# Equation Foundations

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-20260924-equation-foundations · 2026-09-24 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-12)=-8 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(-12)=-7, while the right side is -8; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0001/)

Equation · qf-equation-solution-verification · candidate





## 2. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-11)=-7 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(-11)=-7, while the right side is -7; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0002/)

Equation · qf-equation-solution-verification · candidate





## 3. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-10)=-6 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(-10)=-5, while the right side is -6; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0003/)

Equation · qf-equation-solution-verification · candidate





## 4. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-9)=-5 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(-9)=-5, while the right side is -5; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0004/)

Equation · qf-equation-solution-verification · candidate





## 5. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-8)=-4 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(-8)=-3, while the right side is -4; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0005/)

Equation · qf-equation-solution-verification · candidate





## 6. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-7)=-3 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(-7)=-3, while the right side is -3; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0006/)

Equation · qf-equation-solution-verification · candidate





## 7. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-6)=-2 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(-6)=-1, while the right side is -2; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0007/)

Equation · qf-equation-solution-verification · candidate





## 8. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-5)=-1 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(-5)=-1, while the right side is -1; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0008/)

Equation · qf-equation-solution-verification · candidate





## 9. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-4)=0 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(-4)=1, while the right side is 0; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0009/)

Equation · qf-equation-solution-verification · candidate





## 10. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-3)=1 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(-3)=1, while the right side is 1; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0010/)

Equation · qf-equation-solution-verification · candidate





## 11. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-2)=2 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(-2)=3, while the right side is 2; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0011/)

Equation · qf-equation-solution-verification · candidate





## 12. Checking a Candidate Solution by Substitution

Over the integers, consider x+(-1)=3 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(-1)=3, while the right side is 3; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0012/)

Equation · qf-equation-solution-verification · candidate





## 13. Checking a Candidate Solution by Substitution

Over the integers, consider x+(0)=4 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(0)=5, while the right side is 4; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0013/)

Equation · qf-equation-solution-verification · candidate





## 14. Checking a Candidate Solution by Substitution

Over the integers, consider x+(1)=5 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(1)=5, while the right side is 5; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0014/)

Equation · qf-equation-solution-verification · candidate





## 15. Checking a Candidate Solution by Substitution

Over the integers, consider x+(2)=6 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(2)=7, while the right side is 6; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0015/)

Equation · qf-equation-solution-verification · candidate





## 16. Checking a Candidate Solution by Substitution

Over the integers, consider x+(3)=7 with candidate x=4. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is 4+(3)=7, while the right side is 7; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0016/)

Equation · qf-equation-solution-verification · candidate





## 17. Checking a Candidate Solution by Substitution

Over the integers, consider x+(4)=8 with candidate x=5. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is 5+(4)=9, while the right side is 8; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0017/)

Equation · qf-equation-solution-verification · candidate





## 18. Checking a Candidate Solution by Substitution

Over the integers, consider x+(5)=-8 with candidate x=-13. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is -13+(5)=-8, while the right side is -8; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0018/)

Equation · qf-equation-solution-verification · candidate





## 19. Checking a Candidate Solution by Substitution

Over the integers, consider x+(6)=-7 with candidate x=-12. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is -12+(6)=-6, while the right side is -7; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0019/)

Equation · qf-equation-solution-verification · candidate





## 20. Checking a Candidate Solution by Substitution

Over the integers, consider x+(7)=-6 with candidate x=-13. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is -13+(7)=-6, while the right side is -6; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0020/)

Equation · qf-equation-solution-verification · candidate





## 21. Checking a Candidate Solution by Substitution

Over the integers, consider x+(8)=-5 with candidate x=-12. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is -12+(8)=-4, while the right side is -5; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0021/)

Equation · qf-equation-solution-verification · candidate





## 22. Checking a Candidate Solution by Substitution

Over the integers, consider x+(9)=-4 with candidate x=-13. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is -13+(9)=-4, while the right side is -4; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0022/)

Equation · qf-equation-solution-verification · candidate





## 23. Checking a Candidate Solution by Substitution

Over the integers, consider x+(10)=-3 with candidate x=-12. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is -12+(10)=-2, while the right side is -3; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0023/)

Equation · qf-equation-solution-verification · candidate





## 24. Checking a Candidate Solution by Substitution

Over the integers, consider x+(11)=-2 with candidate x=-13. After substitution into the original equation, is the candidate a solution?

