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Question 31563: Classifying a General 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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R0062-31563 · q-20260924-equation-linear-solution-class-0013

Original question and givens

Over the rational numbers, classify the solution set of 14x+(12)=17x+(6).

  • G001 — Over the rational numbers, classify the solution set of 14x+(12)=17x+(6).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-equation-linear-solution-class, rationals, no-operation-field, a*x+b=c*x+d]

After

derived = [[all_Q_no_solution_or_unique_compact_rational_STRING]]

Conditions: none

Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.

Step identifiers and dependencies

Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001

S002 · implies

Before

source roles = [14, 12, a*x+b=c*x+d, rationals]

After

derived = [Yes]

Conditions: none

The BASE bank checks integers,not an added SAFE limit.This candidate checks the frozen exact integer slice,not arbitrary unsafe JavaScript-number behavior.Distinguish solutions over Z and Q.

Step identifiers and dependencies

Rule: Equation_BASE_integer_coefficients_declared_Z_OR_Q_NOT_blanket_SAFE · Uses: G001, S001

S003 · calculation

Before

Equation class ORIGINAL a minus c := sub(14, 17)

After

Equation class ORIGINAL a minus c = -3

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Equation class ORIGINAL d minus b := sub(6, 12)

After

Equation class ORIGINAL d minus b = -6

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [14, 12, 17, 6]

After

derived = [-3, -6]

Conditions: none

Subtract cx and b from BOTH sides to obtain the equivalent(a−c)x=d−b.IMPORTANT:do not divide by a−c until degeneracy is classified.

Step identifiers and dependencies

Rule: Equation_CLASS_both_sides_move_terms_equivalent_Q · Uses: G001, S004

S006 · calculation

Before

Equation class ORIGINAL constant over nonzero coefficient := div(-6, -3)

After

Equation class ORIGINAL constant over nonzero coefficient = 2

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Equation class reverse ORIGINAL a times x := mul(14, 2)

After

Equation class reverse ORIGINAL a times x = 28

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Equation class reverse ORIGINAL ax plus b := sum(28, 12)

After

Equation class reverse ORIGINAL ax plus b = 40

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Equation class reverse ORIGINAL c times x := mul(17, 2)

After

Equation class reverse ORIGINAL c times x = 34

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Equation class reverse ORIGINAL cx plus d := sum(34, 6)

After

Equation class reverse ORIGINAL cx plus d = 40

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Equation class reverse ORIGINAL sides difference := sub(40, 40)

After

Equation class reverse ORIGINAL sides difference = 0

Conditions: none

Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [14, 12, 17, 6, -3, -6]

After

derived = [2/1, 40, 40, 0, unique solution x=2]

Conditions: none

ONLY the nonzero coefficient permits division,giving a UNIQUE Q solution.Reduce to a positive denominator and OMIT/1 for integer display.Substitute termwise into BOTH ORIGINAL sides,not only the rearranged equation.

Step identifiers and dependencies

Rule: Equation_CLASS_NONZERO_unique_Q_root_positive_den_OMIT1 · Uses: G001, S011

Reverse checks

  • S012 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S011 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL sides difference with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL cx plus d with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL c times x with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL ax plus b with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Equation class reverse ORIGINAL a times x with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Equation class ORIGINAL constant over nonzero coefficient with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S004 · premise_and_result_audit — Recompute Equation class ORIGINAL d minus b with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Equation class ORIGINAL a minus c with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

unique solution x=2

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

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: 807d8bba0230370b5d65f1d38c4eb84fef4497659c368139d8e46f78666bd7be
  • candidate record SHA-256: f7ce06b426b31775aaeee2d32ec7f1b32ba10c0b4e86d456743efa05113f3e14
  • semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd
  • question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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