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R0012-06617 · q-20261010-advance-20261014-r03-equation-0092
Over Q, L=(1)x+(-16), R=(-1)x+(17), x=16/2, common added quantity e=0. 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+(-16), R=(-1)x+(17), x=16/2, common added quantity e=0. 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.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.
Before
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.
Rule: original_rational_equation_scope · Uses: G001
Before
original left numerator linear product := mul(1, 8)
After
original left numerator linear product = 8
Conditions: none
Substitute the specified x into this side's coefficient product.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
original left numerator affine value := sum(8, -16)
After
original left numerator affine value = -8
Conditions: none
Then add this side's own constant,not the other side's.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
original right linear product := mul(-1, 8)
After
original right linear product = -8
Conditions: none
Substitute the specified x into this side's coefficient product.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original right affine value := sum(-8, 17)
After
original right affine value = 9
Conditions: none
Then add this side's own constant,not the other side's.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
original L−R := sub(-8, 9)
After
original L−R = -17
Conditions: none
Record the original difference of the two sides first.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left plus common e := sum(-8, 0)
After
left plus common e = -8
Conditions: none
Add the same defined rational e to the left.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
right plus common e := sum(9, 0)
After
right plus common e = 9
Conditions: none
Add the identical e to the right,not another constant.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
shifted two-side residual := sub(-8, 9)
After
shifted two-side residual = -17
Conditions: none
Subtracting cancels the two copies of e and must recover the original difference.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
original difference=-17;shifted difference=-17
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.
Rule: common_shift_reversible_equality · Uses: G001, S009
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.(-17/1, -8/1, 9/1, -17/1, No)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 79d723742b75c9389b02301fb8d7e7776d99c3ccf2daef5438f46080658b0da3candidate record SHA-256: 043911e58434ca06466ad67a9df85ff67b1c17bdbfcb8210c88a54179424fffesemantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9Lean: NOT_RUN · human review: NOT_PERFORMED