Question 32942: Solution Classification of a Linear System

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-system-linear-equations

Step-by-step derivation

source roles = [qf-linear-system-solution-class, solution_class, reals, []]

derived = [[SOURCE_solution_class_STRING]]

source roles = [[[1, 10, 12], [2, 20, 25]]]

derived = [[coefficient_x, coefficient_y, right_side]]

System ORIGINAL a times e main product := mul(1, 20)

System ORIGINAL a times e main product = 20

System ORIGINAL b times d other product := mul(10, 2)

System ORIGINAL b times d other product = 20

System ORIGINAL signed determinant ae minus bd := sub(20, 20)

System ORIGINAL signed determinant ae minus bd = 0

source roles = [[[1, 10, 12], [2, 20, 25]]]

derived = [20, 20, 0]

source roles = [[[1, 10, 12], [2, 20, 25]], 0]

derived = [[]]

System rank1 ORIGINAL a times f augmented minor := mul(1, 25)

System rank1 ORIGINAL a times f augmented minor = 25

System rank1 ORIGINAL c times d augmented minor := mul(12, 2)

System rank1 ORIGINAL c times d augmented minor = 24

System rank1 ORIGINAL augmented x minor := sub(25, 24)

System rank1 ORIGINAL augmented x minor = 1

System rank1 ORIGINAL b times f augmented minor := mul(10, 25)

System rank1 ORIGINAL b times f augmented minor = 250

System rank1 ORIGINAL c times e augmented minor := mul(12, 20)

System rank1 ORIGINAL c times e augmented minor = 240

System rank1 ORIGINAL augmented y minor := sub(250, 240)

System rank1 ORIGINAL augmented y minor = 10

source roles = [[[1, 10, 12], [2, 20, 25]]]

derived = [1, 10, No]

Reverse checks

• S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S013 · premise_and_result_audit — Recompute System rank1 ORIGINAL augmented y minor with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute System rank1 ORIGINAL c times e augmented minor with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute System rank1 ORIGINAL b times f augmented minor with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute System rank1 ORIGINAL augmented x minor with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute System rank1 ORIGINAL c times d augmented minor with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute System rank1 ORIGINAL a times f augmented minor with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S005 · premise_and_result_audit — Recompute System ORIGINAL signed determinant ae minus bd with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute System ORIGINAL b times d other product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute System ORIGINAL a times e main product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

no solution

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.