View answer and explanationYes

The left side is -13+(11)=-2, while the right side is -2; the two sides are equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0024/)

Equation · qf-equation-solution-verification · candidate





## 25. Checking a Candidate Solution by Substitution

Over the integers, consider x+(12)=-1 with candidate x=-12. After substitution into the original equation, is the candidate a solution?

View answer and explanationNo

The left side is -12+(12)=0, while the right side is -1; the two sides are not equal.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-solution-verification-0025/)

Equation · qf-equation-solution-verification · candidate





## 26. Additive Equation

Solve x+(-15)=20 over the integers.

View answer and explanation35

x=20−(-15)=35; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0001/)

Equation · qf-equation-additive-solve · candidate





## 27. Additive Equation

Solve x+(-14)=19 over the integers.

View answer and explanation33

x=19−(-14)=33; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0002/)

Equation · qf-equation-additive-solve · candidate





## 28. Additive Equation

Solve x+(-13)=18 over the integers.

View answer and explanation31

x=18−(-13)=31; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0003/)

Equation · qf-equation-additive-solve · candidate





## 29. Additive Equation

Solve x+(-12)=17 over the integers.

View answer and explanation29

x=17−(-12)=29; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0004/)

Equation · qf-equation-additive-solve · candidate





## 30. Additive Equation

Solve x+(-11)=16 over the integers.

View answer and explanation27

x=16−(-11)=27; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0005/)

Equation · qf-equation-additive-solve · candidate





## 31. Additive Equation

Solve x+(-10)=15 over the integers.

View answer and explanation25

x=15−(-10)=25; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0006/)

Equation · qf-equation-additive-solve · candidate





## 32. Additive Equation

Solve x+(-9)=14 over the integers.

View answer and explanation23

x=14−(-9)=23; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0007/)

Equation · qf-equation-additive-solve · candidate





## 33. Additive Equation

Solve x+(-8)=13 over the integers.

View answer and explanation21

x=13−(-8)=21; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0008/)

Equation · qf-equation-additive-solve · candidate





## 34. Additive Equation

Solve x+(-7)=12 over the integers.

View answer and explanation19

x=12−(-7)=19; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0009/)

Equation · qf-equation-additive-solve · candidate





## 35. Additive Equation

Solve x+(-6)=11 over the integers.

View answer and explanation17

x=11−(-6)=17; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0010/)

Equation · qf-equation-additive-solve · candidate





## 36. Additive Equation

Solve x+(-5)=10 over the integers.

View answer and explanation15

x=10−(-5)=15; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0011/)

Equation · qf-equation-additive-solve · candidate





## 37. Additive Equation

Solve x+(-4)=9 over the integers.

View answer and explanation13

x=9−(-4)=13; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0012/)

Equation · qf-equation-additive-solve · candidate





## 38. Additive Equation

Solve x+(-3)=8 over the integers.

View answer and explanation11

x=8−(-3)=11; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0013/)

Equation · qf-equation-additive-solve · candidate





## 39. Additive Equation

Solve x+(-2)=7 over the integers.

View answer and explanation9

x=7−(-2)=9; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0014/)

Equation · qf-equation-additive-solve · candidate





## 40. Additive Equation

Solve x+(-1)=6 over the integers.

View answer and explanation7

x=6−(-1)=7; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0015/)

Equation · qf-equation-additive-solve · candidate





## 41. Additive Equation

Solve x+(0)=5 over the integers.

View answer and explanation5

x=5−(0)=5; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0016/)

Equation · qf-equation-additive-solve · candidate





## 42. Additive Equation

Solve x+(1)=4 over the integers.

