# Two-Equation Ranks, Residuals, and Row-Operation Guards

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261012-advance-20261016-r05-system-linear-equations-foundations · 2026-10-12 · 100 questions

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



## 1. Two Rational Substitution Residuals for a System

Over R, rows=[[2, 1, 1], [1, -1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[-1/-3, 1/3], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([0/1, 0/1], Yes)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0001/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 2. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 0], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([0/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0002/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 3. Two Rational Substitution Residuals for a System

Over R, rows=[[0, 0, 0], [0, 0, 2]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([0/1, -2/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, -2/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0003/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 4. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -9], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([9/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0004/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 5. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -8], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([8/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0005/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 6. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -7], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([7/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0006/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 7. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -6], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([6/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0007/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 8. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -5], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([5/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0008/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 9. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -4], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([4/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0009/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 10. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -3], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([3/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0010/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 11. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -2], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([2/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0011/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 12. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, -1], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([1/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0012/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 13. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 0], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 2/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([0/1, 2/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 2/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0013/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 14. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 1], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-1/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0014/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 15. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 2], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-2/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0015/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 16. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 3], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-3/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0016/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 17. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 4], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-4/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0017/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 18. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 5], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-5/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0018/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 19. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 6], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-6/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0019/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 20. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 7], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-7/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0020/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 21. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 8], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-8/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0021/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 22. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 9], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-9/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0022/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 23. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 10], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-10/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0023/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 24. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 11], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 1/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-11/1, 1/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-11/1, 1/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0024/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 25. Two Rational Substitution Residuals for a System

Over R, rows=[[1, 0, 12], [0, 1, 0]], where each [a,b,c] means a*x+b*y=c. Ordered candidate (x,y)=[0/1, 0/1], with nonzero representation denominators. Give (two exact rational a*x+b*y−c residuals, are both zero?).

View answer and explanation([-12/1, 0/1], No)

Normalize valid candidate fractions, including negative denominators, then substitute into both equations; passing one equation alone is insufficient. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-12/1, 0/1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0025/)

Linear systems in two variables and degenerate constraints · qf-density16-system-candidate-two-residuals · candidate





## 26. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[0, 0, 0], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(0, 0, 0, infinitely-many-solutions, not-unique)

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, infinitely-many-solutions, not-unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0026/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 27. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[0, 0, 1], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(0, 0, 1, no-solution, not-applicable)

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 1, no-solution, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0027/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 28. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 1], [2, 2, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(0, 1, 1, infinitely-many-solutions, not-unique)

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 1, infinitely-many-solutions, not-unique).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0028/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 29. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 1], [2, 2, 3]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(0, 1, 2, no-solution, not-applicable)

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 2, no-solution, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0029/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 30. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[2, 1, 1], [1, -1, 0]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(-3, 2, 2, unique-solution, [1/3, 1/3])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-3, 2, 2, unique-solution, [1/3, 1/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0030/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 31. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, -7], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-9/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-9/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0031/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 32. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, -6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-8/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-8/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0032/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 33. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, -5], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-7/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-7/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0033/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 34. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, -4], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-6/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-6/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0034/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 35. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, -3], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-5/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-5/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0035/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 36. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, -2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-4/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-4/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0036/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 37. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, -1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-3/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-3/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0037/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 38. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 0], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-2/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-2/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0038/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 39. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [-1/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [-1/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0039/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 40. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [0/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [0/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0040/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 41. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 3], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [1/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [1/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0041/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 42. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 4], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [2/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [2/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0042/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 43. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 5], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [3/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [3/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0043/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 44. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [4/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [4/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0044/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 45. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 7], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [5/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [5/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0045/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 46. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 8], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [6/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [6/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0046/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 47. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 9], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [7/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [7/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0047/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 48. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 10], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [8/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [8/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0048/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 49. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 11], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [9/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [9/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0049/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 50. Signed Determinant, Both Ranks, and System Class

Over R, rows=[[1, 1, 12], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.

