Question 32929: Solution Classification of a Linear System

Original question and givens

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.

• G001 — 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.

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, 2, 3], [2, 4, 6]]]

derived = [[coefficient_x, coefficient_y, right_side]]

System ORIGINAL a times e main product := mul(1, 4)

System ORIGINAL a times e main product = 4

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

System ORIGINAL b times d other product = 4

System ORIGINAL signed determinant ae minus bd := sub(4, 4)

System ORIGINAL signed determinant ae minus bd = 0

source roles = [[[1, 2, 3], [2, 4, 6]]]

derived = [4, 4, 0]

source roles = [[[1, 2, 3], [2, 4, 6]], 0]

derived = [[]]

System rank1 ORIGINAL a times f augmented minor := mul(1, 6)

System rank1 ORIGINAL a times f augmented minor = 6

System rank1 ORIGINAL c times d augmented minor := mul(3, 2)

System rank1 ORIGINAL c times d augmented minor = 6

System rank1 ORIGINAL augmented x minor := sub(6, 6)

System rank1 ORIGINAL augmented x minor = 0

System rank1 ORIGINAL b times f augmented minor := mul(2, 6)

System rank1 ORIGINAL b times f augmented minor = 12

System rank1 ORIGINAL c times e augmented minor := mul(3, 4)

System rank1 ORIGINAL c times e augmented minor = 12

System rank1 ORIGINAL augmented y minor := sub(12, 12)

System rank1 ORIGINAL augmented y minor = 0

source roles = [[[1, 2, 3], [2, 4, 6]]]

derived = [0, 0, Yes]

System free parameter ORIGINAL rhs over x pivot := div(3, 1)

System free parameter ORIGINAL rhs over x pivot = 3

System free parameter ORIGINAL negative y coefficient := sub(0, 2)

System free parameter ORIGINAL negative y coefficient = -2

System free parameter ORIGINAL x slope in free y := div(-2, 1)

System free parameter ORIGINAL x slope in free y = -2

System free parameter reverse OTHER x coefficient times constant := mul(2, 3)

System free parameter reverse OTHER x coefficient times constant = 6

System free parameter reverse OTHER x coefficient times slope := mul(2, -2)

System free parameter reverse OTHER x coefficient times slope = -4

System free parameter reverse OTHER add y coefficient := sum(-4, 4)

System free parameter reverse OTHER add y coefficient = 0

source roles = [[[1, 2, 3], [2, 4, 6]], 0]

derived = [[x = 3/1 + (-2/1)*t, y = t], 6/1, 0/1, t arbitrary in R]

Reverse checks

• S021 · 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.
• S020 · premise_and_result_audit — Recompute System free parameter reverse OTHER add y coefficient with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute System free parameter reverse OTHER x coefficient times slope with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute System free parameter reverse OTHER x coefficient times constant with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute System free parameter ORIGINAL x slope in free y with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute System free parameter ORIGINAL negative y coefficient with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute System free parameter ORIGINAL rhs over x pivot with exact rational arithmetic, not rounded decimals.
• 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

infinitely many solutions

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.