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P1000-0301 · q-20260927-linear-equation-solution-verification-0001
Over the rational numbers, consider -4x+(-14)=0 and the candidate x=-7/2. Is the candidate a solution?
G001 — Over the rational numbers, consider -4x+(-14)=0 and the candidate x=-7/2. Is the candidate a solution?Use the rational domain. A candidate p/q needs q≠0; a zero residual is equivalent to the cross-multiplied identity.
Before
a=-4, b=-14, x∈Q
After
equation: -4x+(-14)=0
Conditions: none
Read the linear coefficient, constant and rational domain explicitly; a name alone does not establish nondegeneracy.
Rule: read_coefficients · Uses: G001
Before
candidate x := div(-7, 2)
After
candidate x = -7/2
Conditions: none
Read the candidate as an exact rational; its denominator must be nonzero.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
a·x := mul(-4, -7/2)
After
a·x = 14
Conditions: none
Use the original coefficient in the linear term.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
a·x+b := sum(14, -14)
After
a·x+b = 0
Conditions: none
The candidate solves the original equation exactly when the residual is0.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a·p := mul(-4, -7)
After
a·p = 28
Conditions: none
Independently form the first cross-multiplied integer term.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
b·q := mul(-14, 2)
After
b·q = -28
Conditions: none
Multiply the constant by the same nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
a·p+b·q := sum(28, -28)
After
a·p+b·q = 0
Conditions: none
Nonzero q makes this equality equivalent to a zero original residual.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
residual=0; cross=0
After
candidate solves equation: Yes
Conditions: none
The two exact checks agree on the candidate's validity.
Rule: substitution_verdict · Uses: G001, S007
S008 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.S007 · premise_and_result_audit — Recompute a·p+b·q with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute b·q with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute a·p with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute a·x+b with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute a·x with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute candidate x with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.Yes
The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.
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.
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