# Two-Variable Linear System 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-20261003-advance-20261004-r06-system-linear-equations-foundations · 2026-10-03 · 100 questions

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



## 1. Determinant of a Two-Variable Linear System

Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation0

The coefficient determinant is D=0, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0001/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 2. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-11)x+(1)y=-1 and (1)x+(-1)y=-2 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation10

The coefficient determinant is D=10, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0002/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 3. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-10)x+(1)y=0 and (1)x+(0)y=-1 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-1

The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0003/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 4. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-9)x+(1)y=1 and (1)x+(1)y=0 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-10

The coefficient determinant is D=-10, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0004/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 5. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-8)x+(1)y=2 and (1)x+(2)y=1 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-17

The coefficient determinant is D=-17, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0005/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 6. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-7)x+(1)y=-2 and (1)x+(-2)y=2 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation13

The coefficient determinant is D=13, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0006/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 7. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-6)x+(1)y=-1 and (1)x+(-1)y=3 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation5

The coefficient determinant is D=5, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0007/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 8. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-5)x+(1)y=0 and (1)x+(0)y=-3 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-1

The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0008/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 9. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-4)x+(1)y=1 and (1)x+(1)y=-2 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-5

The coefficient determinant is D=-5, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0009/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 10. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-3)x+(1)y=2 and (1)x+(2)y=-1 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-7

The coefficient determinant is D=-7, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0010/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 11. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-2)x+(1)y=-2 and (1)x+(-2)y=0 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation3

The coefficient determinant is D=3, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0011/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 12. Determinant of a Two-Variable Linear System

Over the reals, the constraints (-1)x+(1)y=-1 and (1)x+(-1)y=1 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation0

The coefficient determinant is D=0, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0012/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 13. Determinant of a Two-Variable Linear System

Over the reals, the constraints (0)x+(1)y=0 and (1)x+(0)y=2 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-1

The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0013/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 14. Determinant of a Two-Variable Linear System

Over the reals, the constraints (1)x+(1)y=1 and (1)x+(1)y=3 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation0

The coefficient determinant is D=0, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0014/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 15. Determinant of a Two-Variable Linear System

Over the reals, the constraints (2)x+(1)y=2 and (1)x+(2)y=-3 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation3

The coefficient determinant is D=3, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0015/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 16. Determinant of a Two-Variable Linear System

Over the reals, the constraints (3)x+(1)y=-2 and (1)x+(-2)y=-2 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-7

The coefficient determinant is D=-7, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0016/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 17. Determinant of a Two-Variable Linear System

Over the reals, the constraints (4)x+(1)y=-1 and (1)x+(-1)y=-1 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-5

The coefficient determinant is D=-5, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0017/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 18. Determinant of a Two-Variable Linear System

Over the reals, the constraints (5)x+(1)y=0 and (1)x+(0)y=0 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-1

The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0018/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 19. Determinant of a Two-Variable Linear System

Over the reals, the constraints (6)x+(1)y=1 and (1)x+(1)y=1 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation5

The coefficient determinant is D=5, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0019/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 20. Determinant of a Two-Variable Linear System

Over the reals, the constraints (7)x+(1)y=2 and (1)x+(2)y=2 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation13

The coefficient determinant is D=13, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0020/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 21. Determinant of a Two-Variable Linear System

Over the reals, the constraints (8)x+(1)y=-2 and (1)x+(-2)y=3 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-17

The coefficient determinant is D=-17, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0021/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 22. Determinant of a Two-Variable Linear System

Over the reals, the constraints (9)x+(1)y=-1 and (1)x+(-1)y=-3 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-10

The coefficient determinant is D=-10, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0022/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 23. Determinant of a Two-Variable Linear System

Over the reals, the constraints (10)x+(1)y=0 and (1)x+(0)y=-2 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation-1

The coefficient determinant is D=-1, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0023/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 24. Determinant of a Two-Variable Linear System

Over the reals, the constraints (11)x+(1)y=1 and (1)x+(1)y=-1 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation10

The coefficient determinant is D=10, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0024/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 25. Determinant of a Two-Variable Linear System

Over the reals, the constraints (12)x+(1)y=2 and (1)x+(2)y=0 must both hold. Find the coefficient determinant D=ae−bd.

