Question 32719: Inverse Coefficients of a Nonzero-Slope Affine Function

Original question and givens

Over R, let f:R→R be f(x)=(19)x+(6), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

• G001 — Over R, let f:R→R be f(x)=(19)x+(6), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

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-inverse-function

Step-by-step derivation

source roles = [qf-inverse-affine-coefficients, affine_inverse_coefficients, reals, []]

derived = [[SLOPE_FIRST_pair_fractions_RETAIN1]]

Affine inverse ORIGINAL reciprocal nonzero slope := div(1, 19)

Affine inverse ORIGINAL reciprocal nonzero slope = 1/19

Affine inverse ORIGINAL negative constant := sub(0, 6)

Affine inverse ORIGINAL negative constant = -6

Affine inverse ORIGINAL negative b over m := div(-6, 19)

Affine inverse ORIGINAL negative b over m = -6/19

Affine inverse reverse g after f slope := mul(1/19, 19)

Affine inverse reverse g after f slope = 1

Affine inverse reverse u times original b := mul(1/19, 6)

Affine inverse reverse u times original b = 6/19

Affine inverse reverse g after f constant := sum(6/19, -6/19)

Affine inverse reverse g after f constant = 0

Affine inverse reverse f after g slope := mul(19, 1/19)

Affine inverse reverse f after g slope = 1

Affine inverse reverse m times inverse constant := mul(19, -6/19)

Affine inverse reverse m times inverse constant = -6

Affine inverse reverse f after g constant := sum(-6, 6)

Affine inverse reverse f after g constant = 0

source roles = [19, 6]

derived = [1/19, -6/19, 1, 0, 1, 0]

Reverse checks

• S011 · 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.
• S010 · premise_and_result_audit — Recompute Affine inverse reverse f after g constant with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Affine inverse reverse m times inverse constant with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Affine inverse reverse f after g slope with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Affine inverse reverse g after f constant with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Affine inverse reverse u times original b with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Affine inverse reverse g after f slope with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Affine inverse ORIGINAL negative b over m with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Affine inverse ORIGINAL negative constant with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Affine inverse ORIGINAL reciprocal nonzero slope with exact rational arithmetic, not rounded decimals.
• 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

(1/19, -6/19)

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.