S001 · implies
Before
0x+(5)=0, x∈Q
After
a=0; b=5
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 31: Solution Classification of a Linear Equation
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P250-0031 · q-20260927-linear-equation-solution-class-0006
Original question and givens
Over the rational numbers, classify the solution set of 0x+(5)=0.
G001— Over the rational numbers, classify the solution set of 0x+(5)=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+(5)=5
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
5=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.
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