View answer and explanation3

x=4−(1)=3; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0017/)

Equation · qf-equation-additive-solve · candidate





## 43. Additive Equation

Solve x+(2)=3 over the integers.

View answer and explanation1

x=3−(2)=1; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0018/)

Equation · qf-equation-additive-solve · candidate





## 44. Additive Equation

Solve x+(3)=2 over the integers.

View answer and explanation-1

x=2−(3)=-1; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0019/)

Equation · qf-equation-additive-solve · candidate





## 45. Additive Equation

Solve x+(4)=1 over the integers.

View answer and explanation-3

x=1−(4)=-3; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0020/)

Equation · qf-equation-additive-solve · candidate





## 46. Additive Equation

Solve x+(5)=0 over the integers.

View answer and explanation-5

x=0−(5)=-5; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0021/)

Equation · qf-equation-additive-solve · candidate





## 47. Additive Equation

Solve x+(6)=-1 over the integers.

View answer and explanation-7

x=-1−(6)=-7; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0022/)

Equation · qf-equation-additive-solve · candidate





## 48. Additive Equation

Solve x+(7)=-2 over the integers.

View answer and explanation-9

x=-2−(7)=-9; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0023/)

Equation · qf-equation-additive-solve · candidate





## 49. Additive Equation

Solve x+(8)=-3 over the integers.

View answer and explanation-11

x=-3−(8)=-11; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0024/)

Equation · qf-equation-additive-solve · candidate





## 50. Additive Equation

Solve x+(9)=-4 over the integers.

View answer and explanation-13

x=-4−(9)=-13; substituting back satisfies the original equation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-additive-solve-0025/)

Equation · qf-equation-additive-solve · candidate





## 51. Multiplicative Equation

Solve 1×x=-5 over the integers; the coefficient 1 is explicitly nonzero.

View answer and explanation-5

1≠0, so x=-5/1=-5; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0001/)

Equation · qf-equation-multiplicative-solve · candidate





## 52. Multiplicative Equation

Solve 2×x=-8 over the integers; the coefficient 2 is explicitly nonzero.

View answer and explanation-4

2≠0, so x=-8/2=-4; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0002/)

Equation · qf-equation-multiplicative-solve · candidate





## 53. Multiplicative Equation

Solve 3×x=-9 over the integers; the coefficient 3 is explicitly nonzero.

View answer and explanation-3

3≠0, so x=-9/3=-3; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0003/)

Equation · qf-equation-multiplicative-solve · candidate





## 54. Multiplicative Equation

Solve 4×x=-8 over the integers; the coefficient 4 is explicitly nonzero.

View answer and explanation-2

4≠0, so x=-8/4=-2; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0004/)

Equation · qf-equation-multiplicative-solve · candidate





## 55. Multiplicative Equation

Solve 5×x=-5 over the integers; the coefficient 5 is explicitly nonzero.

View answer and explanation-1

5≠0, so x=-5/5=-1; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0005/)

Equation · qf-equation-multiplicative-solve · candidate





## 56. Multiplicative Equation

Solve 6×x=0 over the integers; the coefficient 6 is explicitly nonzero.

View answer and explanation0

6≠0, so x=0/6=0; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0006/)

Equation · qf-equation-multiplicative-solve · candidate





## 57. Multiplicative Equation

Solve 7×x=7 over the integers; the coefficient 7 is explicitly nonzero.

View answer and explanation1

7≠0, so x=7/7=1; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0007/)

Equation · qf-equation-multiplicative-solve · candidate





## 58. Multiplicative Equation

Solve 8×x=16 over the integers; the coefficient 8 is explicitly nonzero.

View answer and explanation2

8≠0, so x=16/8=2; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0008/)

Equation · qf-equation-multiplicative-solve · candidate





## 59. Multiplicative Equation

Solve 9×x=27 over the integers; the coefficient 9 is explicitly nonzero.

