Original Equation Domains and Noninvertible Scaling
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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-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Equation · qf-density14-equation-common-shift-check · candidate
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