# Linear Eligibility and Degenerate Constant Cases

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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-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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r04-linear-equation-0100/)

Linear equation in one variable · qf-density14-linear-two-row-coincidence · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r04-linear-equation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261010-advance-20261014-r04-linear-equation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r04-linear-equation-foundations/validation-report.json)