Problem 82 - Elementary and Intermediate Algebra

Question

Evaluate each expression, for \(x=3, y=-2,\) and \(z=-4\) See Example 10. $$ -z^{2}-z-12 $$

Step-by-Step Solution

Answer

The evaluated expression is -24.

Step 1: Substitute the Values

Given the expression is \[ -z^2 - z - 12 \]Substitute \( z = -4 \) into the expression:\[ -(-4)^2 - (-4) - 12 \]

Step 2: Calculate the Square

Calculate \[ (-4)^2 = 16 \] Substituting this back into the expression gives:\[ -16 - (-4) - 12 \]

Step 3: Simplify the Negatives

The expression \[ -16 - (-4) \] simplifies to:\[ -16 + 4 = -12 \].

Step 4: Final Calculation

Add the numbers in the expression:\[ -12 - 12 = -24 \].

Key Concepts

Evaluating Expressions

Evaluating an algebraic expression means calculating its value by substituting numbers for any variables and performing the necessary arithmetic operations. In our exercise, we're given the expression \(-z^2 - z - 12\). To find its value, we start by substituting the value of \(z\) into the expression. This involves numerous processes, such as squaring numbers and dealing with negative signs. Evaluating expressions correctly is crucial, as it often sets the foundation for solving more complex problems.The process ensures understanding and practice of arithmetic operations. Properly knowing how to evaluate expressions can make algebra less daunting. When you replace letters with numbers, remember to respect mathematical operations' rules and be accurate every step of the way.

Substituting Values

Substitution is pivotal when valuing expressions in algebra. It involves replacing variables with given numbers. In the given problem, we substituted \(z = -4\) into the expression \(-z^2 - z - 12\). This means wherever we see \(z\), we replace it with \(-4\). Substitution requires precision:

• Find the variables in the expression.
• Replace each variable with its corresponding number.
• Make sure to place the substituted numbers within parentheses, especially when dealing with negative values or exponents to avoid mistakes.

With proper substitution, we transform abstract expressions into tangible numerical problems that we can calculate confidently.

Order of Operations

The order of operations ensures that expressions are calculated systematically and correctly. In mathematics, we follow the PEMDAS/BODMAS rule: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right). This common rule guides us to evaluate any mathematical expression effectively.In our exercise, after substituting \(z = -4\) into \(-z^2 - z - 12\), the next step is applying the order of operations. We first handle the exponent due to \((-4)^2\), resulting in \(16\). The negative sign before \(z^2\) remains, changing our focus to \(-16 - (-4) - 12\).

• Calculate powers and exponents first. For example, \((-4)^2\).
• Handle negative signs carefully, simplifying expressions like \(-(-4)\) to \(+4\).
• Proceed with subtraction and addition from left to right as the last step.

Following these steps maintains consistency and accuracy when evaluating any expression.