View answer and explanation23

The coefficient determinant is D=23, the main-diagonal product minus the other diagonal product. Right-side constants are not included.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0025/)

Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate





## 26. Solution Classification of a Linear System

Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0026/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 27. Solution Classification of a Linear System

Over the reals, the constraints (0)x+(0)y=1 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0027/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 28. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(0)y=2 and (0)x+(0)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0028/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 29. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(2)y=3 and (2)x+(4)y=6 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0029/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 30. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(2)y=3 and (2)x+(4)y=7 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0030/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 31. Solution Classification of a Linear System

Over the reals, the constraints (0)x+(1)y=2 and (0)x+(2)y=5 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0031/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 32. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(0)y=0 and (0)x+(-1)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationunique solution

D=-1 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0032/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 33. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(2)y=0 and (0)x+(0)y=4 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0033/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 34. Solution Classification of a Linear System

Over the reals, the constraints (-2)x+(1)y=1 and (1)x+(-1)y=-2 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationunique solution

D=1 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0034/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 35. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(3)y=5 and (2)x+(6)y=10 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0035/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 36. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(4)y=6 and (2)x+(8)y=13 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0036/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 37. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(1)y=-1 and (1)x+(-1)y=1 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationunique solution

D=-2 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0037/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 38. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(6)y=8 and (2)x+(12)y=16 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0038/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 39. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(7)y=9 and (2)x+(14)y=19 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0039/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 40. Solution Classification of a Linear System

Over the reals, the constraints (4)x+(1)y=2 and (1)x+(-1)y=-3 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationunique solution

D=-5 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0040/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 41. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(9)y=11 and (2)x+(18)y=22 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0041/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 42. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(10)y=12 and (2)x+(20)y=25 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0042/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 43. Solution Classification of a Linear System

Over the reals, the constraints (7)x+(1)y=0 and (1)x+(-1)y=0 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationunique solution

D=-8 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0043/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 44. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(12)y=14 and (2)x+(24)y=28 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0044/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 45. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(13)y=15 and (2)x+(26)y=31 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0045/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 46. Solution Classification of a Linear System

Over the reals, the constraints (10)x+(1)y=-2 and (1)x+(-1)y=3 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationunique solution

D=-11 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0046/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 47. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(15)y=17 and (2)x+(30)y=34 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0047/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 48. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(16)y=18 and (2)x+(32)y=37 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationno solution

D=0 and the classification is no solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0048/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 49. Solution Classification of a Linear System

Over the reals, the constraints (13)x+(1)y=1 and (1)x+(-1)y=-1 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationunique solution

D=-14 and the classification is unique solution. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0049/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 50. Solution Classification of a Linear System

Over the reals, the constraints (1)x+(18)y=20 and (2)x+(36)y=40 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.

View answer and explanationinfinitely many solutions

D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0050/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate





## 51. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (0)x+(0)y=0 and (0)x+(0)y=0 must both hold. Does the reduced positive-denominator candidate x=0/1, y=0/1 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0051/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 52. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (0)x+(0)y=1 and (0)x+(0)y=0 must both hold. Does the reduced positive-denominator candidate x=0/1, y=0/1 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (-1, 0). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0052/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 53. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (1)x+(2)y=0 and (2)x+(4)y=0 must both hold. Does the reduced positive-denominator candidate x=2/1, y=-1/1 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0053/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 54. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (1)x+(0)y=0 and (0)x+(1)y=1 must both hold. Does the reduced positive-denominator candidate x=0/1, y=0/1 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (0, -1). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0054/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 55. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-7 and (4)x+(-3)y=-17 must both hold. Does the reduced positive-denominator candidate x=-4/1, y=1/3 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0055/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 56. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-5 and (4)x+(-3)y=-16 must both hold. Does the reduced positive-denominator candidate x=-3/1, y=2/3 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0056/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 57. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-3 and (4)x+(-3)y=-15 must both hold. Does the reduced positive-denominator candidate x=-3/1, y=1/1 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0057/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 58. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-8 and (4)x+(-3)y=-7 must both hold. Does the reduced positive-denominator candidate x=-2/1, y=-1/1 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0058/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 59. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-6 and (4)x+(-3)y=-6 must both hold. Does the reduced positive-denominator candidate x=-2/1, y=-2/3 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0059/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 60. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-4 and (4)x+(-3)y=-5 must both hold. Does the reduced positive-denominator candidate x=-1/1, y=-1/3 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0060/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 61. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-2 and (4)x+(-3)y=-4 must both hold. Does the reduced positive-denominator candidate x=-1/1, y=0/1 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0061/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 62. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=0 and (4)x+(-3)y=-3 must both hold. Does the reduced positive-denominator candidate x=0/1, y=1/3 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0062/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 63. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=2 and (4)x+(-3)y=-2 must both hold. Does the reduced positive-denominator candidate x=0/1, y=2/3 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0063/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 64. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=4 and (4)x+(-3)y=-1 must both hold. Does the reduced positive-denominator candidate x=1/1, y=1/1 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0064/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 65. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=-1 and (4)x+(-3)y=7 must both hold. Does the reduced positive-denominator candidate x=1/1, y=-1/1 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0065/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 66. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=1 and (4)x+(-3)y=8 must both hold. Does the reduced positive-denominator candidate x=2/1, y=-2/3 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0066/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 67. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=3 and (4)x+(-3)y=9 must both hold. Does the reduced positive-denominator candidate x=2/1, y=-1/3 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0067/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 68. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=5 and (4)x+(-3)y=10 must both hold. Does the reduced positive-denominator candidate x=3/1, y=0/1 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0068/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 69. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=7 and (4)x+(-3)y=11 must both hold. Does the reduced positive-denominator candidate x=3/1, y=1/3 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0069/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 70. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=9 and (4)x+(-3)y=12 must both hold. Does the reduced positive-denominator candidate x=4/1, y=2/3 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0070/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 71. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=11 and (4)x+(-3)y=13 must both hold. Does the reduced positive-denominator candidate x=4/1, y=1/1 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0071/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 72. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=6 and (4)x+(-3)y=21 must both hold. Does the reduced positive-denominator candidate x=5/1, y=-1/1 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (1, 2). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0072/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 73. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=8 and (4)x+(-3)y=22 must both hold. Does the reduced positive-denominator candidate x=5/1, y=-2/3 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0073/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 74. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=10 and (4)x+(-3)y=23 must both hold. Does the reduced positive-denominator candidate x=6/1, y=-1/3 solve the entire system?

