Question 32972: Checking a Rational Candidate in Both Rows

Original question and givens

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?

• G001 — 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?

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-verification, verify_solution, reals, []]

derived = [[ALL_rows_YesNo_STRING]]

source roles = [[[2, 3, 6], [4, -3, 21]]]

derived = [[coefficient_x, coefficient_y, right_side]]

System ORIGINAL a times e main product := mul(2, -3)

System ORIGINAL a times e main product = -6

System ORIGINAL b times d other product := mul(3, 4)

System ORIGINAL b times d other product = 12

System ORIGINAL signed determinant ae minus bd := sub(-6, 12)

System ORIGINAL signed determinant ae minus bd = -18

source roles = [[[2, 3, 6], [4, -3, 21]]]

derived = [-6, 12, -18]

source roles = [x candidate, [5, 1]]

derived = [5/1]

source roles = [y candidate, [-1, 1]]

derived = [-1/1]

System candidate ORIGINAL two positive denominators product := mul(1, 1)

System candidate ORIGINAL two positive denominators product = 1

System cleared row0 ORIGINAL coefficient x times numerator := mul(2, 5)

System cleared row0 ORIGINAL coefficient x times numerator = 10

System cleared row0 ORIGINAL x term times y denominator := mul(10, 1)

System cleared row0 ORIGINAL x term times y denominator = 10

System cleared row0 ORIGINAL coefficient y times numerator := mul(3, -1)

System cleared row0 ORIGINAL coefficient y times numerator = -3

System cleared row0 ORIGINAL y term times x denominator := mul(-3, 1)

System cleared row0 ORIGINAL y term times x denominator = -3

System cleared row0 ORIGINAL rhs times both denominators := mul(6, 1)

System cleared row0 ORIGINAL rhs times both denominators = 6

System cleared row0 ORIGINAL two left terms sum := sum(10, -3)

System cleared row0 ORIGINAL two left terms sum = 7

System cleared row0 ORIGINAL residual := sub(7, 6)

System cleared row0 ORIGINAL residual = 1

source roles = [0, 2, 3, 6, 5, 1, -1, 1]

derived = [10, -3, 6, 1, No]

System cleared row1 ORIGINAL coefficient x times numerator := mul(4, 5)

System cleared row1 ORIGINAL coefficient x times numerator = 20

System cleared row1 ORIGINAL x term times y denominator := mul(20, 1)

System cleared row1 ORIGINAL x term times y denominator = 20

System cleared row1 ORIGINAL coefficient y times numerator := mul(-3, -1)

System cleared row1 ORIGINAL coefficient y times numerator = 3

System cleared row1 ORIGINAL y term times x denominator := mul(3, 1)

System cleared row1 ORIGINAL y term times x denominator = 3

System cleared row1 ORIGINAL rhs times both denominators := mul(21, 1)

System cleared row1 ORIGINAL rhs times both denominators = 21

System cleared row1 ORIGINAL two left terms sum := sum(20, 3)

System cleared row1 ORIGINAL two left terms sum = 23

System cleared row1 ORIGINAL residual := sub(23, 21)

System cleared row1 ORIGINAL residual = 2

source roles = [1, 4, -3, 21, 5, 1, -1, 1]

derived = [20, 3, 21, 2, No]

Candidate direct reverse ORIGINAL row0 x term := mul(2, 5)

Candidate direct reverse ORIGINAL row0 x term = 10

Candidate direct reverse ORIGINAL row0 y term := mul(3, -1)

Candidate direct reverse ORIGINAL row0 y term = -3

Candidate direct reverse ORIGINAL row0 terms sum := sum(10, -3)

Candidate direct reverse ORIGINAL row0 terms sum = 7

Candidate direct reverse ORIGINAL row0 residual minus rhs := sub(7, 6)

Candidate direct reverse ORIGINAL row0 residual minus rhs = 1

source roles = [Candidate direct reverse, 0, 2, 3, 6, 5/1, -1/1]

derived = [10, -3, 7, 1/1]

Candidate direct reverse ORIGINAL row1 x term := mul(4, 5)

Candidate direct reverse ORIGINAL row1 x term = 20

Candidate direct reverse ORIGINAL row1 y term := mul(-3, -1)

Candidate direct reverse ORIGINAL row1 y term = 3

Candidate direct reverse ORIGINAL row1 terms sum := sum(20, 3)

Candidate direct reverse ORIGINAL row1 terms sum = 23

Candidate direct reverse ORIGINAL row1 residual minus rhs := sub(23, 21)

Candidate direct reverse ORIGINAL row1 residual minus rhs = 2

source roles = [Candidate direct reverse, 1, 4, -3, 21, 5/1, -1/1]

derived = [20, 3, 23, 2/1]

Reverse checks

• S035 · 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.
• S034 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 residual minus rhs with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 terms sum with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 y term with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row1 x term with exact rational arithmetic, not rounded decimals.
• S030 · 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.
• S029 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 residual minus rhs with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 terms sum with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 y term with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Candidate direct reverse ORIGINAL row0 x term with exact rational arithmetic, not rounded decimals.
• S025 · 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.
• S024 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL residual with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL two left terms sum with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL rhs times both denominators with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL y term times x denominator with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL coefficient y times numerator with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL x term times y denominator with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute System cleared row1 ORIGINAL coefficient x times numerator with exact rational arithmetic, not rounded decimals.
• S017 · 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.
• S016 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL residual with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL two left terms sum with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL rhs times both denominators with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL y term times x denominator with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL coefficient y times numerator with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL x term times y denominator with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute System cleared row0 ORIGINAL coefficient x times numerator with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute System candidate ORIGINAL two positive denominators product with exact rational arithmetic, not rounded decimals.
• S008 · 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.
• 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

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.