# Linear 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-20260927-linear-equation-foundations · 2026-09-27 · 100 questions

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



## 1. Validity of a Linear-Equation Form

Over the rational numbers, consider 0x+(-12)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationNo

a=0, so this is a degenerate constant equation rather than a linear equation in one variable.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0001/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 2. Validity of a Linear-Equation Form

Over the rational numbers, consider -3x+(-11)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-3≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0002/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 3. Validity of a Linear-Equation Form

Over the rational numbers, consider -2x+(-10)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-2≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0003/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 4. Validity of a Linear-Equation Form

Over the rational numbers, consider -1x+(-9)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0004/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 5. Validity of a Linear-Equation Form

Over the rational numbers, consider 1x+(-8)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0005/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 6. Validity of a Linear-Equation Form

Over the rational numbers, consider 0x+(-7)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationNo

a=0, so this is a degenerate constant equation rather than a linear equation in one variable.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0006/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 7. Validity of a Linear-Equation Form

Over the rational numbers, consider 2x+(-6)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=2≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0007/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 8. Validity of a Linear-Equation Form

Over the rational numbers, consider 3x+(-5)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=3≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0008/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 9. Validity of a Linear-Equation Form

Over the rational numbers, consider 4x+(-4)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=4≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0009/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 10. Validity of a Linear-Equation Form

Over the rational numbers, consider -4x+(-3)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-4≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0010/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 11. Validity of a Linear-Equation Form

Over the rational numbers, consider 0x+(-2)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationNo

a=0, so this is a degenerate constant equation rather than a linear equation in one variable.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0011/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 12. Validity of a Linear-Equation Form

Over the rational numbers, consider -2x+(-1)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-2≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0012/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 13. Validity of a Linear-Equation Form

Over the rational numbers, consider -1x+(0)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0013/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 14. Validity of a Linear-Equation Form

Over the rational numbers, consider 1x+(1)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0014/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 15. Validity of a Linear-Equation Form

Over the rational numbers, consider 1x+(2)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0015/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 16. Validity of a Linear-Equation Form

Over the rational numbers, consider 0x+(3)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationNo

a=0, so this is a degenerate constant equation rather than a linear equation in one variable.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0016/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 17. Validity of a Linear-Equation Form

Over the rational numbers, consider 3x+(4)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=3≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0017/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 18. Validity of a Linear-Equation Form

Over the rational numbers, consider 4x+(5)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=4≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0018/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 19. Validity of a Linear-Equation Form

Over the rational numbers, consider -4x+(6)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-4≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0019/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 20. Validity of a Linear-Equation Form

Over the rational numbers, consider -3x+(7)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-3≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0020/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 21. Validity of a Linear-Equation Form

Over the rational numbers, consider 0x+(8)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationNo

a=0, so this is a degenerate constant equation rather than a linear equation in one variable.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0021/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 22. Validity of a Linear-Equation Form

Over the rational numbers, consider -1x+(9)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=-1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0022/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 23. Validity of a Linear-Equation Form

Over the rational numbers, consider 1x+(10)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0023/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 24. Validity of a Linear-Equation Form

Over the rational numbers, consider 1x+(11)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=1≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0024/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 25. Validity of a Linear-Equation Form

Over the rational numbers, consider 2x+(12)=0. Under this MKO, is it a linear equation in one variable?

View answer and explanationYes

a=2≠0, so this is a linear equation in one variable under the MKO definition.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-form-validity-0025/)

Linear equation in one variable · qf-linear-equation-form-validity · candidate





## 26. Exact Solution of a Linear Equation

Solve -5x+(-15)=0 exactly over the rational numbers; a=-5 is explicitly nonzero.

View answer and explanation-3

x=−(-15)/-5=-3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0001/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 27. Exact Solution of a Linear Equation

Solve -4x+(-8)=0 exactly over the rational numbers; a=-4 is explicitly nonzero.

View answer and explanation-2

x=−(-8)/-4=-2; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0002/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 28. Exact Solution of a Linear Equation

Solve -3x+(-1)=0 exactly over the rational numbers; a=-3 is explicitly nonzero.

View answer and explanation-1/3

x=−(-1)/-3=-1/3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0003/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 29. Exact Solution of a Linear Equation

Solve -2x+(6)=0 exactly over the rational numbers; a=-2 is explicitly nonzero.

