Explore related knowledge and questions →
← Question derivation index · Original question batch
P1000-0311 · q-20260927-linear-equation-solution-verification-0011
Over the rational numbers, consider -3x+(7)=0 and the candidate x=7/3. Is the candidate a solution?
G001 — Over the rational numbers, consider -3x+(7)=0 and the candidate x=7/3. 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=-3, b=7, x∈Q
After
equation: -3x+(7)=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, 3)
After
candidate x = 7/3
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(-3, 7/3)
After
a·x = -7
Conditions: none
Use the original coefficient in the linear term.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
a·x+b := sum(-7, 7)
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(-3, 7)
After
a·p = -21
Conditions: none
Independently form the first cross-multiplied integer term.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
b·q := mul(7, 3)
After
b·q = 21
Conditions: none
Multiply the constant by the same nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
a·p+b·q := sum(-21, 21)
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a7a75505b52bbba8b00df4867dab7082dcc92eead25ccdca43df76da0b20d799candidate record SHA-256: 7e0d66b95af023dc4350274ec6aa9c61a9ba7d3253001e09691bef98e9ebaf6fsemantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612bLean: NOT_RUN · human review: NOT_PERFORMED