Problem 51 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (2.3-7 w)(-6) $$

Step-by-Step Solution

Answer

The expression \( (2.3 - 7w)(-6) \) simplified using the distributive property is \( -13.8 + 42w \) .

Step 1: Distribute -6 to each term inside the parentheses

We will start by multiplying -6 by 2.3 which gives us \( -6 * 2.3 = -13.8 \) . After that, distribute -6 to -7w which gives us \( -6 * -7w = 42w \) .

Step 2: Combine the results

Now, put these two results together. The multiplication of a positive number and a negative number gives a negative result, and the multiplication of two negative numbers gives a positive result. Thus, the expression becomes \( -13.8 + 42w \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental aspect of algebra and constitute a combination of variables, numbers, and arithmetic operations. A core feature of algebraic expressions is that they can contain unknown values represented by letters, also known as variables, such as the 'w' in our exercise.

To work with algebraic expressions, it's crucial to know the rules that govern arithmetic operations including addition, subtraction, multiplication, and division. For example, in our exercise, we encounter an expression within parentheses which needs to be simplified. This requires us to apply the distributive property, a key algebraic rule that allows us to remove the parentheses by distributing a factor across terms inside the parentheses.

The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In the context of our exercise, this means multiplying each term inside the parentheses by -6, which we will explore in more detail in the next sections.

Multiplying Negative Numbers

Multiplying numbers with different signs can often cause confusion, but remembering a simple rule can help: When multiplying two numbers with different signs, the result is always negative. Conversely, if the two numbers have like signs, the result is positive.

In our exercise, we are faced with multiplying negative numbers when we distribute -6 across the terms inside the parentheses. When -6 is multiplied by 2.3, a positive number, the result is -13.8, adhering to the rule that a negative times a positive gives a negative outcome. In contrast, when we multiply -6 by -7w, both negative, the product is a positive 42w, because a negative times a negative results in a positive. Acknowledging this rule helps us predict the sign of our results, allowing us to proceed with confidence as we combine like terms to reach the final expression.

Combining Like Terms

Once the distributive property has been applied, combining like terms is often the final step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In our exercise, after distributing -6 across the terms, we get two results: -13.8 and 42w, which we then combine into a single expression.

However, in this case, -13.8 and 42w do not actually combine because they are not like terms; one is a constant and the other is a variable term. When combining like terms, only the coefficients of the same variables are added or subtracted while the variable part remains unchanged. Here, since we only have one variable term, 42w, and one constant, -13.8, they are simply written side by side in the final result of the expression, which is \( -13.8 + 42w \). Understanding how to identify and combine like terms is critical for simplifying expressions and solving equations throughout algebra.