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Working with negative numbers might initially seem daunting, but understanding a few simple rules can make operations with negatives as clear as positive number arithmetic. When evaluating expressions involving negative numbers, remember that subtracting a negative is equivalent to adding its positive counterpart. This is often visualized on a number line where moving left is subtraction and right is addition.

For example, when we have \( (-8.21) - (-6.798) \) from our exercise, the operation changes to \( (-8.21) + 6.798 \) because subtracting \( -6.798 \) is the same as adding \( +6.798 \) due to the sign change. The key rule to remember is 'two negatives make a positive' when concerning multiplication and division as well.

Using this concept ensures that students are not misled by the signs and can evaluate expressions like \( -a - (-b) \) as \( -a + b \) correctly, leading to a proper understanding of negative number operations.

A pivotal skill when working with algebraic expressions is the ability to correctly substitute values for variables. This skill transforms an abstract algebraic expression into a concrete arithmetic problem that can be calculated.

In the context of our exercise, once the values of \( x = -8.21 \) and \( y = -6.798 \) are given, the expression \( x - y \) becomes \( (-8.21) - (-6.798) \) upon substitution. It is crucial to substitute the values exactly as they are, including their signs, to avoid errors.

Students often find this step simple yet can make careless mistakes if they rush. Double-checking the substituted values against the original variables helps prevent such errors, ensuring a smooth subsequent evaluation.

Prealgebra arithmetic forms the foundation of all algebraic concepts that students will encounter in their mathematical journey. It involves basic operations such as addition, subtraction, multiplication, and division. The trick lies in mastering these operations with both positive and negative numbers.

With the given exercise, after substitution, students are faced with an arithmetic problem: \( (-8.21) + 6.798 \). This requires them to add two numbers with different signs—a situation where we subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value. Hence, \( -8.21 + 6.798 \) results in \( -1.412 \), as \( -8.21 \) carries the larger absolute value.

Understanding the basic arithmetic involved in handling numbers of different signs leads to computational fluency in prealgebra and beyond, a valuable skill for all future mathematical endeavors.