22.6 Rearranged Formula Verification

Rearranged Formula Verification is the process of confirming that a literal equation rearranged to isolate a chosen target variable, whether through Additive Formula Rearrangement, Multiplicative Formula Rearrangement, Multi-Step Formula Rearrangement, or Grouped Formula Rearrangement, remains equivalent to the original equation by testing both forms against a shared set of specific numerical values.

Original Formula Retention is the preliminary requirement of this verification: the original, unrearranged equation must be kept available in its exact original form throughout the verification process, since the entire purpose of this check is to compare the rearranged formula against the original rather than against any intermediate step produced during rearrangement.

Shared Numerical Data Selection is the action of choosing specific numerical values for every symbol in the equation other than the target variable, values that will be substituted consistently into both the original formula and the rearranged formula. These chosen values need not correspond to any real-world scenario; their purpose is purely to provide a concrete numerical test of the algebraic equivalence between the two forms.

Original Formula Evaluation is the action of substituting the Shared Numerical Data Selection into the original, unrearranged equation and solving for the target variable using whatever method applies to that specific numerical instance, producing a specific numerical value for the target under this particular choice of data.

Rearranged Formula Evaluation is the corresponding action of substituting the identical Shared Numerical Data Selection into the rearranged formula produced by isolating the target variable, and directly computing the value of the target by evaluating the expression on the side opposite the isolated target.

Target Value Agreement Check is the central comparison of this verification: the numerical value for the target produced by Original Formula Evaluation is compared against the numerical value produced by Rearranged Formula Evaluation. Exact agreement between these two values, for the chosen shared numerical data, provides evidence that the rearrangement was performed correctly and that the rearranged formula is a valid equivalent of the original.

Nonzero Factor Constraint Check is the action of reviewing whether the Shared Numerical Data Selection assigns a nonzero value to every symbolic factor that was divided by during Multiplicative Formula Rearrangement or Multi-Step Formula Rearrangement, confirming that the Nonzero Symbolic Factor Condition assumed during rearrangement is actually satisfied by the chosen test values. Selecting a value of zero for such a factor would make the division performed during rearrangement invalid for that particular test case, undermining the verification rather than confirming it.

Formula Unit Compatibility Check applies specifically when the literal equation being rearranged represents a formula drawn from a real-world context, as described in Formula and Literal Equation Distinction, confirming that the units associated with each symbol remain consistent and meaningful once the target has been isolated. This check ensures that the rearranged formula, beyond being algebraically correct, still produces a result expressed in units appropriate to the quantity the target variable represents, which is a consideration specific to formulas with physical or applied meaning rather than to purely symbolic literal equations.