S001 · implies
Before
4x+(5)=0, x∈Q
After
a=4; 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 43: Solution Classification of a Linear Equation
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P250-0043 · q-20260927-linear-equation-solution-class-0018
Original question and givens
Over the rational numbers, classify the solution set of 4x+(5)=0.
G001— Over the rational numbers, classify the solution set of 4x+(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 · equivalent
Before
4x+(5)=0
After
4x=-5
Conditions: none
Subtract 5 from both sides, preserving equivalence.
Step identifiers and dependencies
Rule: subtract_same_constant · Uses: G001, S001
S003 · calculation
Before
x := div(-5, 4)
After
x = -5/4
Conditions: none
Now a=4 is known to be nonzero, so division by it is valid.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
a·x := mul(4, -5/4)
After
a·x = -5
Conditions: none
Substitute the candidate and calculate the linear term.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
a·x+b := sum(-5, 5)
After
a·x+b = 0
Conditions: none
Add the constant back; a result of 0 verifies the original equation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
a x₁+b=0 and a x₂+b=0
After
a(x₁−x₂)=0 ⇒ x₁=x₂
Conditions: a≠0
Subtract two proposed solution equations and cancel the nonzero a; no distinct second solution exists.
Step identifiers and dependencies
Rule: nonzero_cancellation · Uses: G001, S005
Reverse checks
S006· premise_and_result_audit — The hypothesis a≠0 is essential to this uniqueness argument.S005· premise_and_result_audit — Recompute a·x+b with exact rational arithmetic, not rounded decimals.S004· premise_and_result_audit — Recompute a·x with exact rational arithmetic, not rounded decimals.S003· premise_and_result_audit — Recompute x with exact rational arithmetic, not rounded decimals.S002· premise_and_result_audit — Adding b back to both sides recovers the original equation.S001· premise_and_result_audit — When a=0 this is a degenerate constant equation, not a nondegenerate linear equation.
Result
unique solution x=-5/4
The unique rational solution is x=-5/4.
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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