19.4 Constant-Term Consolidation

Constant-Term Consolidation is the action of transferring the constant term from the side of the equation that retains the variable to the opposite side, through an operation applied identically to both sides, so that a single combined constant occupies the side free of the variable while only the variable term remains on the other. This action follows immediately after Variable-Term Consolidation and mirrors that earlier step, but operates on the constant terms rather than the variable terms.

Constant Destination Side Selection is the preliminary recognition that, once Variable-Term Consolidation has produced a One-Sided Variable-Term Result, the constant term still attached to the variable's side must be moved to the side that no longer contains any variable term. Unlike Variable Destination Side Selection, this choice is not a matter of preference, since only one side remains free of the variable after consolidation, and that side is necessarily the destination for the constant.

Additive Constant Removal applies when the constant term remaining on the variable's side is added to the variable term, requiring the subtraction property of equality to remove it: that same constant is subtracted from both sides, cancelling it from the variable's side and combining it, with its sign reversed, into the constant already present on the opposite side.

Subtractive Constant Removal applies when the constant term remaining on the variable's side is instead subtracted from the variable term, requiring the addition property of equality to remove it: that same constant is added to both sides, cancelling it from the variable's side and combining it, with its sign reversed, into the constant on the opposite side.

Signed Constant Combination is the arithmetic action of combining the constant transferred from the variable's side with the constant already present on the destination side, carrying out the addition or subtraction indicated by Additive Constant Removal or Subtractive Constant Removal with correct attention to sign. Because two constants, each potentially positive or negative, are combined into one at this step, careless sign handling here directly produces an incorrect right-hand value even when every earlier step was performed correctly.

Isolated Variable Term is the resulting expression on one side of the equation once Constant-Term Consolidation is complete: a single coefficient multiplying the variable, with no constant term remaining attached to it, set equal to the single combined constant produced by Signed Constant Combination, shown in the general form below.

This form is identical in structure to the result of Additive Term Removal in an ordinary multi-step equation, and it marks the point at which the two-sided equation has been fully reduced to a one-step multiplicative form.

Remaining Coefficient Removal is the action of eliminating the coefficient attached to the isolated variable term by dividing both sides of the equation by that coefficient, invoking the division property of equality exactly as in Multiplicative Layer Removal, with the same care given to sign as described in Positive Net Variable Coefficient and Negative Net Variable Coefficient.

Final Variable Value is the numerical result obtained once Remaining Coefficient Removal has been carried out, representing the candidate solution to the original two-sided variable equation. This value must still be substituted into the original equation, in its unreduced, two-sided form with the variable present on both the left and right sides, to confirm through Independent Side Evaluation and Final Solution Confirmation that it satisfies the equation as originally written.