View answer and explanation(1, 2, 2, unique-solution, [10/1, 2/1])

Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [10/1, 2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0050/)

Linear systems in two variables and degenerate constraints · qf-density16-system-classify-solve · candidate





## 51. Reversible Row Addition and Before/After Classification

Over R, rows=[[0, 0, 0], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[0, 0, 0], [0, 0, 0]], [0, 0, 0, infinitely-many-solutions, not-unique], [0, 0, 0, infinitely-many-solutions, not-unique], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0, 0], [0, 0, 0]], [0, 0, 0, infinitely-many-solutions, not-unique], [0, 0, 0, infinitely-many-solutions, not-unique], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0051/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 52. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 1, 1], [2, 2, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 1, 1], [0, 0, 0]], [0, 1, 1, infinitely-many-solutions, not-unique], [0, 1, 1, infinitely-many-solutions, not-unique], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1, 1], [0, 0, 0]], [0, 1, 1, infinitely-many-solutions, not-unique], [0, 1, 1, infinitely-many-solutions, not-unique], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0052/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 53. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 1, 1], [2, 2, 3]], where each [a,b,c] means a*x+b*y=c. t=-2; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 1, 1], [0, 0, 1]], [0, 1, 2, no-solution, not-applicable], [0, 1, 2, no-solution, not-applicable], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1, 1], [0, 0, 1]], [0, 1, 2, no-solution, not-applicable], [0, 1, 2, no-solution, not-applicable], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0053/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 54. Reversible Row Addition and Before/After Classification