View answer and explanation3

x=−(6)/-2=3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0004/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 30. Exact Solution of a Linear Equation

Solve -1x+(13)=0 exactly over the rational numbers; a=-1 is explicitly nonzero.

View answer and explanation13

x=−(13)/-1=13; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0005/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 31. Exact Solution of a Linear Equation

Solve 1x+(-11)=0 exactly over the rational numbers; a=1 is explicitly nonzero.

View answer and explanation11

x=−(-11)/1=11; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0006/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 32. Exact Solution of a Linear Equation

Solve 1x+(-4)=0 exactly over the rational numbers; a=1 is explicitly nonzero.

View answer and explanation4

x=−(-4)/1=4; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0007/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 33. Exact Solution of a Linear Equation

Solve 2x+(3)=0 exactly over the rational numbers; a=2 is explicitly nonzero.

View answer and explanation-3/2

x=−(3)/2=-3/2; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0008/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 34. Exact Solution of a Linear Equation

Solve 3x+(10)=0 exactly over the rational numbers; a=3 is explicitly nonzero.

View answer and explanation-10/3

x=−(10)/3=-10/3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0009/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 35. Exact Solution of a Linear Equation

Solve 4x+(-14)=0 exactly over the rational numbers; a=4 is explicitly nonzero.

View answer and explanation7/2

x=−(-14)/4=7/2; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0010/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 36. Exact Solution of a Linear Equation

Solve 5x+(-7)=0 exactly over the rational numbers; a=5 is explicitly nonzero.

View answer and explanation7/5

x=−(-7)/5=7/5; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0011/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 37. Exact Solution of a Linear Equation

Solve -5x+(0)=0 exactly over the rational numbers; a=-5 is explicitly nonzero.

View answer and explanation0

x=−(0)/-5=0; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0012/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 38. Exact Solution of a Linear Equation

Solve -4x+(7)=0 exactly over the rational numbers; a=-4 is explicitly nonzero.

View answer and explanation7/4

x=−(7)/-4=7/4; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0013/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 39. Exact Solution of a Linear Equation

Solve -3x+(14)=0 exactly over the rational numbers; a=-3 is explicitly nonzero.

View answer and explanation14/3

x=−(14)/-3=14/3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0014/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 40. Exact Solution of a Linear Equation

Solve -2x+(-10)=0 exactly over the rational numbers; a=-2 is explicitly nonzero.

View answer and explanation-5

x=−(-10)/-2=-5; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0015/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 41. Exact Solution of a Linear Equation

Solve -1x+(-3)=0 exactly over the rational numbers; a=-1 is explicitly nonzero.

View answer and explanation-3

x=−(-3)/-1=-3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0016/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 42. Exact Solution of a Linear Equation

Solve 1x+(4)=0 exactly over the rational numbers; a=1 is explicitly nonzero.

View answer and explanation-4

x=−(4)/1=-4; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0017/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 43. Exact Solution of a Linear Equation

Solve 1x+(11)=0 exactly over the rational numbers; a=1 is explicitly nonzero.

View answer and explanation-11

x=−(11)/1=-11; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0018/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 44. Exact Solution of a Linear Equation

Solve 2x+(-13)=0 exactly over the rational numbers; a=2 is explicitly nonzero.

View answer and explanation13/2

x=−(-13)/2=13/2; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0019/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 45. Exact Solution of a Linear Equation

Solve 3x+(-6)=0 exactly over the rational numbers; a=3 is explicitly nonzero.

View answer and explanation2

x=−(-6)/3=2; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0020/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 46. Exact Solution of a Linear Equation

Solve 4x+(1)=0 exactly over the rational numbers; a=4 is explicitly nonzero.

View answer and explanation-1/4

x=−(1)/4=-1/4; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0021/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 47. Exact Solution of a Linear Equation

Solve 5x+(8)=0 exactly over the rational numbers; a=5 is explicitly nonzero.

View answer and explanation-8/5

x=−(8)/5=-8/5; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0022/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 48. Exact Solution of a Linear Equation

Solve -5x+(15)=0 exactly over the rational numbers; a=-5 is explicitly nonzero.

