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Evaluating expressions is a key part of algebra that involves simplifying a given arithmetic or algebraic expression to find its value. To evaluate an expression, you substitute the given values into it and perform arithmetic operations to simplify it. In this case, you need to evaluate the expression \(|x| - |y|\) for specific values of \(x\) and \(y\).

Consider that expressions may include variables and constants, and you need to determine their individual values before doing any arithmetic. In our example, by substituting \(x = -1\) and \(y = -2\) into the expression, we rewrite it as \(|-1| - |-2|\). This builds the foundation for further simplification using both substitution and arithmetic.

Substitution in algebra is the process of replacing variables in an expression with their specific values. This step is crucial for evaluating expressions, as it transforms abstract variables into concrete numbers which can be easily computed.

To substitute correctly:

• Identify the variables in the expression, which are placeholders for numbers.
• Use the given values to replace these variables, ensuring accuracy in substitution.
• Always rewrite the expression with substituted values before proceeding to arithmetic operations.

In our example, substituting \(x = -1\) and \(y = -2\) into \(|x| - |y|\) changes it to \(|-1| - |-2|\). This critical step allows us to move forward to evaluate the absolute values.

Basic arithmetic operations are the fundamental mathematical processes used in simplifying expressions. These include addition, subtraction, multiplication, and division. When evaluating expressions with absolute values, basic arithmetic is applied after substituting and resolving the absolute values.

Here’s a brief on handling the arithmetic in our example:

• Evaluate absolute values: Find the non-negative value of a number. For \(-1\), its absolute value \(|-1|\) is 1, and for \(-2\), its absolute value \(|-2|\) is 2.
• Subtract: Once we have the absolute values, perform the subtraction: \(1 - 2\).
• Result: The subtraction yields \(-1\), which is the final evaluated result.

This systematic use of arithmetic ensures accurate simplification and leads us to the final answer of the expression for the given values.