Problem 13 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$-6(x-5)$$

Step-by-Step Solution

Answer

The equivalent algebraic expression is \(-6x + 30\).

Step 1: Understanding the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equivalent to \(ab + ac\). It also applies to subtraction, so \(a(b - c)\) is equivalent to \(ab - ac\). Our task is to use this property to simplify \(-6(x - 5)\).

Step 2: Apply the Distributive Property to the Expression

We'll apply the Distributive Property to \(-6(x - 5)\). Here, \(a = -6\), \(b = x\), and \(c = 5\). So, the expression becomes \(-6 \times x - (-6) \times 5\).

Step 3: Perform the Multiplication

Now calculate each product separately: \(-6 \times x = -6x\) and \(-6 \times 5 = -30\).

Step 4: Combine the Results

Combine the results from the previous step. The expression \(-6x - 30\) is the equivalent expression after distributing the term \(-6\) through the terms inside the parentheses.

Key Concepts

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and arithmetic operations. It is an essential part of algebra and can consist of terms or components linked by operations such as addition, subtraction, multiplication, or division. Variables within these expressions denote unknown or changeable values, often represented by letters like \(x\) or \(y\). For example, in the expression \(-6(x-5)\), \(x\) is the variable, while \(-6\) and \(-5\) are coefficients and constants respectively.

Understanding the structure of algebraic expressions allows us to manipulate them using different algebraic rules, such as the Distributive Property, to find equivalent forms or solve equations. Recognizing these components is the first step toward simplifying expressions and making calculations easier.

Equivalent Expression

Equivalent expressions are expressions that might look different at first glance but represent the same quantity. For example, \(-6x - 30\) is equivalent to \(-6(x - 5)\) after applying the Distributive Property.

Finding an equivalent expression often involves simplifying or rearranging terms using properties like:

• Distributive Property
• Combining like terms
• Factoring

These transformations do not change the value but make manipulation easier and provide insight into the problem. Recognizing equivalent expressions helps in solving equations, comparing values, and optimizing algebraic problems.

Subtraction in Algebra

Subtraction in algebra involves removing one amount from another and can be thought of similarly to addition, by adding negative terms. The expression \(a - b\) can be rewritten as \(a + (-b)\), where \(-b\) is the additive inverse of \(b\). This perspective is particularly useful when applying the Distributive Property.

In the expression \(-6(x - 5)\), subtraction is visible inside the parentheses, and when expanded, it demonstrates the distribution of the negative factor \(-6\) across subtraction, turning negative terms into positives and vice-versa.

• \(-6 \times x = -6x\)
• \(-6 \times (-5) = 30\)

Understanding subtraction within algebraic contexts allows us to manipulate expressions with precision and apply different algebraic rules efficiently.