View answer and explanation3

9≠0, so x=27/9=3; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0009/)

Equation · qf-equation-multiplicative-solve · candidate





## 60. Multiplicative Equation

Solve 1×x=4 over the integers; the coefficient 1 is explicitly nonzero.

View answer and explanation4

1≠0, so x=4/1=4; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0010/)

Equation · qf-equation-multiplicative-solve · candidate





## 61. Multiplicative Equation

Solve 2×x=10 over the integers; the coefficient 2 is explicitly nonzero.

View answer and explanation5

2≠0, so x=10/2=5; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0011/)

Equation · qf-equation-multiplicative-solve · candidate





## 62. Multiplicative Equation

Solve 3×x=-15 over the integers; the coefficient 3 is explicitly nonzero.

View answer and explanation-5

3≠0, so x=-15/3=-5; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0012/)

Equation · qf-equation-multiplicative-solve · candidate





## 63. Multiplicative Equation

Solve 4×x=-16 over the integers; the coefficient 4 is explicitly nonzero.

View answer and explanation-4

4≠0, so x=-16/4=-4; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0013/)

Equation · qf-equation-multiplicative-solve · candidate





## 64. Multiplicative Equation

Solve 5×x=-15 over the integers; the coefficient 5 is explicitly nonzero.

View answer and explanation-3

5≠0, so x=-15/5=-3; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0014/)

Equation · qf-equation-multiplicative-solve · candidate





## 65. Multiplicative Equation

Solve 6×x=-12 over the integers; the coefficient 6 is explicitly nonzero.

View answer and explanation-2

6≠0, so x=-12/6=-2; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0015/)

Equation · qf-equation-multiplicative-solve · candidate





## 66. Multiplicative Equation

Solve 7×x=-7 over the integers; the coefficient 7 is explicitly nonzero.

View answer and explanation-1

7≠0, so x=-7/7=-1; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0016/)

Equation · qf-equation-multiplicative-solve · candidate





## 67. Multiplicative Equation

Solve 8×x=0 over the integers; the coefficient 8 is explicitly nonzero.

View answer and explanation0

8≠0, so x=0/8=0; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0017/)

Equation · qf-equation-multiplicative-solve · candidate





## 68. Multiplicative Equation

Solve 9×x=9 over the integers; the coefficient 9 is explicitly nonzero.

View answer and explanation1

9≠0, so x=9/9=1; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0018/)

Equation · qf-equation-multiplicative-solve · candidate





## 69. Multiplicative Equation

Solve 1×x=2 over the integers; the coefficient 1 is explicitly nonzero.

View answer and explanation2

1≠0, so x=2/1=2; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0019/)

Equation · qf-equation-multiplicative-solve · candidate





## 70. Multiplicative Equation

Solve 2×x=6 over the integers; the coefficient 2 is explicitly nonzero.

View answer and explanation3

2≠0, so x=6/2=3; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0020/)

Equation · qf-equation-multiplicative-solve · candidate





## 71. Multiplicative Equation

Solve 3×x=12 over the integers; the coefficient 3 is explicitly nonzero.

View answer and explanation4

3≠0, so x=12/3=4; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0021/)

Equation · qf-equation-multiplicative-solve · candidate





## 72. Multiplicative Equation

Solve 4×x=20 over the integers; the coefficient 4 is explicitly nonzero.

View answer and explanation5

4≠0, so x=20/4=5; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0022/)

Equation · qf-equation-multiplicative-solve · candidate





## 73. Multiplicative Equation

Solve 5×x=-25 over the integers; the coefficient 5 is explicitly nonzero.

View answer and explanation-5

5≠0, so x=-25/5=-5; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0023/)

Equation · qf-equation-multiplicative-solve · candidate





## 74. Multiplicative Equation

Solve 6×x=-24 over the integers; the coefficient 6 is explicitly nonzero.

View answer and explanation-4

6≠0, so x=-24/6=-4; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0024/)

Equation · qf-equation-multiplicative-solve · candidate





## 75. Multiplicative Equation

Solve 7×x=-21 over the integers; the coefficient 7 is explicitly nonzero.

