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Solution verification in algebra involves checking whether a proposed solution satisfies the original equation. This is an important step in solving equations as it ensures accuracy and correctness. By verifying solutions, we are essentially asking, "Does this value of the variable make the equation true?" In the given example, we have the equation \( \frac{r}{9} = 7 \) and the number 63 is proposed as a solution. To verify this, we substitute 63 into the place of \( r \), transforming the equation into \( \frac{63}{9} = 7 \). Next, we simplify the left side of the equation to see if both sides are equal. After simplification, which results in \( 7 = 7 \), since both sides match, 63 is a solution. Verification gives you confidence that the solution is correct and a crucial step to avoid errors.

Fractional equations are equations in which the unknown variable appears in the numerator of a fraction. Solving such equations often involves manipulating the equation to eliminate fractions. The key idea is to perform operations that make the fractions manageable or remove them entirely by considering equal parts on the opposite side. Using our example, \( \frac{r}{9} = 7 \), the fraction involves dividing \( r \) by 9. To solve for \( r \), typically, you'd multiply each term by 9 to clear the fraction, giving you \( r = 63 \). Although the task here was to verify a solution and not to solve the equation, understanding the operation of managing fractional parts is central in algebra. Keeping an eye on denominators and ensuring operations are applied equally to every term are crucial for maintaining the equality of an equation.

The substitution method involves replacing a variable in an equation with a given value to simplify and assess the truth of the equation. This method is especially useful for checking if a certain number is a solution. In the presented exercise, the given number 63 is substituted for \( r \) in the equation \( \frac{r}{9} = 7 \). Consequently, the equation becomes \( \frac{63}{9} = 7 \), which simplifies to \( 7 = 7 \). Through substitution, calculations are simplified, and one can directly verify if the substituted value results in a true statement. This method reinforces understanding by illustrating how numbers and operations on either side of the equation maintain balance. Substitution serves both as a tool for solving equations and verifying potential solutions, emphasizing how variables act as placeholders for numbers.