S001 · implies
Before
0x+(7)=0, x∈Q
After
a=0; b=7
Conditions: none
Read the rational solution domain and check the leading coefficient before dividing by a.
Step identifiers and dependencies
Rule: equation_domain · Uses: G001
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Question 33: Solution Classification of a Linear Equation
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P250-0033 · q-20260927-linear-equation-solution-class-0008
Original question and givens
Over the rational numbers, classify the solution set of 0x+(7)=0.
G001— Over the rational numbers, classify the solution set of 0x+(7)=0.
Definition and scope
Use the rational domain. Treat constant equations separately when a=0; divide by a only when a is nonzero.
Step-by-step derivation
S002 · implies
Before
a=0
After
0x+(7)=7
Conditions: a=0
Zero times any rational number is zero, so the unknown term disappears.
Step identifiers and dependencies
Rule: zero_coefficient · Uses: G001, S001
S003 · implies
Before
7=0
After
false for every x∈Q
Conditions: none
The remaining constant equality is independent of x; check its truth directly.
Step identifiers and dependencies
Rule: constant_equality · Uses: G001, S002
S004 · implies
Before
constant equality has fixed truth value
After
Solution set = ∅
Conditions: none
If the equality is always true every allowed value is a solution; if always false there is no solution.
Step identifiers and dependencies
Rule: solution_set_definition · Uses: G001, S003
S005 · implies
Before
a=0 and b=0 or b≠0
After
no solution
Conditions: none
These two constant branches exhaust the case a=0.
Step identifiers and dependencies
Rule: exhaustive_cases · Uses: G001, S004
Reverse checks
S005· premise_and_result_audit — In the original equation the truth of 0x+b=0 is unchanged for every rational x.S004· premise_and_result_audit — All solutions and no solutions are both distinct from a unique solution.S003· premise_and_result_audit — The classification depends on b=0, not on sampling a few values of x.S002· premise_and_result_audit — No reciprocal of a or division by zero is introduced.S001· premise_and_result_audit — When a=0 this is a degenerate constant equation, not a nondegenerate linear equation.
Result
no solution
There is no solution.
This question review and source binding
Review date: 2026-10-24
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: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 1a9f35ae455bb4c91d1cb1064d39784445e82097b4e34951e9acf4e3efe14b75candidate record SHA-256: 336c5fe30bbce01ad12bfbc86e120a609f6eb65f6ef5d42798cb3b1bd2a31b11semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36bquestion-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937deeLean: NOT_RUN · human review: NOT_PERFORMED