Problem 63 - Prealgebra

Question

Use the Distributive Property to write expression as an equivalent expression. \(-2(y-8)\)

Step-by-Step Solution

Answer

The equivalent expression is \(-2y + 16\).

Step 1: Identify the Distributive Property

The Distributive Property allows you to distribute a multiplication over addition or subtraction inside parentheses. The property states: \[ a(b + c) = ab + ac \] In this context, we will distribute \(-2\) across the terms inside the parentheses \(y - 8\).

Step 2: Apply the Distributive Property

Use the Distributive Property to expand the expression by multiplying each term inside the parentheses by \(-2\): \[ -2(y - 8) = -2 \cdot y + (-2) \cdot (-8) \]

Step 3: Perform the Multiplications

Calculate the products for each term:1. \(-2 \cdot y = -2y\)2. \(-2 \cdot (-8) = 16\)

Step 4: Write the Equivalent Expression

Combine the results from the multiplications to reformulate the expression: \[ -2y + 16 \]This is the equivalent expression after applying the Distributive Property.

Key Concepts

Understanding Equivalent Expressions

The concept of equivalent expressions involves different algebraic expressions that simplify to the same overall value. Even though they may look different at a glance, they represent the same quantity.

When you use the Distributive Property, you create an equivalent expression by rewriting an expression in a different form. In our exercise, we start with \(-2(y-8)\) and end with \(-2y + 16\). Both expressions are equivalent because no matter what value you substitute for \(y\), both expressions will yield the same result after simplification.

• Original Expression Example: \(-2(y - 8)\)
• Equivalent Form: \(-2y + 16\)

Through this process, we can manipulate expressions to make them easier to work with without changing their values. This is particularly useful in solving equations and understanding algebraic relationships.

Exploring Multiplication in Distributive Property

Multiplication plays a crucial role in the application of the Distributive Property, which is fundamental in simplifying algebraic expressions. This property allows you to multiply a single term by each term inside the parentheses.

In our example, we applied multiplication like this:

• Multiply \(-2\) by \(y\) to get \(-2y\)
• Multiply \(-2\) by \(-8\) to get \(16\)

This systematic multiplication ensures that each component of the original expression is correctly distributed, maintaining the proportional balance of the expression. By addressing each term separately, you effectively preserve the integrity of the original expression while transforming it into an equivalent one.

The Role of Subtraction Within Expressions

Subtraction is a fundamental operation in algebra that can sometimes complicate expressions, but it can also be simplified using the Distributive Property. In our example, subtraction appears inside the parentheses, with \(y - 8\).

When distributing \(-2\) over the subtraction:

• The term \(y\) becomes \(-2y\).
• The subtraction of \(8\) becomes positive \(16\) due to multiplication by \(-2\).

Notice how subtraction is transformed through multiplication: the negative signs are addressed, leading to a new additive term. This shows how subtraction can be managed by careful application of the Distributive Property, leading to simpler and more manageable expressions.

Understanding Algebraic Expressions and Their Components

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, and multiplication. These expressions are the building blocks of algebra and are used to express relationships and solve equations.

In the given example, the expression involves a variable \(y\) and numbers -2 and -8. Here’s the breakdown:

• Variables: \(y\) represents an unknown value.
• Constants: Numbers like \(-2\) and \(-8\) are fixed.
• Operators: The expression uses both multiplication and subtraction.

By understanding each part of an algebraic expression, you can better manipulate and simplify them. Not only is this useful in creating equivalent expressions, but it's also foundational for solving more complex algebraic problems.