Problem 23 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. $$-3(9-2)$$

Step-by-Step Solution

Answer

The equivalent expression is \(-27 + 6\), which evaluates to \(-21\).

Step 1: Apply the Distributive Property

To use the Distributive Property, we multiply the term outside the parenthesis by each term inside the parenthesis. Here, we have \(-3(9-2)\). Apply the Distributive Property:\(-3 \times 9 - 3 \times (-2)\).

Step 2: Simplify the Expression

Now, calculate the products from Step 1:\(-3 \times 9 = -27\) and \(-3 \times (-2) = 6\). So, the expression becomes \(-27 + 6\).

Step 3: Evaluate the Expression

Add the numbers from the simplified expression: \(-27 + 6 = -21\). This is the evaluated equivalent expression.

Key Concepts

Equivalent Expressions

In mathematics, when two expressions are equivalent, they have the same value even if they appear different. This concept is fundamental, as it helps us to understand that different formats of numbers or expressions can represent the same quantity. One common way to identify equivalent expressions is by using the Distributive Property. For instance, in the expression \(-3(9-2)\), applying the Distributive Property helps transform it to \(-3 \times 9 - 3 \times (-2)\), simplifying the process of evaluating it.

• Expressions are equivalent if they have the same numerical value.
• Using properties like distribution shows different formatted expressions meet this equality.

Understanding this helps in perceiving relationships between numbers and expressions, allowing for flexibility in problem-solving.

Simplifying Expressions

Simplifying expressions is the process of condensing an expression to its simplest form without changing its value. Effectively simplifying requires one to perform operations to reduce complexity. In the initial form \(-3 \times 9 - 3 \times (-2)\), the products are calculated individually. Simplifying further gives us \(-27 + 6\).

• It involves performing arithmetic operations such as addition, subtraction, multiplication, or division.
• The goal is to transform the expression to its simplest equivalent form.

The simplified expression retains the same value as the original, just expressed in a less complicated way, aiding in subsequent steps such as evaluation.

Evaluating Expressions

Evaluating expressions involves calculating their value by performing the arithmetic indicated. After simplifying, the expression \(-27 + 6\) can be evaluated by adding the numbers together. This yields the result \(-21\).

• To evaluate, solve any remaining arithmetic operations as per the rules of precedence.
• It involves operations like addition, subtraction, and multiplication to finalize the expression to a single value.

This process directly provides the numeric result of the given expressions and is often the last step to confirm an equivalent expression has been thoroughly simplified and interpreted.