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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-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.
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.
0
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
13
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
3
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
0
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
0
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
3
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
13
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
-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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
23
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
Linear systems in two variables and degenerate constraints · qf-linear-system-determinant · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
unique 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
unique 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
unique 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
unique 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
unique 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
unique 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
no 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
unique 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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.
infinitely 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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-class · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
No
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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?
Yes
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
Linear systems in two variables and degenerate constraints · qf-linear-system-solution-verification · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(-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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
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.
(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
Linear systems in two variables and degenerate constraints · qf-linear-system-unique-solve · candidate
Canonical question JSONL · English JSONL · Validation report