# Original Equation Domains and Noninvertible Scaling

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-r03-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. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((0)x+(0))/(1), R(x)=(0)x+(0), q≠0, x=0/1. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

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

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (0/1, 0/1, 0/1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0001/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 2. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((2)x+(1))/(-3), R(x)=(1)x+(0), q≠0, x=1/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2/3, 1/2, -7/6, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-2/3, 1/2, -7/6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0002/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 3. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-10)x+(3))/(3), R(x)=(1)x+(-2), q≠0, x=3/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-4/1, -1/2, -7/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-4/1, -1/2, -7/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0003/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 4. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-9)x+(4))/(-2), R(x)=(-1)x+(-3), q≠0, x=4/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(7/1, -5/1, 12/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (7/1, -5/1, 12/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0004/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 5. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-8)x+(5))/(3), R(x)=(0)x+(-4), q≠0, x=5/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/1, -4/1, -1/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-5/1, -4/1, -1/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0005/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 6. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-7)x+(6))/(-2), R(x)=(1)x+(-5), q≠0, x=6/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(15/2, -2/1, 19/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (15/2, -2/1, 19/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0006/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 7. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-6)x+(7))/(3), R(x)=(-1)x+(-6), q≠0, x=7/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-14/3, -19/2, 29/6, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-14/3, -19/2, 29/6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0007/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 8. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-5)x+(8))/(-2), R(x)=(0)x+(-7), q≠0, x=8/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(6/1, -7/1, 13/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (6/1, -7/1, 13/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0008/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 9. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-4)x+(9))/(3), R(x)=(1)x+(-8), q≠0, x=9/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/1, -7/2, 1/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-3/1, -7/2, 1/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0009/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 10. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-3)x+(10))/(-2), R(x)=(-1)x+(-9), q≠0, x=10/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(5/2, -14/1, 33/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (5/2, -14/1, 33/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0010/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 11. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-2)x+(11))/(3), R(x)=(0)x+(-10), q≠0, x=11/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

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

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (0/1, -10/1, 10/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0011/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 12. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((-1)x+(12))/(-2), R(x)=(1)x+(-11), q≠0, x=12/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/1, -5/1, 2/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-3/1, -5/1, 2/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0012/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 13. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((0)x+(13))/(3), R(x)=(-1)x+(-12), q≠0, x=13/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(13/3, -37/2, 137/6, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (13/3, -37/2, 137/6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0013/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 14. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((1)x+(14))/(-2), R(x)=(0)x+(-13), q≠0, x=14/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-21/2, -13/1, 5/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-21/2, -13/1, 5/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0014/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 15. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((2)x+(15))/(3), R(x)=(1)x+(-14), q≠0, x=15/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(10/1, -13/2, 33/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (10/1, -13/2, 33/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0015/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 16. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((3)x+(16))/(-2), R(x)=(-1)x+(-15), q≠0, x=16/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-20/1, -23/1, 3/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-20/1, -23/1, 3/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0016/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 17. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((4)x+(17))/(3), R(x)=(0)x+(-16), q≠0, x=17/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(17/1, -16/1, 33/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (17/1, -16/1, 33/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0017/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 18. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((5)x+(18))/(-2), R(x)=(1)x+(-17), q≠0, x=18/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-63/2, -8/1, -47/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-63/2, -8/1, -47/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0018/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 19. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((6)x+(19))/(3), R(x)=(-1)x+(-18), q≠0, x=19/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(76/3, -55/2, 317/6, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (76/3, -55/2, 317/6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0019/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 20. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((7)x+(20))/(-2), R(x)=(0)x+(-19), q≠0, x=20/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-45/1, -19/1, -26/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-45/1, -19/1, -26/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0020/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 21. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((8)x+(21))/(3), R(x)=(1)x+(-20), q≠0, x=21/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(35/1, -19/2, 89/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (35/1, -19/2, 89/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0021/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 22. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((9)x+(22))/(-2), R(x)=(-1)x+(-21), q≠0, x=22/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-121/2, -32/1, -57/2, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-121/2, -32/1, -57/2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0022/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 23. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((10)x+(23))/(3), R(x)=(0)x+(-22), q≠0, x=23/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(46/1, -22/1, 68/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (46/1, -22/1, 68/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0023/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 24. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((11)x+(24))/(-2), R(x)=(1)x+(-23), q≠0, x=24/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-78/1, -11/1, -67/1, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (-78/1, -11/1, -67/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0024/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 25. Original Rational Equation Sides and Defect

