Problem 55 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. $$-x y z, \text { for } x=-6, y=2, \text { and } z=-5$$

Step-by-Step Solution

Answer

The evaluated expression equals 5.

Step 1: Substitution

Substitute the given values into the expression. This means replacing \(a\) with \(-8\) and \(b\) with \(-3\) so the expression becomes -(-8) + -3.

Step 2: Simplify the expression

Now, perform the operations. Remember, subtracting a negative is the same as addition, so -(-8) becomes +8. Thus, our expression turns into 8 + -3.

Step 3: Final computation

Finally, add 8 and -3 to find the result. After completing the calculation, the final answer is 5.

Key Concepts

Substitution Method

In mathematics, the substitution method is a helpful tool when evaluating expressions. This is particularly useful when dealing with algebraic expressions containing variables. To apply the substitution method, follow these steps:

• Identify the variables in the expression. In our example, these are \(a\) and \(b\).
• Replace each variable with the given values. Substitute \(a = -8\) and \(b = -3\) into the expression \(-a + b\).
• The expression then transforms from \(-a + b\) to \(-(-8) + (-3)\).

Replacing variables with specific values allows you to simplify and solve the expression more easily. It's like replacing placeholders with actual numbers to find the final answer.

Negative Numbers

Understanding negative numbers is crucial when working with mathematical expressions. Negative numbers are numbers less than zero; they often appear with a minus sign (-) before them. When adding, subtracting, or dealing with negative numbers, remember:

• Subtracting a negative is the same as adding the number's positive counterpart. Hence, \(-(-8)\) becomes \(+8\), because the two negatives cancel each other.
• Adding negative numbers is akin to subtraction. For instance, adding \(-3\) is the same as subtracting 3 from the total.

These concepts help you manipulate and simplify expressions accurately. They simplify the expression \(-(-8) + (-3)\) to \(8 - 3\), making it easier to calculate.

Mathematical Operations

Mathematical operations are the basic building blocks that allow you to manipulate and solve expressions. They include addition, subtraction, multiplication, and division. Each operation obeys specific rules and properties, particularly when negative numbers are involved.For our expression, the operations involved are addition and subtraction:

• Addition and subtraction with negatives: In the expression \(8 + (-3)\), you are essentially finding the difference since adding a negative is like subtracting. So, \(8 + (-3)\) simplifies to \(8 - 3\).
• When performing these operations, always pay close attention to signs to ensure your final result is accurate.

The outcome of the expression in this example is obtained through careful execution of these operations, arriving at the answer \(5\). Understanding these operations ensures a correct and straightforward calculation process.