# Question 6625: Both Equation Sides after a Common Shift [Return to catalogue page 34](https://ocme.evemisslab.com/en/questions/page/34/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r03-equation-foundations/#q-20261010-advance-20261014-r03-equation-0100) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0012-06625 · q-20261010-advance-20261014-r03-equation-0100 ## Original question and givens Over Q, L=(1)x+(-24), R=(-1)x+(25), x=24/2, 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+(-24), R=(-1)x+(25), x=24/2, 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](https://ocme.evemisslab.com/en/mko/mko-equation/) ## Step-by-step derivation S001 · impliesBefore common rational domain with original equation definitions After evaluate/classify the given model without enlarging its original domain Conditions: none Original definedness precedes substitution and transformations.Check the declared model,not arbitrary equation solving inferred from cross multiplication or one-point values. Step identifiers and dependenciesRule: original_rational_equation_scope · Uses: G001 S002 · calculationBefore original left numerator linear product := mul(1, 12) After original left numerator linear product = 12 Conditions: none Substitute the specified x into this side's coefficient product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore original left numerator affine value := sum(12, -24) After original left numerator affine value = -12 Conditions: none Then add this side's own constant,not the other side's. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore original right linear product := mul(-1, 12) After original right linear product = -12 Conditions: none Substitute the specified x into this side's coefficient product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore original right affine value := sum(-12, 25) After original right affine value = 13 Conditions: none Then add this side's own constant,not the other side's. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore original L−R := sub(-12, 13) After original L−R = -25 Conditions: none Record the original difference of the two sides first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore left plus common e := sum(-12, -1) After left plus common e = -13 Conditions: none Add the same defined rational e to the left. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore right plus common e := sum(13, -1) After right plus common e = 12 Conditions: none Add the identical e to the right,not another constant. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore shifted two-side residual := sub(-13, 12) After shifted two-side residual = -25 Conditions: none Subtracting cancels the two copies of e and must recover the original difference. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore original difference=-25;shifted difference=-25 After shifted equality=No Conditions: none Adding e to both sides is reversed by subtracting e from both sides,so it preserves equality on the common defined rational domain. Step identifiers and dependenciesRule: common_shift_reversible_equality · Uses: G001, S009 ## 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 (-25/1, -13/1, 12/1, -25/1, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 18c1e0cccfeaaff72fb30e7d668822f2ba5bbd2e662238159747f4d7aa2098b9 - candidate record SHA-256: 5002b65a6f4e52d22d2a4d861c3e8f74b19e6ba6639129fa916d56b15d711a30 - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r03-equation-0100.json)