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Substitution is a foundational concept in algebra that involves replacing variables with their given numerical values. This process allows you to transform abstract expressions into concrete numbers, making it easier to perform calculations.
This is particularly useful when dealing with expressions containing variables such as \(x, y, \) and \(z\). For the given exercise, substitution is used to replace the variables according to their provided values: \(x = -3, y = 8,\) and \(z = -12\).

• Find the expression: \(z-y\).
• Substitute \(z = -12\) and \(y = 8\) into the expression to get \(-12 - 8\).

By performing substitution, the problem transitions from dealing with variables to handling actual numbers, which sets the stage for the arithmetic operation to follow. Remember, the correct application of substitution is crucial, as it sets the foundation for solving the equation comprehensively.

Subtraction is an arithmetic operation that indicates the removal or deduction of a quantity from another. When applied to expressions, it plays a crucial role in finding the difference between numbers.
After substituting the values for the variables in \(z-y\), we reach the expression \(-12 - 8\). Here, subtraction is performed by recognizing the combination of these negative and positive components.

• Rewrite the expression \(-12 - 8\) as \(-12 + (-8)\).
• This aligns the operation as a sum of two negative numbers, which simplifies the understanding of subtraction in this context.

In essence, simplifying \(-12 - 8\) to \(-12 + (-8)\) reflects the idea that subtracting 8 from \(-12\) is equivalent to adding a negative 8. The final result of \(-12 + (-8) = -20\) is attained by combining similar terms, providing a straightforward outcome.

Understanding negative numbers is vital when dealing with algebraic expressions, especially when the operation involves them. A negative number is less than zero, and it is represented with a minus sign (\(-\)). They represent values that are opposite to positive numbers on the number line.
When solving \(-12 - 8\), the focus on negative numbers becomes apparent.

• The number \(-12\) indicates a position 12 units below zero on the number line.
• Adding further negativity by subtracting 8, or equivalently thinking of it as adding \(-8\), extends this distance further into the negative.
• Thus, combining \(-12\) and \(-8\) reaches \(-20\), which is an integration of their negative contributions.

One key takeaway is the understanding that adding two negative numbers results in a more negative value. This concept underscores the importance of grasping negative numbers when operating with subtraction and algebraic expressions.