Two-Equation Ranks, Residuals, and Row-Operation Guards
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Linear systems in two variables and degenerate constraints · qf-density16-system-row-scaling-solution · candidate
Canonical question JSONL · English JSONL · Validation report