Over R, rows=[[2, 1, 1], [1, -1, 0]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[2, 1, 1], [-1, -2, -1]], [-3, 2, 2, unique-solution, [1/3, 1/3]], [-3, 2, 2, unique-solution, [1/3, 1/3]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1, 1], [-1, -2, -1]], [-3, 2, 2, unique-solution, [1/3, 1/3]], [-3, 2, 2, unique-solution, [1/3, 1/3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0054/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 55. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -8], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -8], [0, 1, 2]], [1, 2, 2, unique-solution, [-8/1, 2/1]], [1, 2, 2, unique-solution, [-8/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -8], [0, 1, 2]], [1, 2, 2, unique-solution, [-8/1, 2/1]], [1, 2, 2, unique-solution, [-8/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0055/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 56. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -7], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -7], [1, 1, -5]], [1, 2, 2, unique-solution, [-7/1, 2/1]], [1, 2, 2, unique-solution, [-7/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -7], [1, 1, -5]], [1, 2, 2, unique-solution, [-7/1, 2/1]], [1, 2, 2, unique-solution, [-7/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0056/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 57. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -6], [-1, 1, 8]], [1, 2, 2, unique-solution, [-6/1, 2/1]], [1, 2, 2, unique-solution, [-6/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -6], [-1, 1, 8]], [1, 2, 2, unique-solution, [-6/1, 2/1]], [1, 2, 2, unique-solution, [-6/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0057/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 58. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -5], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -5], [0, 1, 2]], [1, 2, 2, unique-solution, [-5/1, 2/1]], [1, 2, 2, unique-solution, [-5/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -5], [0, 1, 2]], [1, 2, 2, unique-solution, [-5/1, 2/1]], [1, 2, 2, unique-solution, [-5/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0058/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 59. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -4], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -4], [1, 1, -2]], [1, 2, 2, unique-solution, [-4/1, 2/1]], [1, 2, 2, unique-solution, [-4/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -4], [1, 1, -2]], [1, 2, 2, unique-solution, [-4/1, 2/1]], [1, 2, 2, unique-solution, [-4/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0059/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 60. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -3], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -3], [-1, 1, 5]], [1, 2, 2, unique-solution, [-3/1, 2/1]], [1, 2, 2, unique-solution, [-3/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -3], [-1, 1, 5]], [1, 2, 2, unique-solution, [-3/1, 2/1]], [1, 2, 2, unique-solution, [-3/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0060/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 61. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -2], [0, 1, 2]], [1, 2, 2, unique-solution, [-2/1, 2/1]], [1, 2, 2, unique-solution, [-2/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -2], [0, 1, 2]], [1, 2, 2, unique-solution, [-2/1, 2/1]], [1, 2, 2, unique-solution, [-2/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0061/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 62. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, -1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, -1], [1, 1, 1]], [1, 2, 2, unique-solution, [-1/1, 2/1]], [1, 2, 2, unique-solution, [-1/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -1], [1, 1, 1]], [1, 2, 2, unique-solution, [-1/1, 2/1]], [1, 2, 2, unique-solution, [-1/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0062/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 63. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 0], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 0], [-1, 1, 2]], [1, 2, 2, unique-solution, [0/1, 2/1]], [1, 2, 2, unique-solution, [0/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 0], [-1, 1, 2]], [1, 2, 2, unique-solution, [0/1, 2/1]], [1, 2, 2, unique-solution, [0/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0063/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 64. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 1], [0, 1, 2]], [1, 2, 2, unique-solution, [1/1, 2/1]], [1, 2, 2, unique-solution, [1/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 1], [0, 1, 2]], [1, 2, 2, unique-solution, [1/1, 2/1]], [1, 2, 2, unique-solution, [1/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0064/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 65. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 2], [1, 1, 4]], [1, 2, 2, unique-solution, [2/1, 2/1]], [1, 2, 2, unique-solution, [2/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 2], [1, 1, 4]], [1, 2, 2, unique-solution, [2/1, 2/1]], [1, 2, 2, unique-solution, [2/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0065/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 66. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 3], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 3], [-1, 1, -1]], [1, 2, 2, unique-solution, [3/1, 2/1]], [1, 2, 2, unique-solution, [3/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 3], [-1, 1, -1]], [1, 2, 2, unique-solution, [3/1, 2/1]], [1, 2, 2, unique-solution, [3/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0066/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 67. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 4], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 4], [0, 1, 2]], [1, 2, 2, unique-solution, [4/1, 2/1]], [1, 2, 2, unique-solution, [4/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 4], [0, 1, 2]], [1, 2, 2, unique-solution, [4/1, 2/1]], [1, 2, 2, unique-solution, [4/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0067/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 68. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 5], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 5], [1, 1, 7]], [1, 2, 2, unique-solution, [5/1, 2/1]], [1, 2, 2, unique-solution, [5/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 5], [1, 1, 7]], [1, 2, 2, unique-solution, [5/1, 2/1]], [1, 2, 2, unique-solution, [5/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0068/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 69. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 6], [-1, 1, -4]], [1, 2, 2, unique-solution, [6/1, 2/1]], [1, 2, 2, unique-solution, [6/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 6], [-1, 1, -4]], [1, 2, 2, unique-solution, [6/1, 2/1]], [1, 2, 2, unique-solution, [6/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0069/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 70. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 7], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 7], [0, 1, 2]], [1, 2, 2, unique-solution, [7/1, 2/1]], [1, 2, 2, unique-solution, [7/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 7], [0, 1, 2]], [1, 2, 2, unique-solution, [7/1, 2/1]], [1, 2, 2, unique-solution, [7/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0070/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 71. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 8], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 8], [1, 1, 10]], [1, 2, 2, unique-solution, [8/1, 2/1]], [1, 2, 2, unique-solution, [8/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 8], [1, 1, 10]], [1, 2, 2, unique-solution, [8/1, 2/1]], [1, 2, 2, unique-solution, [8/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0071/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 72. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 9], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 9], [-1, 1, -7]], [1, 2, 2, unique-solution, [9/1, 2/1]], [1, 2, 2, unique-solution, [9/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 9], [-1, 1, -7]], [1, 2, 2, unique-solution, [9/1, 2/1]], [1, 2, 2, unique-solution, [9/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0072/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 73. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 10], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 10], [0, 1, 2]], [1, 2, 2, unique-solution, [10/1, 2/1]], [1, 2, 2, unique-solution, [10/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 10], [0, 1, 2]], [1, 2, 2, unique-solution, [10/1, 2/1]], [1, 2, 2, unique-solution, [10/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0073/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 74. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 11], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 11], [1, 1, 13]], [1, 2, 2, unique-solution, [11/1, 2/1]], [1, 2, 2, unique-solution, [11/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 11], [1, 1, 13]], [1, 2, 2, unique-solution, [11/1, 2/1]], [1, 2, 2, unique-solution, [11/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0074/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 75. Reversible Row Addition and Before/After Classification

Over R, rows=[[1, 0, 12], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-1; replace only row2 by row2+t*row1, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, invertible-row-operation eligibility).