View answer and explanation3

x=−(15)/-5=3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0023/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 49. Exact Solution of a Linear Equation

Solve -4x+(-9)=0 exactly over the rational numbers; a=-4 is explicitly nonzero.

View answer and explanation-9/4

x=−(-9)/-4=-9/4; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0024/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 50. Exact Solution of a Linear Equation

Solve -3x+(-2)=0 exactly over the rational numbers; a=-3 is explicitly nonzero.

View answer and explanation-2/3

x=−(-2)/-3=-2/3; the result is reduced by the gcd to a canonical rational with a positive denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-exact-solve-0025/)

Linear equation in one variable · qf-linear-equation-exact-solve · candidate





## 51. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -4x+(-14)=0 and the candidate x=-7/2. Is the candidate a solution?

View answer and explanationYes

q=2≠0 and -4×-7+(-14)×2=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0001/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 52. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -3x+(-9)=0 and the candidate x=-2/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and -3×-2+(-9)×1=-3, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0002/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 53. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -2x+(-4)=0 and the candidate x=-2/1. Is the candidate a solution?

View answer and explanationYes

q=1≠0 and -2×-2+(-4)×1=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0003/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 54. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -1x+(1)=0 and the candidate x=2/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and -1×2+(1)×1=-1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0004/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 55. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 2x+(6)=0 and the candidate x=-3/1. Is the candidate a solution?

View answer and explanationYes

q=1≠0 and 2×-3+(6)×1=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0005/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 56. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 1x+(11)=0 and the candidate x=-10/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and 1×-10+(11)×1=1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0006/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 57. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 2x+(-13)=0 and the candidate x=13/2. Is the candidate a solution?

View answer and explanationYes

q=2≠0 and 2×13+(-13)×2=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0007/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 58. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 3x+(-8)=0 and the candidate x=3/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and 3×3+(-8)×1=1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0008/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 59. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 4x+(-3)=0 and the candidate x=3/4. Is the candidate a solution?

View answer and explanationYes

q=4≠0 and 4×3+(-3)×4=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0009/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 60. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -4x+(2)=0 and the candidate x=1/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and -4×1+(2)×1=-2, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0010/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 61. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -3x+(7)=0 and the candidate x=7/3. Is the candidate a solution?

View answer and explanationYes

q=3≠0 and -3×7+(7)×3=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0011/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 62. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -2x+(12)=0 and the candidate x=7/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and -2×7+(12)×1=-2, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0012/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 63. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -1x+(-12)=0 and the candidate x=-12/1. Is the candidate a solution?

View answer and explanationYes

q=1≠0 and -1×-12+(-12)×1=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0013/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 64. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 2x+(-7)=0 and the candidate x=4/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and 2×4+(-7)×1=1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0014/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 65. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 1x+(-2)=0 and the candidate x=2/1. Is the candidate a solution?

View answer and explanationYes

q=1≠0 and 1×2+(-2)×1=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0015/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 66. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 2x+(3)=0 and the candidate x=-1/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and 2×-1+(3)×1=1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0016/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 67. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 3x+(8)=0 and the candidate x=-8/3. Is the candidate a solution?

View answer and explanationYes

q=3≠0 and 3×-8+(8)×3=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0017/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 68. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 4x+(13)=0 and the candidate x=-3/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and 4×-3+(13)×1=1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0018/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 69. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -4x+(-11)=0 and the candidate x=-11/4. Is the candidate a solution?

View answer and explanationYes

q=4≠0 and -4×-11+(-11)×4=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0019/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 70. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -3x+(-6)=0 and the candidate x=-1/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and -3×-1+(-6)×1=-3, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0020/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 71. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -2x+(-1)=0 and the candidate x=-1/2. Is the candidate a solution?

View answer and explanationYes

q=2≠0 and -2×-1+(-1)×2=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0021/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 72. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider -1x+(4)=0 and the candidate x=5/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and -1×5+(4)×1=-1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0022/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 73. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 2x+(9)=0 and the candidate x=-9/2. Is the candidate a solution?

View answer and explanationYes

q=2≠0 and 2×-9+(9)×2=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0023/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 74. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 1x+(14)=0 and the candidate x=-13/1. Is the candidate a solution?

View answer and explanationNo

q=1≠0 and 1×-13+(14)×1=1, so the candidate is not a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0024/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 75. Cross-Multiplication Check of a Rational Solution

Over the rational numbers, consider 2x+(-10)=0 and the candidate x=5/1. Is the candidate a solution?

