16.2 Candidate Solution Verification

Candidate Solution Verification is the step-by-step procedure for testing whether a proposed value actually solves a given equation, by substituting it into both sides independently, evaluating each side on its own, and comparing the two results.

Starting the Verification

Candidate Value Identification

Verification begins by clearly identifying the specific numerical value being proposed as a solution, the one that will be substituted for the variable throughout the equation.

Substitution into Every Variable Occurrence

The candidate value is substituted into every occurrence of the variable across both sides of the equation, with no occurrence left unreplaced, exactly as with substituting into any expression.

Evaluating Each Side Separately

Independent Left-Side Evaluation

The left side of the equation, with the candidate value substituted in, is evaluated on its own to a single number, following the standard order of operations, without reference to what the right side evaluates to.

Independent Right-Side Evaluation

The right side of the equation, with the candidate value substituted in if it contains the variable, is likewise evaluated on its own to a single number, kept entirely separate from the left side's evaluation until the comparison step.

Comparing the Results

Side-Value Comparison

Once both sides have been evaluated independently, their two resulting numbers are compared directly; whether the two values are equal determines the outcome of the verification.

Candidate Solution Confirmation

If the two independently evaluated sides produce the identical number, the candidate value is confirmed as a solution, since the equation holds true under that specific assignment.

Candidate Solution Rejection

If the two independently evaluated sides produce different numbers, the candidate value is rejected as a non-solution, since the equation does not hold true under that specific assignment.

A Special Outcome

Undefined Side Value during Verification

If substituting the candidate value causes one side of the equation to become undefined, such as by creating a zero denominator, the candidate value cannot be a solution regardless of what the other side evaluates to, since an undefined side has no numerical value to compare.

Completing the Check

Verification Statement Completion

A complete verification states the candidate value, shows the independent evaluation of each side, and concludes explicitly whether the candidate is confirmed as a solution or rejected as a non-solution, leaving no ambiguity about the outcome of the test.