Linear Eligibility and Degenerate Constant Cases
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261010-advance-20261014-r04-linear-equation-foundations · 2026-10-10 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=0, b=0. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(No, undefined, all Q)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (No, undefined, all Q).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
2. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=0, b=1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(No, undefined, no-solution)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (No, undefined, no-solution).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
3. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-10. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -5/1], single-solution 5/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -5/1], single-solution 5/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
4. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=9. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -9/2], single-solution 9/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -9/2], single-solution 9/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
5. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-8. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -4/1], single-solution 4/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -4/1], single-solution 4/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
6. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=7. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -7/2], single-solution 7/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -7/2], single-solution 7/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
7. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-6. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -3/1], single-solution 3/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -3/1], single-solution 3/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
8. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=5. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -5/2], single-solution 5/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -5/2], single-solution 5/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
9. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-4. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -2/1], single-solution 2/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -2/1], single-solution 2/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
10. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=3. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -3/2], single-solution 3/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -3/2], single-solution 3/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
11. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=-2. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -1/1], single-solution 1/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -1/1], single-solution 1/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
12. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, -1/2], single-solution 1/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, -1/2], single-solution 1/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
13. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=0. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 0/1], single-solution 0/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 0/1], single-solution 0/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
14. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-1. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 1/2], single-solution -1/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 1/2], single-solution -1/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
15. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=2. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 1/1], single-solution -1/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 1/1], single-solution -1/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
16. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-3. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 3/2], single-solution -3/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 3/2], single-solution -3/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
17. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=4. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 2/1], single-solution -2/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 2/1], single-solution -2/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
18. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-5. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 5/2], single-solution -5/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 5/2], single-solution -5/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
19. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=6. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 3/1], single-solution -3/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 3/1], single-solution -3/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
20. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-7. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 7/2], single-solution -7/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 7/2], single-solution -7/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
21. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=8. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 4/1], single-solution -4/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 4/1], single-solution -4/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
22. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-9. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 9/2], single-solution -9/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 9/2], single-solution -9/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
23. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=10. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 5/1], single-solution -5/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 5/1], single-solution -5/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
24. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=-2, b=-11. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 11/2], single-solution -11/2)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 11/2], single-solution -11/2).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
25. Linear Eligibility, Normalized Row, and Solution Class
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Candidate a=2, b=12. Give (is a nonzero?, [1/1,b/a] or undefined, original solution class). Use all Q, no-solution, or single-solution n/d.
View answer and explanation
(Yes, [1/1, 6/1], single-solution -6/1)
Only divide by a after proving it nonzero; otherwise classify the constant equation separately, without calling it linear. The exact result, retaining the requested order, is (Yes, [1/1, 6/1], single-solution -6/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-normalized-row · candidate
26. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=0, ordered ts=[0, 1, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([0, 1, -1], [No, Yes, Yes], [all Q, single-solution 0/1, single-solution 0/1])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([0, 1, -1], [No, Yes, Yes], [all Q, single-solution 0/1, single-solution 0/1]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
27. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=1, ordered ts=[2, 3, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([0, 1, -1], [No, Yes, Yes], [no-solution, single-solution -1/1, single-solution 1/1])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([0, 1, -1], [No, Yes, Yes], [no-solution, single-solution -1/1, single-solution 1/1]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
28. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=-10, ordered ts=[-9, 4, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-10, 3, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 10/3, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-10, 3, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 10/3, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
29. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=-9, ordered ts=[-7, 6, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-9, 4, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 9/4, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-9, 4, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 9/4, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
30. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=-8, ordered ts=[-9, 4, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-8, 5, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 8/5, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-8, 5, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 8/5, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
31. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=-7, ordered ts=[-7, 6, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-7, 6, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 7/6, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-7, 6, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 7/6, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
32. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=-6, ordered ts=[-5, 8, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-6, 7, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 6/7, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-6, 7, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 6/7, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
33. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=-5, ordered ts=[-3, 10, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-5, 8, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 5/8, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-5, 8, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 5/8, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
34. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=-4, ordered ts=[-5, 8, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-4, 9, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 4/9, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-4, 9, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 4/9, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
35. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=-3, ordered ts=[-3, 10, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-3, 10, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 3/10, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-3, 10, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 3/10, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
36. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=-2, ordered ts=[-1, 12, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-2, 11, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 2/11, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-2, 11, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 2/11, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
37. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=-1, ordered ts=[1, 14, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([-1, 12, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 1/12, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([-1, 12, 0], [Yes, Yes, No], [single-solution -1/1, single-solution 1/12, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
38. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=0, ordered ts=[-1, 12, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([0, 13, 0], [No, Yes, No], [all Q, single-solution 0/1, all Q])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([0, 13, 0], [No, Yes, No], [all Q, single-solution 0/1, all Q]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
39. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=1, ordered ts=[1, 14, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([1, 14, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -1/14, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([1, 14, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -1/14, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
40. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=2, ordered ts=[3, 16, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([2, 15, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -2/15, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([2, 15, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -2/15, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
41. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=3, ordered ts=[5, 18, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([3, 16, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -3/16, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([3, 16, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -3/16, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
42. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=4, ordered ts=[3, 16, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([4, 17, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -4/17, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([4, 17, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -4/17, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
43. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=5, ordered ts=[5, 18, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([5, 18, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -5/18, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([5, 18, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -5/18, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
44. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=6, ordered ts=[7, 20, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([6, 19, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -6/19, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([6, 19, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -6/19, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
45. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=7, ordered ts=[9, 22, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([7, 20, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -7/20, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([7, 20, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -7/20, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
46. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=8, ordered ts=[7, 20, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([8, 21, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -8/21, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([8, 21, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -8/21, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
47. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=0, b=9, ordered ts=[9, 22, 0]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([9, 22, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -9/22, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([9, 22, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -9/22, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
48. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=1, b=10, ordered ts=[11, 24, 1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([10, 23, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -10/23, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([10, 23, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -10/23, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
49. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=2, b=11, ordered ts=[13, 26, 2]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([11, 24, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -11/24, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([11, 24, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -11/24, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