Use the original equation’s common rational domain. L(x)=((12)x+(25))/(3), R(x)=(-1)x+(-24), q≠0, x=25/2. Give (L, R, L−R, does the original equation hold?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(175/3, -73/2, 569/6, No)

Evaluate both legal original sides independently and test whether the exact difference is zero; this is a finite substitution check, not a general solving proof. The exact result, retaining the requested order, is (175/3, -73/2, 569/6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0025/)

Equation · qf-density14-equation-fraction-side-check · candidate





## 26. Solution Classes Before and After Scaling

Over Q, the original equation is (0)x+(0)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(all Q, all Q, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (all Q, all Q, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0026/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 27. Solution Classes Before and After Scaling

Over Q, the original equation is (0)x+(1)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(no-solution, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (no-solution, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0027/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 28. Solution Classes Before and After Scaling

Over Q, the original equation is (0)x+(1)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(no-solution, no-solution, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (no-solution, no-solution, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0028/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 29. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(0)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 0/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 0/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0029/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 30. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-1)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 1/2, single-solution 1/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 1/2, single-solution 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0030/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 31. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-2)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 1/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 1/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0031/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 32. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-3)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 3/2, single-solution 3/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 3/2, single-solution 3/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0032/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 33. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-4)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 2/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 2/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0033/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 34. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-5)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 5/2, single-solution 5/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 5/2, single-solution 5/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0034/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 35. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-6)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 3/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 3/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0035/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 36. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-7)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 7/2, single-solution 7/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 7/2, single-solution 7/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0036/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 37. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-8)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 4/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 4/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0037/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 38. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-9)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 9/2, single-solution 9/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 9/2, single-solution 9/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0038/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 39. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-10)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 5/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 5/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0039/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 40. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-11)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 11/2, single-solution 11/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 11/2, single-solution 11/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0040/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 41. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-12)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 6/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 6/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0041/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 42. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-13)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 13/2, single-solution 13/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 13/2, single-solution 13/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0042/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 43. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-14)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 7/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 7/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0043/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 44. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-15)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 15/2, single-solution 15/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 15/2, single-solution 15/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0044/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 45. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-16)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 8/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 8/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0045/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 46. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-17)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 17/2, single-solution 17/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 17/2, single-solution 17/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0046/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 47. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-18)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 9/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 9/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0047/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 48. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-19)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 19/2, single-solution 19/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 19/2, single-solution 19/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0048/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 49. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-20)=0; multiply both sides by k=0. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 10/1, all Q, No)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 10/1, all Q, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0049/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 50. Solution Classes Before and After Scaling

Over Q, the original equation is (2)x+(-21)=0; multiply both sides by k=2. Give (original solution class, scaled solution class, are the classes equal?). Use all Q, no-solution, or single-solution n/d.