View answer and explanationNo

After clearing the nonzero denominators, the two residuals are (3, 6). Both must be zero, giving No; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0074/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 75. Checking a Rational Candidate in Both Rows

Over the reals, the constraints (2)x+(3)y=12 and (4)x+(-3)y=24 must both hold. Does the reduced positive-denominator candidate x=6/1, y=0/1 solve the entire system?

View answer and explanationYes

After clearing the nonzero denominators, the two residuals are (0, 0). Both must be zero, giving Yes; passing only one row is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0075/)

Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate





## 76. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(0)y=0 and (0)x+(1)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

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

D=1≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (0/1, 0/1). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0076/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 77. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=1 and (1)x+(-1)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(1/2, 1/2)

D=-2≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (1/2, 1/2). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0077/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 78. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (-2)x+(1)y=3 and (1)x+(2)y=-4 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-2/1, -1/1)

D=-5≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-2/1, -1/1). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0078/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 79. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-9 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-17/3, -10/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-17/3, -10/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0079/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 80. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-8 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-14/3, -10/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-14/3, -10/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0080/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 81. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-7 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-16/3, -5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-16/3, -5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0081/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 82. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-6 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-13/3, -5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-13/3, -5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0082/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 83. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-5 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-10/3, -5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-10/3, -5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0083/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 84. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-4 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-7/3, -5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-7/3, -5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0084/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 85. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-3 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-4/3, -5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-4/3, -5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0085/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 86. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-2 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(-2/1, 0/1)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-2/1, 0/1). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0086/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 87. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=-1 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

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

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (-1/1, 0/1). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0087/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 88. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=0 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

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

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (0/1, 0/1). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0088/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 89. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=1 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

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

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (1/1, 0/1). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0089/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 90. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=2 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(2/1, 0/1)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (2/1, 0/1). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0090/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 91. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=3 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(4/3, 5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (4/3, 5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0091/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 92. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=4 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(7/3, 5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (7/3, 5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0092/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 93. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=5 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(10/3, 5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (10/3, 5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0093/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 94. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=6 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(13/3, 5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (13/3, 5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0094/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 95. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=7 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(16/3, 5/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (16/3, 5/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0095/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 96. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=8 and (1)x+(-2)y=-2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(14/3, 10/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (14/3, 10/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0096/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 97. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=9 and (1)x+(-2)y=-1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(17/3, 10/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (17/3, 10/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0097/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 98. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=10 and (1)x+(-2)y=0 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(20/3, 10/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (20/3, 10/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0098/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 99. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=11 and (1)x+(-2)y=1 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(23/3, 10/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (23/3, 10/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0099/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate





## 100. Exact Unique Solution with a Nonzero Determinant

Over the reals, the constraints (1)x+(1)y=12 and (1)x+(-2)y=2 must both hold, with D≠0. Find the exact pair (x,y), writing each component as a reduced fraction p/q with positive denominator.

View answer and explanation(26/3, 10/3)

D=-3≠0, so use x=(ce−bf)/D and y=(af−cd)/D, reduce the fractions, and obtain (26/3, 10/3). The D=0 cases are outside this unique-solution exercise.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r06-system-linear-equations-0100/)

Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r06-system-linear-equations-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-advance-20261004-r06-system-linear-equations-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r06-system-linear-equations-foundations/validation-report.json)