# Linear systems in two variables and degenerate constraints

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

A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks.

mko-system-linear-equations · definition · v0.10.0



## Mathematical statement

For real unknowns x and y, require a x+b y=c and d x+e y=f simultaneously. Define L(x,y)=(a x+b y−c,d x+e y−f) and z=(0,0), so the solution set satisfies L(x,y)=z. If D=ae−bd≠0, the unique solution is x=(ce−bf)/D and y=(af−cd)/D. If D=0, check consistency: there may be no solution or infinitely many solutions.





## Representative formula (exact TeX)

L(x,y)=z





## Assumptions


- All coefficients and right-side constants are real. Exercises use integer coefficients and exact rational candidates.
- Zero-coefficient rows are allowed as linear-system constraints: 0=0 imposes no restriction, while 0=c≠0 is inconsistent. Do not apply the unique-solution division when D=0.





## Symbol scope


- L — vector of the two linear-constraint residuals
- x — first real unknown
- y — second real unknown
- z — zero vector (0,0)





## Intuition, definition, and misconceptions

To verify a rational candidate pair, first check nonzero denominators and substitute into both rows. Passing only one row does not make it a solution of the system.

Use the unique-solution formula only for D≠0. When D=0, coefficient rank and consistency of the augmented data determine no solution or infinitely many solutions; first check inconsistent zero rows. If all coefficients and right sides are zero, all of R² is the solution set; if all coefficients are zero and any right side is nonzero, there is no solution.

The exercises use exact two-by-two minors and zero-row cases to check consistency. D and both Cramer numerators being zero are insufficient by themselves to establish infinitely many solutions. Exact solutions are not guessed from intersections in a sketch.

Common errors include substituting into only one row, dividing when D=0, or overlooking an inconsistent row 0=nonzero.





## Adopted derivation guide

[Linear systems in two variables: the unique-solution formula, checking both rows, and consistency when D=0](https://ocme.evemisslab.com/en/derivations/derivation-mko-system-linear-equations-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Linear equation in one variable](https://ocme.evemisslab.com/en/mko/mko-linear-equation/) — Distinguish nonzero coefficients, substitution checks, and degenerate constraints.
- [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/mko/mko-vector/) — Solutions and residuals use ordered pairs of components.





## Verification boundary

Explicit two-row systems in two variables can be checked exactly; general rank theorems and formal correctness of a solver await Evidence.

not_formalized

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





## Next obligation

Bind solution sets of linear constraints, zero rows, the two-by-two determinant, rank and augmented consistency, and the solution formula for nonzero D.





## 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/system-linear-equations.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-system-linear-equations/index.md)