Question 32980: Exact Unique Solution with a Nonzero Determinant

Original question and givens

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.

• G001 — 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.

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-unique-solve, unique_solve, reals, []]

derived = [[solution_pair_fractions_RETAIN1]]

source roles = [[[1, 1, -8], [1, -2, 2]]]

derived = [[coefficient_x, coefficient_y, right_side]]

System ORIGINAL a times e main product := mul(1, -2)

System ORIGINAL a times e main product = -2

System ORIGINAL b times d other product := mul(1, 1)

System ORIGINAL b times d other product = 1

System ORIGINAL signed determinant ae minus bd := sub(-2, 1)

System ORIGINAL signed determinant ae minus bd = -3

source roles = [[[1, 1, -8], [1, -2, 2]]]

derived = [-2, 1, -3]

Cramer ORIGINAL c times e := mul(-8, -2)

Cramer ORIGINAL c times e = 16

Cramer ORIGINAL b times f := mul(1, 2)

Cramer ORIGINAL b times f = 2

Cramer ORIGINAL numerator x ce minus bf := sub(16, 2)

Cramer ORIGINAL numerator x ce minus bf = 14

Cramer ORIGINAL a times f := mul(1, 2)

Cramer ORIGINAL a times f = 2

Cramer ORIGINAL c times d := mul(-8, 1)

Cramer ORIGINAL c times d = -8

Cramer ORIGINAL numerator y af minus cd := sub(2, -8)

Cramer ORIGINAL numerator y af minus cd = 10

Cramer ORIGINAL x numerator over NONZERO D := div(14, -3)

Cramer ORIGINAL x numerator over NONZERO D = -14/3

Cramer ORIGINAL y numerator over NONZERO D := div(10, -3)

Cramer ORIGINAL y numerator over NONZERO D = -10/3

Cramer reverse ORIGINAL row0 x term := mul(1, -14/3)

Cramer reverse ORIGINAL row0 x term = -14/3

Cramer reverse ORIGINAL row0 y term := mul(1, -10/3)

Cramer reverse ORIGINAL row0 y term = -10/3

Cramer reverse ORIGINAL row0 terms sum := sum(-14/3, -10/3)

Cramer reverse ORIGINAL row0 terms sum = -8

Cramer reverse ORIGINAL row0 residual minus rhs := sub(-8, -8)

Cramer reverse ORIGINAL row0 residual minus rhs = 0

source roles = [Cramer reverse, 0, 1, 1, -8, -14/3, -10/3]

derived = [-14/3, -10/3, -8, 0/1]

Cramer reverse ORIGINAL row1 x term := mul(1, -14/3)

Cramer reverse ORIGINAL row1 x term = -14/3

Cramer reverse ORIGINAL row1 y term := mul(-2, -10/3)

Cramer reverse ORIGINAL row1 y term = 20/3

Cramer reverse ORIGINAL row1 terms sum := sum(-14/3, 20/3)

Cramer reverse ORIGINAL row1 terms sum = 2

Cramer reverse ORIGINAL row1 residual minus rhs := sub(2, 2)

Cramer reverse ORIGINAL row1 residual minus rhs = 0

source roles = [Cramer reverse, 1, 1, -2, 2, -14/3, -10/3]

derived = [-14/3, 20/3, 2, 0/1]

source roles = [[[1, 1, -8], [1, -2, 2]], -3]

derived = [14, 10, -14/3, -10/3]

Reverse checks

• 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 — 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.
• S023 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 residual minus rhs with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 terms sum with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 y term with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row1 x term with exact rational arithmetic, not rounded decimals.
• S019 · 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.
• S018 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 residual minus rhs with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 terms sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 y term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Cramer reverse ORIGINAL row0 x term with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Cramer ORIGINAL y numerator over NONZERO D with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Cramer ORIGINAL x numerator over NONZERO D with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Cramer ORIGINAL numerator y af minus cd with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Cramer ORIGINAL c times d with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Cramer ORIGINAL a times f with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Cramer ORIGINAL numerator x ce minus bf with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Cramer ORIGINAL b times f with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Cramer ORIGINAL c times e with exact rational arithmetic, not rounded decimals.
• 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

(-14/3, -10/3)

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.