View answer and explanation(single-solution 21/2, single-solution 21/2, Yes)

Classify the original and scaled equations separately, including constant degeneracies; multiplication by zero does not generally preserve the solution set. The exact result, retaining the requested order, is (single-solution 21/2, single-solution 21/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0050/)

Equation · qf-density14-equation-zero-scale-solution-loss · candidate





## 51. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((0)x+(0))/(x−(0))=0. Exclude x=h=0; ordered queries=[0/1, 1/1, -1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

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

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 0/1, 0/1], [0/1, 0/1, 0/1], [No, Yes, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0051/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 52. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((2)x+(-2))/(x−(1))=0. Exclude x=h=1; ordered queries=[1/1, 2/1, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

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

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 2/1, 2/1], [0/1, 2/1, -2/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0052/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 53. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-10)x+(3))/(x−(0))=1. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, -5/1, -19/2], [3/1, -5/2, -19/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, -5/1, -19/2], [3/1, -5/2, -19/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0053/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 54. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-9)x+(4))/(x−(1))=-1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, -18/1, -21/2], [-5/1, -9/1, -21/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, -18/1, -21/2], [-5/1, -9/1, -21/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0054/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 55. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-8)x+(5))/(x−(2))=0. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, -30/1, -27/2], [-11/1, -15/1, -27/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, -30/1, -27/2], [-11/1, -15/1, -27/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0055/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 56. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-7)x+(6))/(x−(-2))=1. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 32/1, 2/1], [20/1, 16/1, 4/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 32/1, 2/1], [20/1, 16/1, 4/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0056/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 57. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-6)x+(7))/(x−(-1))=-1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 21/1, 3/2], [13/1, 21/2, 3/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 21/1, 3/2], [13/1, 21/2, 3/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0057/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 58. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-5)x+(8))/(x−(0))=0. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 11/1, -1/1], [8/1, 11/2, -2/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 11/1, -1/1], [8/1, 11/2, -2/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0058/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 59. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-4)x+(9))/(x−(1))=1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 5/1, -5/2], [5/1, 5/2, -5/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 5/1, -5/2], [5/1, 5/2, -5/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0059/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 60. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-3)x+(10))/(x−(2))=-1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 6/1, 0/1], [4/1, 3/1, 0/1], [No, No, Yes])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 6/1, 0/1], [4/1, 3/1, 0/1], [No, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0060/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 61. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-2)x+(11))/(x−(-2))=0. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 28/1, 11/2], [15/1, 14/1, 11/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 28/1, 11/2], [15/1, 14/1, 11/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0061/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 62. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((-1)x+(12))/(x−(-1))=1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 24/1, 9/2], [13/1, 12/1, 9/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 24/1, 9/2], [13/1, 12/1, 9/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0062/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 63. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((0)x+(13))/(x−(0))=-1. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 27/1, 15/2], [13/1, 27/2, 15/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 27/1, 15/2], [13/1, 27/2, 15/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0063/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 64. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((1)x+(14))/(x−(1))=0. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 31/1, 17/2], [15/1, 31/2, 17/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 31/1, 17/2], [15/1, 31/2, 17/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0064/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 65. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((2)x+(15))/(x−(2))=1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 39/1, 21/2], [19/1, 39/2, 21/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 39/1, 21/2], [19/1, 39/2, 21/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0065/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 66. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((3)x+(16))/(x−(-2))=-1. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 24/1, 9/1], [10/1, 12/1, 18/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 24/1, 9/1], [10/1, 12/1, 18/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0066/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 67. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((4)x+(17))/(x−(-1))=0. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 30/1, 21/2], [13/1, 15/1, 21/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 30/1, 21/2], [13/1, 15/1, 21/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0067/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 68. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((5)x+(18))/(x−(0))=1. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 40/1, 13/1], [18/1, 20/1, 26/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 40/1, 13/1], [18/1, 20/1, 26/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0068/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 69. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((6)x+(19))/(x−(1))=-1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 57/1, 39/2], [25/1, 57/2, 39/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 57/1, 39/2], [25/1, 57/2, 39/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0069/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 70. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((7)x+(20))/(x−(2))=0. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 75/1, 24/1], [34/1, 75/2, 48/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 75/1, 24/1], [34/1, 75/2, 48/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0070/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 71. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((8)x+(21))/(x−(-2))=1. Exclude x=h=-2; ordered queries=[-2/1, -3/2, 0/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 17/1, 19/2], [5/1, 17/2, 19/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 17/1, 19/2], [5/1, 17/2, 19/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0071/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 72. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((9)x+(22))/(x−(-1))=-1. Exclude x=h=-1; ordered queries=[-1/1, -1/2, 1/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 36/1, 33/2], [13/1, 18/1, 33/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 36/1, 33/2], [13/1, 18/1, 33/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0072/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 73. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((10)x+(23))/(x−(0))=0. Exclude x=h=0; ordered queries=[0/1, 1/2, 2/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 56/1, 43/2], [23/1, 28/1, 43/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 56/1, 43/2], [23/1, 28/1, 43/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0073/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 74. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((11)x+(24))/(x−(1))=1. Exclude x=h=1; ordered queries=[1/1, 3/2, 3/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 80/1, 55/2], [35/1, 40/1, 55/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 80/1, 55/2], [35/1, 40/1, 55/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0074/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 75. Original Denominator Guard versus Cross-Multiplication

