# Question 7011: Complete Three-Point Proportion Law and Ratio Values [Return to catalogue page 36](https://ocme.evemisslab.com/en/questions/page/36/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r02-ratio-proportion-foundations/#q-20261011-advance-20261015-r02-ratio-proportion-0036) 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. R0013-07011 · q-20261011-advance-20261015-r02-ratio-proportion-0036 ## Original question and givens Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-2/2, complete ordered pairs (x,y)=[[0, 0], [1, 10], [2, -9]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0. - G001 — Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-2/2, complete ordered pairs (x,y)=[[0, 0], [1, 10], [2, -9]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0. ## 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-ratio-proportion](https://ocme.evemisslab.com/en/mko/mko-ratio-proportion/) ## Step-by-step derivation S001 · impliesBefore exact rational numerical ratios;each denominator needs its own guard After cross-multiplied equality does not legalize a zero ratio denominator Conditions: none A numerical ratio is not an automatic causal or geometric-similarity proof.Cross residual0 does not legalize a zero denominator,and the reverse ratio needs its own denominator guard. Step identifiers and dependenciesRule: rational_ratio_definition_scope · Uses: G001 S002 · calculationBefore point0 kx := mul(-1, 0) After point0 kx = 0 Conditions: none First evaluate k·x at this point,without assuming y/x is defined. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore point0 law residual := sub(0, 0) After point0 law residual = 0 Conditions: none The point's proportional-law residual is y−kx. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore point=(0,0);k=-1;residual=0 After law matches=Yes;ratio=undefined Conditions: none At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs. Step identifiers and dependenciesRule: finite_point_law_and_ratio_guard · Uses: G001, S003 S005 · calculationBefore point1 kx := mul(-1, 1) After point1 kx = -1 Conditions: none First evaluate k·x at this point,without assuming y/x is defined. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore point1 law residual := sub(10, -1) After point1 law residual = 11 Conditions: none The point's proportional-law residual is y−kx. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore point1 y/x := div(10, 1) After point1 y/x = 10 Conditions: none Evaluate y/x only when this point's x is nonzero. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore point=(1,10);k=-1;residual=11 After law matches=No;ratio=10/1 Conditions: none At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs. Step identifiers and dependenciesRule: finite_point_law_and_ratio_guard · Uses: G001, S007 S009 · calculationBefore point2 kx := mul(-1, 2) After point2 kx = -2 Conditions: none First evaluate k·x at this point,without assuming y/x is defined. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore point2 law residual := sub(-9, -2) After point2 law residual = -7 Conditions: none The point's proportional-law residual is y−kx. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore point2 y/x := div(-9, 2) After point2 y/x = -9/2 Conditions: none Evaluate y/x only when this point's x is nonzero. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore point=(2,-9);k=-1;residual=-7 After law matches=No;ratio=-9/2 Conditions: none At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs. Step identifiers and dependenciesRule: finite_point_law_and_ratio_guard · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute point2 y/x with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute point2 law residual with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute point2 kx with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute point1 y/x with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute point1 law residual with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute point1 kx with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute point0 law residual with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute point0 kx 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 ([0/1, 11/1, -7/1], [Yes, No, No], [undefined, 10/1, -9/2]) 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: 8e43b97590e83833769bfa8da95d6f1e38b1edeb58c7a05afb60562ad1e53165 - candidate record SHA-256: acee07ab67abb0e2560c5442b5dc17d0c102ccddcfd3604b3b327b13e328635b - semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59 - question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r02-ratio-proportion-0036.json)