Linear systems in two variables and degenerate constraints

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

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 — Distinguish nonzero coefficients, substitution checks, and degenerate constraints.
• Plane vectors in a fixed basis — 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.