates require equality and admissibility checks; approximate candidates require a stated residual criterion whose scale is understood.
- Does substituting the candidate into the original equality satisfy both sides and every retained restriction? action
- For an approximate candidate, what residual, scaling convention and tolerance determine acceptance? measurement
Solution set and geometry Describes consistency, degrees of freedom and the geometry supported by the chosen domain.
A single linear equation may determine one value, leave free variables, exclude every assignment or impose no effective constraint.
Consistency and parameterisation
Determines the solution set under each domain and parameter branch.
Nonzero field coefficient enables isolation
Over a field, with unrestricted unknowns and at least one nonzero coefficient, one unknown can be expressed in terms of the other n − 1; further restrictions may reduce or empty this set.
- Which unknown can be isolated, and which remaining unknowns are free within the stated domain? action
- After applying all restrictions, is the solution set empty, a singleton, finite with several members or infinite? measurement
Affine geometric interpretation
Relates real solutions to geometric constraints without confusing affine sets with linear maps.
Nonzero coefficient vector defines a hyperplane
In unrestricted real n-space, a nonzero coefficient vector defines an affine hyperplane of dimension n − 1; it passes through the origin exactly when b = 0.
- Do the domain and coefficient conditions support a real hyperplane interpretation, and what is its dimension? boundary
- What coefficient vector supplies a normal direction, and does the solution set contain the origin? measurement
Applied meaning and numerical use Connects an applied equation to quantities, assumptions and the reliability of computed solutions.
Correct algebra alone cannot establish whether terms have compatible units or whether uncertain coefficients support a reliable answer.
Quantity interpretation
Records what variables and coefficients mean when the equation represents an external situation.
Terms require compatible quantity meaning
For a dimensional applied equation, each additive term and the right-hand side must represent compatible quantities, with assumptions supporting the asserted relation.
- What quantities and units do the unknowns, coefficients and constant represent? definition
- Which derivation, observation or modelling assumption supports this particular linear equality and its range of use? provenance
Uncertainty and sensitivity
Separates exact symbolic solvability from numerical reliability under rounded or uncertain inputs.
Input uncertainty propagates through solution
For an isolated unknown, coefficient and constant uncertainty can materially affect the answer, particularly near parameter values where the isolating coefficient vanishes.
- Which coefficients and constants are exact, rounded or uncertain, and what uncertainty information is available? measurement
- How much can the selected solution vary under the stated input uncertainty and assumptions about free variables? measurement
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- This describes the conventional algebraic sense; the supplied domain code INF.MED should be checked against the registry's intended classification.
- The usual elementary setting is real or complex scalars; restricting unknowns to integers or changing the underlying algebraic structure changes solvability.
- Some treatments include degenerate forms with every coefficient zero; these reduce to an identity or a contradiction and are excluded by this definition.
- Which of these check these first hold for the sense of linear equation this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Linear equation in one unknown
- Linear equation in two unknowns
- Linear equation in several unknowns
- Homogeneous linear equation (b = 0)
- Nonhomogeneous linear equation (b ≠ 0)
- Which of these kinds and varieties hold for the sense of linear equation this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Expressing balance constraints in budgets, material flows and electrical circuits
- Representing straight lines and hyperplanes in coordinate geometry
- Formulating constraints in linear programming
- Approximating nonlinear relationships near a chosen operating point
- Constructing systems of equations for engineering and scientific computation
- Which of these real-world use hold for the sense of linear equation this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Treating products, powers or variable denominators involving the unknowns as linear terms
- Dividing by a coefficient without checking whether it can be zero
- Combining quantities with incompatible physical units
- Using a linear approximation outside the region where it is accurate
- Confusing linearity in fitted parameters with linearity in predictor variables
- Which of these failure modes and hazards hold for the sense of linear equation this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Linear expression - An expression has no asserted equality; a linear equation equates an expression to another expression or value.
- System of linear equations - A system imposes multiple linear equations simultaneously on shared unknowns.
- Linear inequality - An inequality uses an ordering relation such as ≤ rather than equality.
- Nonlinear equation - Its dependence on the designated unknowns cannot be expressed in linear form, for example because it includes their products or squares.
- Linear map - A linear map preserves addition and scalar multiplication; a linear equation specifies a constraint, and its solution set need not pass through the origin.
- Linear differential equation - Its unknown is a function acted on linearly by differentiation, rather than a finite collection of scalar unknowns.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of linear equation this model covers, and on what evidence? provenance
What the second pass must settle
- Does the registry intend only elementary scalar equations over real or complex numbers, or also integer, finite-field and more general algebraic domains?
- Should identities and contradictions obtained when every variable coefficient vanishes be classified as linear equations themselves or only as degenerate outcomes?
- Should equations that become linear through domain-restricted cancellation belong to this entry or remain linked instances of a broader equation model?
- Which existing Vercy models already own equation equivalence, scalar domains, quantities and solution sets, so this entry can reuse their boundaries?
- Which authoritative mathematical sources and registry evidence should establish the final definition and explain the INF / INF.MED placement?