Over Q, the original equation is ((12)x+(25))/(x−(2))=-1. Exclude x=h=2; ordered queries=[2/1, 5/2, 4/1]. Give (domain-eligibility mask, original residuals or undefined, cross-multiplied residuals, original-solution mask), keeping each list’s order.

View answer and explanation([No, Yes, Yes], [undefined, 111/1, 75/2], [49/1, 111/2, 75/1], [No, No, No])

Check the original denominator before evaluating; a zero cross-multiplication residual does not restore an excluded undefined point or replace original-equation eligibility. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 111/1, 75/2], [49/1, 111/2, 75/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0075/)

Equation · qf-density14-equation-denominator-exclusion-check · candidate





## 76. Both Equation Sides after a Common Shift

Over Q, L=(0)x+(0), R=(0)x+(0), x=0/1, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

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

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (0/1, 1/1, 1/1, 0/1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0076/)

Equation · qf-density14-equation-common-shift-check · candidate





## 77. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-1), R=(-1)x+(2), x=1/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2/1, -1/2, 3/2, -2/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-2/1, -1/2, 3/2, -2/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0077/)

Equation · qf-density14-equation-common-shift-check · candidate





## 78. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-2), R=(-1)x+(3), x=2/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/1, 0/1, 3/1, -3/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-3/1, 0/1, 3/1, -3/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0078/)

Equation · qf-density14-equation-common-shift-check · candidate





## 79. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-3), R=(-1)x+(4), x=3/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-4/1, -5/2, 3/2, -4/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-4/1, -5/2, 3/2, -4/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0079/)

Equation · qf-density14-equation-common-shift-check · candidate





## 80. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-4), R=(-1)x+(5), x=4/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/1, -2/1, 3/1, -5/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-5/1, -2/1, 3/1, -5/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0080/)

Equation · qf-density14-equation-common-shift-check · candidate





## 81. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-5), R=(-1)x+(6), x=5/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-6/1, -3/2, 9/2, -6/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-6/1, -3/2, 9/2, -6/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0081/)

Equation · qf-density14-equation-common-shift-check · candidate





## 82. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-6), R=(-1)x+(7), x=6/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-7/1, -4/1, 3/1, -7/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-7/1, -4/1, 3/1, -7/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0082/)

Equation · qf-density14-equation-common-shift-check · candidate





## 83. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-7), R=(-1)x+(8), x=7/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-8/1, -7/2, 9/2, -8/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-8/1, -7/2, 9/2, -8/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0083/)

Equation · qf-density14-equation-common-shift-check · candidate





## 84. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-8), R=(-1)x+(9), x=8/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-9/1, -3/1, 6/1, -9/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-9/1, -3/1, 6/1, -9/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0084/)

Equation · qf-density14-equation-common-shift-check · candidate





## 85. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-9), R=(-1)x+(10), x=9/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-10/1, -11/2, 9/2, -10/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-10/1, -11/2, 9/2, -10/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0085/)

