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Question 297: 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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P1000-0297 · q-20260927-linear-equation-exact-solve-0022

Original question and givens

Solve 5x+(8)=0 exactly over the rational numbers; a=5 is explicitly nonzero.

  • G001 — Solve 5x+(8)=0 exactly over the rational numbers; a=5 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=5, b=8, x∈Q

After

equation: 5x+(8)=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

5x+(8)=0

After

5x=-8

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(-8, 5)

After

x = -8/5

Conditions: none

Division by a is allowed because 5≠0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

a·x := mul(5, -8/5)

After

a·x = -8

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(-8, 8)

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

-8/5

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: 77e7054c06ad710ae2b861ed0032acf1ba9432d746e353f727f914d24274f77c
  • candidate record SHA-256: 7046b36e6b0ced76edd4c6170916a4380f6784109fe579fdac6300710ac455af
  • semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
  • question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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