View answer and explanation-3

7≠0, so x=-21/7=-3; substitution verifies the equality.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-multiplicative-solve-0025/)

Equation · qf-equation-multiplicative-solve · candidate





## 76. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 2x+(0)=5x+(-6).

View answer and explanationunique solution x=2

(2−5)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0001/)

Equation · qf-equation-linear-solution-class · candidate





## 77. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 2x+(-3)=2x+(-3).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0002/)

Equation · qf-equation-linear-solution-class · candidate





## 78. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 3x+(2)=3x+(3).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0003/)

Equation · qf-equation-linear-solution-class · candidate





## 79. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 5x+(3)=8x+(-3).

View answer and explanationunique solution x=2

(5−8)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0004/)

Equation · qf-equation-linear-solution-class · candidate





## 80. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 5x+(0)=5x+(0).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0005/)

Equation · qf-equation-linear-solution-class · candidate





## 81. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 6x+(5)=6x+(6).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0006/)

Equation · qf-equation-linear-solution-class · candidate





## 82. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 8x+(6)=11x+(0).

View answer and explanationunique solution x=2

(8−11)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0007/)

Equation · qf-equation-linear-solution-class · candidate





## 83. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 8x+(3)=8x+(3).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0008/)

Equation · qf-equation-linear-solution-class · candidate





## 84. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 9x+(8)=9x+(9).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0009/)

Equation · qf-equation-linear-solution-class · candidate





## 85. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 11x+(9)=14x+(3).

View answer and explanationunique solution x=2

(11−14)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0010/)

Equation · qf-equation-linear-solution-class · candidate





## 86. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 11x+(6)=11x+(6).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0011/)

Equation · qf-equation-linear-solution-class · candidate





## 87. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 12x+(11)=12x+(12).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0012/)

Equation · qf-equation-linear-solution-class · candidate





## 88. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 14x+(12)=17x+(6).

View answer and explanationunique solution x=2

(14−17)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0013/)

Equation · qf-equation-linear-solution-class · candidate





## 89. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 14x+(9)=14x+(9).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0014/)

Equation · qf-equation-linear-solution-class · candidate





## 90. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 15x+(14)=15x+(15).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0015/)

Equation · qf-equation-linear-solution-class · candidate





## 91. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 17x+(15)=20x+(9).

View answer and explanationunique solution x=2

(17−20)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0016/)

Equation · qf-equation-linear-solution-class · candidate





## 92. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 17x+(12)=17x+(12).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0017/)

Equation · qf-equation-linear-solution-class · candidate





## 93. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 18x+(17)=18x+(18).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0018/)

Equation · qf-equation-linear-solution-class · candidate





## 94. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 20x+(18)=23x+(12).

View answer and explanationunique solution x=2

(20−23)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0019/)

Equation · qf-equation-linear-solution-class · candidate





## 95. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 20x+(15)=20x+(15).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0020/)

Equation · qf-equation-linear-solution-class · candidate





## 96. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 21x+(20)=21x+(21).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0021/)

Equation · qf-equation-linear-solution-class · candidate





## 97. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 23x+(21)=26x+(15).

View answer and explanationunique solution x=2

(23−26)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0022/)

Equation · qf-equation-linear-solution-class · candidate





## 98. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 23x+(18)=23x+(18).

View answer and explanationinfinitely many solutions

Both sides reduce to the same expression, so every rational x is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0023/)

Equation · qf-equation-linear-solution-class · candidate





## 99. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 24x+(23)=24x+(24).

View answer and explanationno solution

The variable terms cancel but the constants disagree, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0024/)

Equation · qf-equation-linear-solution-class · candidate





## 100. Classifying a General Linear Equation

Over the rational numbers, classify the solution set of 26x+(24)=29x+(18).

View answer and explanationunique solution x=2

(26−29)x=-6, so the equation has unique solution x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260924-equation-linear-solution-class-0025/)

Equation · qf-equation-linear-solution-class · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260924-equation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260924-equation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260924-equation-foundations/validation-report.json)