View answer and explanation([[1, 0, 12], [-1, 1, -10]], [1, 2, 2, unique-solution, [12/1, 2/1]], [1, 2, 2, unique-solution, [12/1, 2/1]], Yes)

Add the complete rows and recompute each classification; adding −t times the first row reverses this operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 12], [-1, 1, -10]], [1, 2, 2, unique-solution, [12/1, 2/1]], [1, 2, 2, unique-solution, [12/1, 2/1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0075/)

Linear systems in two variables and degenerate constraints · qf-density16-system-eliminate-row · candidate





## 76. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[0, 0, 0], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. t=0; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[0, 0, 0], [0, 0, 0]], [0, 0, 0, infinitely-many-solutions, not-unique], [0, 0, 0, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0, 0], [0, 0, 0]], [0, 0, 0, infinitely-many-solutions, not-unique], [0, 0, 0, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0076/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 77. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 1, 1], [2, 2, 2]], where each [a,b,c] means a*x+b*y=c. t=0; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[0, 0, 0], [2, 2, 2]], [0, 1, 1, infinitely-many-solutions, not-unique], [0, 1, 1, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0, 0], [2, 2, 2]], [0, 1, 1, infinitely-many-solutions, not-unique], [0, 1, 1, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0077/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 78. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[0, 0, 0], [0, 1, 2]], [1, 2, 2, unique-solution, [1/1, 2/1]], [0, 1, 1, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0, 0], [0, 1, 2]], [1, 2, 2, unique-solution, [1/1, 2/1]], [0, 1, 1, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0078/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 79. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 1, 1], [2, 2, 3]], where each [a,b,c] means a*x+b*y=c. t=0; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[0, 0, 0], [2, 2, 3]], [0, 1, 2, no-solution, not-applicable], [0, 1, 1, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0, 0], [2, 2, 3]], [0, 1, 2, no-solution, not-applicable], [0, 1, 1, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0079/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 80. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -8], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, -8], [0, 1, 2]], [1, 2, 2, unique-solution, [-8/1, 2/1]], [1, 2, 2, unique-solution, [-8/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -8], [0, 1, 2]], [1, 2, 2, unique-solution, [-8/1, 2/1]], [1, 2, 2, unique-solution, [-8/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0080/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 81. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -7], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, 14], [0, 1, 2]], [1, 2, 2, unique-solution, [-7/1, 2/1]], [-2, 2, 2, unique-solution, [-7/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, 14], [0, 1, 2]], [1, 2, 2, unique-solution, [-7/1, 2/1]], [-2, 2, 2, unique-solution, [-7/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0081/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 82. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, -6], [0, 1, 2]], [1, 2, 2, unique-solution, [-6/1, 2/1]], [1, 2, 2, unique-solution, [-6/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -6], [0, 1, 2]], [1, 2, 2, unique-solution, [-6/1, 2/1]], [1, 2, 2, unique-solution, [-6/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0082/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 83. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -5], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, 10], [0, 1, 2]], [1, 2, 2, unique-solution, [-5/1, 2/1]], [-2, 2, 2, unique-solution, [-5/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, 10], [0, 1, 2]], [1, 2, 2, unique-solution, [-5/1, 2/1]], [-2, 2, 2, unique-solution, [-5/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0083/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 84. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -4], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, -4], [0, 1, 2]], [1, 2, 2, unique-solution, [-4/1, 2/1]], [1, 2, 2, unique-solution, [-4/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -4], [0, 1, 2]], [1, 2, 2, unique-solution, [-4/1, 2/1]], [1, 2, 2, unique-solution, [-4/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0084/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 85. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -3], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, 6], [0, 1, 2]], [1, 2, 2, unique-solution, [-3/1, 2/1]], [-2, 2, 2, unique-solution, [-3/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, 6], [0, 1, 2]], [1, 2, 2, unique-solution, [-3/1, 2/1]], [-2, 2, 2, unique-solution, [-3/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0085/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 86. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, -2], [0, 1, 2]], [1, 2, 2, unique-solution, [-2/1, 2/1]], [1, 2, 2, unique-solution, [-2/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, -2], [0, 1, 2]], [1, 2, 2, unique-solution, [-2/1, 2/1]], [1, 2, 2, unique-solution, [-2/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0086/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 87. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, -1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, 2], [0, 1, 2]], [1, 2, 2, unique-solution, [-1/1, 2/1]], [-2, 2, 2, unique-solution, [-1/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, 2], [0, 1, 2]], [1, 2, 2, unique-solution, [-1/1, 2/1]], [-2, 2, 2, unique-solution, [-1/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0087/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 88. