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Question 279: Exact Solution of a Linear Equation

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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P1000-0279 · q-20260927-linear-equation-exact-solve-0004

Original question and givens

Solve -2x+(6)=0 exactly over the rational numbers; a=-2 is explicitly nonzero.

  • G001 — Solve -2x+(6)=0 exactly over the rational numbers; a=-2 is explicitly nonzero.

Definition and scope

Use the rational domain with a≠0. Move the constant, divide by nonzero a and give the reduced fraction with positive denominator.

Step-by-step derivation

S001 · implies

Before

a=-2, b=6, x∈Q

After

equation: -2x+(6)=0

Conditions: none

Read the linear coefficient, constant and rational domain explicitly; a name alone does not establish nondegeneracy.

Step identifiers and dependencies

Rule: read_coefficients · Uses: G001

S002 · equivalent

Before

-2x+(6)=0

After

-2x=-6

Conditions: none

Subtract b from both sides, preserving the solution set.

Step identifiers and dependencies

Rule: subtract_constant · Uses: G001, S001

S003 · calculation

Before

x := div(-6, -2)

After

x = 3

Conditions: none

Division by a is allowed because -2≠0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

a·x := mul(-2, 3)

After

a·x = -6

Conditions: none

Substitute into the linear term.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

a·x+b := sum(-6, 6)

After

a·x+b = 0

Conditions: none

Adding the constant must give0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

a(x₁−x₂)=0 and a≠0

After

x₁=x₂

Conditions: none

Subtract equations for any two solutions and cancel nonzero a to prove uniqueness.

Step identifiers and dependencies

Rule: unique_solution · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.
  • 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 — Substitute into the original equation; retain every nonzero condition before division.
  • S001 · premise_and_result_audit — Substitute into the original equation; retain every nonzero condition before division.

Result

3

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

This question review and source binding

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: cb22fef4c873a3a80185532b7fa98b449dae4066b851c5ca768b0c18c22fbe29
  • candidate record SHA-256: 7cbdf2a036e1d9608b9eb717c24411b303247be7e7ef1fb89a165f954ed0f513
  • semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
  • question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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