Equation · qf-density14-equation-common-shift-check · candidate





## 86. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-10), R=(-1)x+(11), x=10/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-11/1, -5/1, 6/1, -11/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-11/1, -5/1, 6/1, -11/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0086/)

Equation · qf-density14-equation-common-shift-check · candidate





## 87. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-11), R=(-1)x+(12), x=11/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-12/1, -9/2, 15/2, -12/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-12/1, -9/2, 15/2, -12/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0087/)

Equation · qf-density14-equation-common-shift-check · candidate





## 88. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-12), R=(-1)x+(13), x=12/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-13/1, -7/1, 6/1, -13/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-13/1, -7/1, 6/1, -13/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0088/)

Equation · qf-density14-equation-common-shift-check · candidate





## 89. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-13), R=(-1)x+(14), x=13/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-14/1, -13/2, 15/2, -14/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-14/1, -13/2, 15/2, -14/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0089/)

Equation · qf-density14-equation-common-shift-check · candidate





## 90. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-14), R=(-1)x+(15), x=14/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-15/1, -6/1, 9/1, -15/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-15/1, -6/1, 9/1, -15/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0090/)

Equation · qf-density14-equation-common-shift-check · candidate





## 91. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-15), R=(-1)x+(16), x=15/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-16/1, -17/2, 15/2, -16/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-16/1, -17/2, 15/2, -16/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0091/)

Equation · qf-density14-equation-common-shift-check · candidate





## 92. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-16), R=(-1)x+(17), x=16/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-17/1, -8/1, 9/1, -17/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-17/1, -8/1, 9/1, -17/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0092/)

Equation · qf-density14-equation-common-shift-check · candidate





## 93. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-17), R=(-1)x+(18), x=17/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-18/1, -15/2, 21/2, -18/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-18/1, -15/2, 21/2, -18/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0093/)

Equation · qf-density14-equation-common-shift-check · candidate





## 94. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-18), R=(-1)x+(19), x=18/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-19/1, -10/1, 9/1, -19/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-19/1, -10/1, 9/1, -19/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0094/)

Equation · qf-density14-equation-common-shift-check · candidate





## 95. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-19), R=(-1)x+(20), x=19/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-20/1, -19/2, 21/2, -20/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-20/1, -19/2, 21/2, -20/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0095/)

Equation · qf-density14-equation-common-shift-check · candidate





## 96. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-20), R=(-1)x+(21), x=20/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-21/1, -9/1, 12/1, -21/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-21/1, -9/1, 12/1, -21/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0096/)

Equation · qf-density14-equation-common-shift-check · candidate





## 97. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-21), R=(-1)x+(22), x=21/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-22/1, -23/2, 21/2, -22/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-22/1, -23/2, 21/2, -22/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0097/)

Equation · qf-density14-equation-common-shift-check · candidate





## 98. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-22), R=(-1)x+(23), x=22/2, common added quantity e=0. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-23/1, -11/1, 12/1, -23/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-23/1, -11/1, 12/1, -23/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0098/)

Equation · qf-density14-equation-common-shift-check · candidate





## 99. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-23), R=(-1)x+(24), x=23/2, common added quantity e=1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-24/1, -21/2, 27/2, -24/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-24/1, -21/2, 27/2, -24/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0099/)

Equation · qf-density14-equation-common-shift-check · candidate





## 100. Both Equation Sides after a Common Shift

Over Q, L=(1)x+(-24), R=(-1)x+(25), x=24/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-25/1, -13/1, 12/1, -25/1, No)

Add the same rational quantity to both sides, then recompute their difference directly; the last field tests equality of the shifted sides. The exact result, retaining the requested order, is (-25/1, -13/1, 12/1, -25/1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r03-equation-0100/)

Equation · qf-density14-equation-common-shift-check · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r03-equation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261010-advance-20261014-r03-equation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r03-equation-foundations/validation-report.json)