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Substitution is a useful technique in algebra. It involves replacing variables in an expression with their specified values. In the given exercise, we have three variables: \(x\), \(y\), and \(z\). Instead of keeping them as unknowns, we substitute \(x = -12\), \(y = 4\), and \(z = -1\).

By performing substitution, we update our initial expression from \(|y| - |x| + |z|\) to \(|4| - |-12| + |-1|\). This makes our task easier as we're working with concrete numbers instead of abstract variables. The clarity and simplicity substitution offers is key in simplifying and subsequently solving algebraic expressions.

Once substitution is complete, the next step is to carry out arithmetic operations on concrete numbers. This comprises addition, subtraction, multiplication, and division of numbers. In this particular problem, we have an arithmetic sequence of operations involving subtraction and addition.

Our main expression becomes: \(4 - 12 + 1\). We proceed with the operations as follows:

• First, perform the subtraction to find \(4 - 12\). This yields \(-8\).
• Next, perform the addition \(-8 + 1\), resulting in \(-7\).

It's essential to follow the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure each step is calculated correctly.

Evaluating expressions is a crucial aspect of algebra where you simplify them to find a solution. After substitution and executing arithmetic operations, you will get a final value, representing the solution of the expression.

For our case, after substituting \(x\), \(y\), and \(z\) with their given values and performing the necessary calculations, our expression evaluates to \(-7\).

Whenever you evaluate an expression, double-check each step, ensuring you transform and calculate correctly. Practicing this makes more complex expressions easier to handle as your skills and confidence grow. Understanding how to apply these foundational concepts ensures you navigate algebraic expressions effectively.