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 0], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, 0], [0, 1, 2]], [1, 2, 2, unique-solution, [0/1, 2/1]], [1, 2, 2, unique-solution, [0/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 0], [0, 1, 2]], [1, 2, 2, unique-solution, [0/1, 2/1]], [1, 2, 2, unique-solution, [0/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0088/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 89. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, -2], [0, 1, 2]], [1, 2, 2, unique-solution, [1/1, 2/1]], [-2, 2, 2, unique-solution, [1/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, -2], [0, 1, 2]], [1, 2, 2, unique-solution, [1/1, 2/1]], [-2, 2, 2, unique-solution, [1/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0089/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 90. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, 2], [0, 1, 2]], [1, 2, 2, unique-solution, [2/1, 2/1]], [1, 2, 2, unique-solution, [2/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 2], [0, 1, 2]], [1, 2, 2, unique-solution, [2/1, 2/1]], [1, 2, 2, unique-solution, [2/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0090/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 91. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 3], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, -6], [0, 1, 2]], [1, 2, 2, unique-solution, [3/1, 2/1]], [-2, 2, 2, unique-solution, [3/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, -6], [0, 1, 2]], [1, 2, 2, unique-solution, [3/1, 2/1]], [-2, 2, 2, unique-solution, [3/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0091/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 92. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 4], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, 4], [0, 1, 2]], [1, 2, 2, unique-solution, [4/1, 2/1]], [1, 2, 2, unique-solution, [4/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 4], [0, 1, 2]], [1, 2, 2, unique-solution, [4/1, 2/1]], [1, 2, 2, unique-solution, [4/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0092/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 93. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 5], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, -10], [0, 1, 2]], [1, 2, 2, unique-solution, [5/1, 2/1]], [-2, 2, 2, unique-solution, [5/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, -10], [0, 1, 2]], [1, 2, 2, unique-solution, [5/1, 2/1]], [-2, 2, 2, unique-solution, [5/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0093/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 94. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 6], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, 6], [0, 1, 2]], [1, 2, 2, unique-solution, [6/1, 2/1]], [1, 2, 2, unique-solution, [6/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 6], [0, 1, 2]], [1, 2, 2, unique-solution, [6/1, 2/1]], [1, 2, 2, unique-solution, [6/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0094/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 95. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 7], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, -14], [0, 1, 2]], [1, 2, 2, unique-solution, [7/1, 2/1]], [-2, 2, 2, unique-solution, [7/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, -14], [0, 1, 2]], [1, 2, 2, unique-solution, [7/1, 2/1]], [-2, 2, 2, unique-solution, [7/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0095/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 96. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 8], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, 8], [0, 1, 2]], [1, 2, 2, unique-solution, [8/1, 2/1]], [1, 2, 2, unique-solution, [8/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 8], [0, 1, 2]], [1, 2, 2, unique-solution, [8/1, 2/1]], [1, 2, 2, unique-solution, [8/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0096/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 97. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 9], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, -18], [0, 1, 2]], [1, 2, 2, unique-solution, [9/1, 2/1]], [-2, 2, 2, unique-solution, [9/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, -18], [0, 1, 2]], [1, 2, 2, unique-solution, [9/1, 2/1]], [-2, 2, 2, unique-solution, [9/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0097/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 98. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 10], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, 10], [0, 1, 2]], [1, 2, 2, unique-solution, [10/1, 2/1]], [1, 2, 2, unique-solution, [10/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 10], [0, 1, 2]], [1, 2, 2, unique-solution, [10/1, 2/1]], [1, 2, 2, unique-solution, [10/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0098/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 99. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 11], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[-2, 0, -22], [0, 1, 2]], [1, 2, 2, unique-solution, [11/1, 2/1]], [-2, 2, 2, unique-solution, [11/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 0, -22], [0, 1, 2]], [1, 2, 2, unique-solution, [11/1, 2/1]], [-2, 2, 2, unique-solution, [11/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0099/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate





## 100. Row Scaling: Classification versus Operation Reversibility

Over R, rows=[[1, 0, 12], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.

View answer and explanation([[1, 0, 12], [0, 1, 2]], [1, 2, 2, unique-solution, [12/1, 2/1]], [1, 2, 2, unique-solution, [12/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)

Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 12], [0, 1, 2]], [1, 2, 2, unique-solution, [12/1, 2/1]], [1, 2, 2, unique-solution, [12/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r05-system-linear-equations-0100/)

Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r05-system-linear-equations-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261016-r05-system-linear-equations-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r05-system-linear-equations-foundations/validation-report.json)