20.8 Original Equation Verification

Original Equation Verification is the process of confirming that a value obtained by solving a fractional or decimal linear equation, whether through Direct Fractional Equation Solving, Numerical Denominator Clearing, Direct Decimal Equation Solving, Decimal Clearing by Scaling, or a Mixed Fractional and Decimal Equations conversion strategy, truly satisfies the equation exactly as it was originally presented, before any clearing, scaling, or conversion was applied.

Solution Substitution into Fractional Terms is the action of placing the obtained solution into every occurrence of the variable within the original equation's fractional terms, exactly as those terms appeared before Numerical Denominator Clearing or any other clearing technique was used. This substitution must respect the original fractional form, including any grouped numerator addressed by Grouped Numerator over a Constant Denominator, rather than an intermediate cleared or scaled version of the equation.

Solution Substitution into Decimal Terms is the corresponding action performed when the original equation, or a portion of it under Mixed Fractional and Decimal Equations, contained decimal coefficients or constants, placing the obtained solution into every occurrence of the variable within those original decimal terms exactly as first written, before any Decimal Clearing by Scaling was applied.

Independent Original-Side Evaluation is the action of carrying out, separately, all arithmetic indicated on the left side and on the right side of the original, unaltered equation after substitution, including the original fraction arithmetic, the original decimal arithmetic, or both together in the mixed case, producing a single numerical result for each side.

Exact Fraction Comparison is the specific standard applied when Independent Original-Side Evaluation produces fractional results on either or both sides: the two resulting fractions must be compared using exact fraction arithmetic, reduced to lowest terms or expressed with a common denominator, rather than converted to rounded decimal approximations for comparison, since an approximation could mask a genuine discrepancy between two fractions that are close in value but not truly equal.

Terminating Decimal Comparison is the corresponding standard applied when Independent Original-Side Evaluation produces decimal results that terminate after a finite number of places: the two resulting decimals can be compared directly, digit by digit, since a terminating decimal carries no ambiguity or loss of precision in its written form.

Approximation and Exact Equality Distinction is the governing principle that separates acceptable verification from unreliable verification: whenever a result produced during Independent Original-Side Evaluation is a repeating or nonterminating decimal, it must be compared either in its exact repeating form or by way of its equivalent exact fraction, following Nonterminating Decimal Avoidance, rather than by a rounded or truncated decimal approximation that could produce a false impression of equality or inequality between the two sides.

Verified Fractional or Decimal Solution is the outcome confirmed once Exact Fraction Comparison or Terminating Decimal Comparison, applied according to Approximation and Exact Equality Distinction, shows that the two sides of the original equation evaluate to the identical exact value. This outcome authorizes the solution to be reported as final and correct; any disagreement between the two sides indicates that an error occurred somewhere in the clearing, scaling, conversion, or isolation process, and the entire solving procedure must be revisited to locate and correct that error before verification can succeed.