Question 296: Exact Solution of a Linear Equation

Original question and givens

Solve 4x+(1)=0 exactly over the rational numbers; a=4 is explicitly nonzero.

• G001 — Solve 4x+(1)=0 exactly over the rational numbers; a=4 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=4, b=1, x∈Q

equation: 4x+(1)=0

4x+(1)=0

4x=-1

x := div(-1, 4)

x = -1/4

a·x := mul(4, -1/4)

a·x = -1

a·x+b := sum(-1, 1)

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

-1/4

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.