View answer and explanationYes

q=1≠0 and 2×5+(-10)×1=0, so the candidate is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-verification-0025/)

Linear equation in one variable · qf-linear-equation-solution-verification · candidate





## 76. Solution Classification of a Linear Equation

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

View answer and explanationall rational numbers are solutions

With a=0 and b=0, the equation is 0=0, so every rational number is a solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0001/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 77. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 0x+(1)=0.

View answer and explanationno solution

With a=0 and b=1≠0, the equation is 1=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0002/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 78. Solution Classification of a Linear Equation

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

View answer and explanationno solution

With a=0 and b=2≠0, the equation is 2=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0003/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 79. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 0x+(3)=0.

View answer and explanationno solution

With a=0 and b=3≠0, the equation is 3=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0004/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 80. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 0x+(4)=0.

View answer and explanationno solution

With a=0 and b=4≠0, the equation is 4=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0005/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 81. Solution Classification of a Linear Equation

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

View answer and explanationno solution

With a=0 and b=5≠0, the equation is 5=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0006/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 82. Solution Classification of a Linear Equation

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

View answer and explanationno solution

With a=0 and b=6≠0, the equation is 6=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0007/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 83. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 0x+(7)=0.

View answer and explanationno solution

With a=0 and b=7≠0, the equation is 7=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0008/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 84. Solution Classification of a Linear Equation

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

View answer and explanationno solution

With a=0 and b=8≠0, the equation is 8=0, so it has no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0009/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 85. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 3x+(-3)=0.

View answer and explanationunique solution x=1

Because a=3≠0, the equation has the unique solution x=1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0010/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 86. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 4x+(-2)=0.

View answer and explanationunique solution x=1/2

Because a=4≠0, the equation has the unique solution x=1/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0011/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 87. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 5x+(-1)=0.

View answer and explanationunique solution x=1/5

Because a=5≠0, the equation has the unique solution x=1/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0012/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 88. Solution Classification of a Linear Equation

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

View answer and explanationunique solution x=0

Because a=6≠0, the equation has the unique solution x=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0013/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 89. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 7x+(1)=0.

View answer and explanationunique solution x=-1/7

Because a=7≠0, the equation has the unique solution x=-1/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0014/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 90. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 1x+(2)=0.

View answer and explanationunique solution x=-2

Because a=1≠0, the equation has the unique solution x=-2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0015/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 91. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 2x+(3)=0.

View answer and explanationunique solution x=-3/2

Because a=2≠0, the equation has the unique solution x=-3/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0016/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 92. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 3x+(4)=0.

View answer and explanationunique solution x=-4/3

Because a=3≠0, the equation has the unique solution x=-4/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0017/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 93. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 4x+(5)=0.

View answer and explanationunique solution x=-5/4

Because a=4≠0, the equation has the unique solution x=-5/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0018/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 94. Solution Classification of a Linear Equation

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

View answer and explanationunique solution x=-6/5

Because a=5≠0, the equation has the unique solution x=-6/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0019/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 95. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 6x+(7)=0.

View answer and explanationunique solution x=-7/6

Because a=6≠0, the equation has the unique solution x=-7/6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0020/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 96. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 7x+(8)=0.

View answer and explanationunique solution x=-8/7

Because a=7≠0, the equation has the unique solution x=-8/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0021/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 97. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 1x+(9)=0.

View answer and explanationunique solution x=-9

Because a=1≠0, the equation has the unique solution x=-9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0022/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 98. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 2x+(10)=0.

View answer and explanationunique solution x=-5

Because a=2≠0, the equation has the unique solution x=-5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0023/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 99. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 3x+(11)=0.

View answer and explanationunique solution x=-11/3

Because a=3≠0, the equation has the unique solution x=-11/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0024/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate





## 100. Solution Classification of a Linear Equation

Over the rational numbers, classify the solution set of 4x+(12)=0.

View answer and explanationunique solution x=-3

Because a=4≠0, the equation has the unique solution x=-3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260927-linear-equation-solution-class-0025/)

Linear equation in one variable · qf-linear-equation-solution-class · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260927-linear-equation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260927-linear-equation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260927-linear-equation-foundations/validation-report.json)