Question 6613: Both Equation Sides after a Common Shift

Original question and givens

Over Q, L=(1)x+(-12), R=(-1)x+(13), x=12/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, retaining /1 for integers.

• G001 — Over Q, L=(1)x+(-12), R=(-1)x+(13), x=12/2, common added quantity e=-1. Give (original L−R, L+e, R+e, shifted difference, are the shifted sides equal?). Use exact reduced fractions with positive denominators, 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-equation

Step-by-step derivation

common rational domain with original equation definitions

evaluate/classify the given model without enlarging its original domain

original left numerator linear product := mul(1, 6)

original left numerator linear product = 6

original left numerator affine value := sum(6, -12)

original left numerator affine value = -6

original right linear product := mul(-1, 6)

original right linear product = -6

original right affine value := sum(-6, 13)

original right affine value = 7

original L−R := sub(-6, 7)

original L−R = -13

left plus common e := sum(-6, -1)

left plus common e = -7

right plus common e := sum(7, -1)

right plus common e = 6

shifted two-side residual := sub(-7, 6)

shifted two-side residual = -13

original difference=-13;shifted difference=-13

shifted equality=No

Reverse checks

• S010 · premise_and_result_audit — Subtract the same e from both sides to recover the original equation without enlarging its domain.
• S009 · premise_and_result_audit — Recompute shifted two-side residual with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute right plus common e with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute left plus common e with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute original L−R with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute original right affine value with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute original right linear product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute original left numerator affine value with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute original left numerator linear product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(-13/1, -7/1, 6/1, -13/1, 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.