Question 280: Exact Solution of a Linear Equation

Original question and givens

Solve -1x+(13)=0 exactly over the rational numbers; a=-1 is explicitly nonzero.

• G001 — Solve -1x+(13)=0 exactly over the rational numbers; a=-1 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.

• mko-linear-equation

Step-by-step derivation

a=-1, b=13, x∈Q

equation: -1x+(13)=0

-1x+(13)=0

-1x=-13

x := div(-13, -1)

x = 13

a·x := mul(-1, 13)

a·x = -13

a·x+b := sum(-13, 13)

a·x+b = 0

a(x₁−x₂)=0 and a≠0

x₁=x₂

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

13

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.