Problem 78 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of the expression \(-z\) is 4 when \(z = -4\).

Step 1: Substitute Values

First, let's substitute the given value of \( z \) into the expression \( -z \). Here, \( z = -4 \), so we replace \( z \) with \(-4\). The expression becomes \(-(-4)\).

Step 2: Apply Negative Sign

Now, address the negative sign in front of the parentheses. The negative of a negative number becomes a positive number. Therefore, \(-(-4) = 4\).

Key Concepts

Understanding Negative Numbers

Negative numbers are fundamental in algebra and mathematics at large. They are numbers less than zero, represented with a minus sign (-). In many contexts, they indicate values like losses, debts, or temperatures below zero. It’s important to grasp how they operate:

• When you add a negative number, you are essentially subtracting its absolute value.
• Subtracting a negative number is like adding its positive counterpart.

When dealing with operations involving negative numbers, remember:

• Negative times negative equals positive, as in the case of multiplying two debts resulting in a positive summation in context.
• Negative times positive equals negative, reflecting the concept of owing more debt.

Understanding these rules helps you manage and manipulate negative numbers accurately in any mathematical expression.

Substitution in Algebra

Substitution is a method used to simplify complex algebraic expressions by replacing variables with their specific numeric values. It's like a puzzle where each variable is a piece that you substitute into the overall picture to see the full image. In algebra, this often helps solve equations or evaluate expressions. When you substitute:

• Ensure you use the correct value for each variable, as one mistake can change the entire outcome of the expression.
• Remember the order of operations (PEMDAS/BODMAS—parentheses, exponents, multiplication and division, addition and subtraction) so that calculations are carried out correctly once the substitution is made.

In our example, the expression \(-z\) was simplified by substituting \(z = -4\), transforming \(-z\) into \(-(-4)\), which is then simplified further.

Integer Operations

Integer operations are mathematical computations involving whole numbers, which include positive numbers, negative numbers, and zero. When performing calculations with integers, it is crucial to remember:

• Addition of two positive integers is straightforward as it increases the sum.
• Adding a negative integer essentially decreases the sum because it is synonymous with subtraction.
• Multiplication and division follow the simple rule of sign multiplication: same signs result in a positive product, whereas different signs give a negative product.

In the provided exercise, the operation \(-(-4)\) was performed, where subtracting a negative resulted in performing an operation similar to subtraction cancellation, thus turning the outcome into a positive integer, which evaluates to 4. Understanding these operations and rules will ensure accurate evaluations and solutions to problems.