Question 7020: Complete Three-Point Proportion Law and Ratio Values

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=7/2, complete ordered pairs (x,y)=[[0, 0], [1, 19], [2, -18]]. 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=7/2, complete ordered pairs (x,y)=[[0, 0], [1, 19], [2, -18]]. 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

Step-by-step derivation

exact rational numerical ratios;each denominator needs its own guard

cross-multiplied equality does not legalize a zero ratio denominator

point0 kx := mul(7/2, 0)

point0 kx = 0

point0 law residual := sub(0, 0)

point0 law residual = 0

point=(0,0);k=7/2;residual=0

law matches=Yes;ratio=undefined

point1 kx := mul(7/2, 1)

point1 kx = 7/2

point1 law residual := sub(19, 7/2)

point1 law residual = 31/2

point1 y/x := div(19, 1)

point1 y/x = 19

point=(1,19);k=7/2;residual=31/2

law matches=No;ratio=19/1

point2 kx := mul(7/2, 2)

point2 kx = 7

point2 law residual := sub(-18, 7)

point2 law residual = -25

point2 y/x := div(-18, 2)

point2 y/x = -9

point=(2,-18);k=7/2;residual=-25

law matches=No;ratio=-9/1

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, 31/2, -25/1], [Yes, No, No], [undefined, 19/1, -9/1])

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.