50. Parameter-Dependent Linear and Constant Cases
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a_i=t_i−k, k=-1, b=12, ordered ts=[11, 24, -1]. Give (three coefficients, three linear-eligibility values, three solution classes), retaining zero-coefficient cases.
View answer and explanation
([12, 25, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -12/25, no-solution])
Form each actual coefficient before separating a unique rational root from all-Q/no-solution degeneracies; no general linear-system formal proof is implied. The exact result, retaining the requested order, is ([12, 25, 0], [Yes, Yes, No], [single-solution -1/1, single-solution -12/25, no-solution]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-parameter-slice · candidate
51. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=1≠0, b=0, ordered offsets=[0/1, 0/2, 0/-1]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, [0/1, 0/1, 0/1], [0/1, 0/1, 0/1], [Yes, Yes, Yes])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (0/1, [0/1, 0/1, 0/1], [0/1, 0/1, 0/1], [Yes, Yes, Yes]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
52. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-2≠0, b=1, ordered offsets=[0/1, 1/2, -1/2]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, [1/2, 1/1, 0/1], [0/1, -1/1, 1/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (1/2, [1/2, 1/1, 0/1], [0/1, -1/1, 1/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
53. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=3≠0, b=-10, ordered offsets=[0/1, 3/2, -2/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(10/3, [10/3, 29/6, 8/3], [0/1, 9/2, -2/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (10/3, [10/3, 29/6, 8/3], [0/1, 9/2, -2/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
54. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-4≠0, b=-9, ordered offsets=[0/1, 4/2, -3/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-9/4, [-9/4, -1/4, -13/4], [0/1, -8/1, 4/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-9/4, [-9/4, -1/4, -13/4], [0/1, -8/1, 4/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
55. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=5≠0, b=-8, ordered offsets=[0/1, 5/2, -4/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(8/5, [8/5, 41/10, 4/15], [0/1, 25/2, -20/3], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (8/5, [8/5, 41/10, 4/15], [0/1, 25/2, -20/3], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
56. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-6≠0, b=-7, ordered offsets=[0/1, 6/2, -5/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-7/6, [-7/6, 11/6, -17/6], [0/1, -18/1, 10/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-7/6, [-7/6, 11/6, -17/6], [0/1, -18/1, 10/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
57. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=7≠0, b=-6, ordered offsets=[0/1, 7/2, -6/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(6/7, [6/7, 61/14, -8/7], [0/1, 49/2, -14/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (6/7, [6/7, 61/14, -8/7], [0/1, 49/2, -14/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
58. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-8≠0, b=-5, ordered offsets=[0/1, 8/2, -7/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-5/8, [-5/8, 27/8, -71/24], [0/1, -32/1, 56/3], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-5/8, [-5/8, 27/8, -71/24], [0/1, -32/1, 56/3], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
59. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=9≠0, b=-4, ordered offsets=[0/1, 9/2, -8/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(4/9, [4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (4/9, [4/9, 89/18, -20/9], [0/1, 81/2, -24/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
60. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-10≠0, b=-3, ordered offsets=[0/1, 10/2, -9/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-3/10, [-3/10, 47/10, -33/10], [0/1, -50/1, 30/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-3/10, [-3/10, 47/10, -33/10], [0/1, -50/1, 30/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
61. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=11≠0, b=-2, ordered offsets=[0/1, 11/2, -10/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(2/11, [2/11, 125/22, -104/33], [0/1, 121/2, -110/3], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (2/11, [2/11, 125/22, -104/33], [0/1, 121/2, -110/3], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
62. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-12≠0, b=-1, ordered offsets=[0/1, 12/2, -11/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/12, [-1/12, 71/12, -15/4], [0/1, -72/1, 44/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-1/12, [-1/12, 71/12, -15/4], [0/1, -72/1, 44/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
63. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=13≠0, b=0, ordered offsets=[0/1, 13/2, -12/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, [0/1, 13/2, -4/1], [0/1, 169/2, -52/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (0/1, [0/1, 13/2, -4/1], [0/1, 169/2, -52/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
64. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-14≠0, b=1, ordered offsets=[0/1, 14/2, -13/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/14, [1/14, 99/14, -179/42], [0/1, -98/1, 182/3], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (1/14, [1/14, 99/14, -179/42], [0/1, -98/1, 182/3], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
65. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=15≠0, b=2, ordered offsets=[0/1, 15/2, -14/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
66. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-16≠0, b=3, ordered offsets=[0/1, 16/2, -15/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(3/16, [3/16, 131/16, -77/16], [0/1, -128/1, 80/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (3/16, [3/16, 131/16, -77/16], [0/1, -128/1, 80/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
67. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=17≠0, b=4, ordered offsets=[0/1, 17/2, -16/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-4/17, [-4/17, 281/34, -284/51], [0/1, 289/2, -272/3], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-4/17, [-4/17, 281/34, -284/51], [0/1, 289/2, -272/3], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
68. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-18≠0, b=5, ordered offsets=[0/1, 18/2, -17/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(5/18, [5/18, 167/18, -97/18], [0/1, -162/1, 102/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (5/18, [5/18, 167/18, -97/18], [0/1, -162/1, 102/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
69. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=19≠0, b=6, ordered offsets=[0/1, 19/2, -18/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-6/19, [-6/19, 349/38, -120/19], [0/1, 361/2, -114/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-6/19, [-6/19, 349/38, -120/19], [0/1, 361/2, -114/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
70. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-20≠0, b=7, ordered offsets=[0/1, 20/2, -19/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(7/20, [7/20, 207/20, -359/60], [0/1, -200/1, 380/3], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (7/20, [7/20, 207/20, -359/60], [0/1, -200/1, 380/3], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
71. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=21≠0, b=8, ordered offsets=[0/1, 21/2, -20/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-8/21, [-8/21, 425/42, -148/21], [0/1, 441/2, -140/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-8/21, [-8/21, 425/42, -148/21], [0/1, 441/2, -140/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
72. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-22≠0, b=9, ordered offsets=[0/1, 22/2, -21/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(9/22, [9/22, 251/22, -145/22], [0/1, -242/1, 154/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (9/22, [9/22, 251/22, -145/22], [0/1, -242/1, 154/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
73. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=23≠0, b=10, ordered offsets=[0/1, 23/2, -22/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-10/23, [-10/23, 509/46, -536/69], [0/1, 529/2, -506/3], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-10/23, [-10/23, 509/46, -536/69], [0/1, 529/2, -506/3], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
74. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=-24≠0, b=11, ordered offsets=[0/1, 24/2, -23/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(11/24, [11/24, 299/24, -173/24], [0/1, -288/1, 184/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (11/24, [11/24, 299/24, -173/24], [0/1, -288/1, 184/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
75. Exact Root Offsets and Original Substitution
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=25≠0, b=12, ordered offsets=[0/1, 25/2, -24/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-12/25, [-12/25, 601/50, -212/25], [0/1, 625/2, -200/1], [Yes, No, No])
Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-12/25, [-12/25, 601/50, -212/25], [0/1, 625/2, -200/1], [Yes, No, No]).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-root-substitution-grid · candidate
76. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[1, 0], [2, 0]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, 0/1, Yes, 0/1)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (0/1, 0/1, Yes, 0/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
77. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[1, -1], [-2, 2]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/1, 1/1, Yes, 1/1)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (1/1, 1/1, Yes, 1/1).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
78. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -2], [-3, -1]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/1, -1/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (1/1, -1/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
79. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -3], [-3, -2]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(3/2, -2/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (3/2, -2/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
80. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -4], [-3, -3]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(2/1, -1/1, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (2/1, -1/1, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
81. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -5], [-3, -4]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(5/2, -4/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (5/2, -4/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
82. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -6], [-3, -5]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(3/1, -5/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (3/1, -5/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
83. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -7], [-3, -6]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(7/2, -2/1, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (7/2, -2/1, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
84. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -8], [-3, -7]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(4/1, -7/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (4/1, -7/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
85. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -9], [-3, -8]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(9/2, -8/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (9/2, -8/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
86. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -10], [-3, -9]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(5/1, -3/1, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (5/1, -3/1, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
87. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -11], [-3, -10]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(11/2, -10/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (11/2, -10/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
88. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -12], [-3, -11]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(6/1, -11/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (6/1, -11/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
89. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -13], [-3, -12]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(13/2, -4/1, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (13/2, -4/1, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
90. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -14], [-3, -13]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(7/1, -13/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (7/1, -13/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
91. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -15], [-3, -14]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(15/2, -14/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (15/2, -14/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
92. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -16], [-3, -15]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(8/1, -5/1, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (8/1, -5/1, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
93. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -17], [-3, -16]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(17/2, -16/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (17/2, -16/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
94. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -18], [-3, -17]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(9/1, -17/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (9/1, -17/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
95. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -19], [-3, -18]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(19/2, -6/1, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (19/2, -6/1, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
96. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -20], [-3, -19]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(10/1, -19/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (10/1, -19/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
97. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -21], [-3, -20]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(21/2, -20/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (21/2, -20/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
98. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -22], [-3, -21]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(11/1, -7/1, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (11/1, -7/1, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
99. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -23], [-3, -22]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(23/2, -22/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (23/2, -22/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
100. Two Affine Roots and a Common Root
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -24], [-3, -23]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(12/1, -23/3, No, none)
Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (12/1, -23/3, No, none).
Reviewed step-by-step derivation
Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate
Canonical question JSONL · English JSONL · Validation report