Problem 35 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$(3+y) 6$$

Step-by-Step Solution

Answer

The expression \((3+y) \times 6\) simplifies to \(18 + 6y\) using the Distributive Property.

Step 1: Identify the Expression

The given expression is \((3 + y) \times 6\). This is a product of a binomial \((3 + y)\) and a number (6).

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b + c) = ab + ac\). In this expression, set \(a = 6\), \(b = 3\), and \(c = y\). Now distribute the 6 to both terms inside the parentheses: \((3+y) \times 6 = 6 \times 3 + 6 \times y\).

Step 3: Simplify the Expression

Calculate the products: \(6 \times 3 = 18\) and \(6 \times y = 6y\). So, the expression \((3+y) \times 6\) simplifies to \(18 + 6y\).

Key Concepts

Equivalent Algebraic Expression

When we talk about an equivalent algebraic expression, we mean a mathematical phrase that represents the same quantity or value in a different way. The distributive property is one tool that helps us transform expressions into equivalent forms. In our example, using the expression \((3+y) \times 6\), we start by applying the distributive property. This property allows us to "distribute" the number outside the parentheses to each term inside.

• First, identify the expression: \((3+y) \times 6\).
• Recognize that the number 6 multiplies each component inside the brackets.
• By distributing, we obtain: \(6 \times 3 + 6 \times y\).

Now, the expression \(18 + 6y\) is equivalent to the original expression. It means they have the same value, just expressed differently. It's like calling the same person by their nickname or their full name—different appearance, same essence!

Binomial

A binomial is a specific type of polynomial that consists of exactly two terms. In the context of our example, the expression inside the parentheses \((3+y)\) is a binomial. Here's why binomials are important in algebra:

• They allow us to perform operations like addition, subtraction, and especially multiplication, which we see when we apply the distributive property.
• Each term in a binomial can be a number, a variable, or a product of numbers and variables.

In the expression \((3+y)\), "3" is a constant term, and "y" is a variable term. Binomials like these appear frequently in algebra because they form the basis of more complex expressions. Understanding binomials is crucial for applying rules like the distributive property, helping simplify and solve algebraic expressions efficiently.

Simplifying Expressions

Simplifying expressions means transforming them into a form that is easier to understand or work with, without changing their value. After applying the distributive property to the expression \((3+y) \times 6\), our goal is to simplify it.

• The distributive property gave us the expression \(6 \times 3 + 6 \times y\).
• Calculate the simple arithmetic: \(6 \times 3 = 18\).
• The other term becomes \(6y\), since \(6 \times y\) can be directly simplified to this form.

Thus, the initial expression of \((3+y) \times 6\) is simplified to \(18 + 6y\). Simplifying expressions is a fundamental skill in algebra because it makes equations easier to solve and work with. By mastering simplification, you can more easily spot patterns, solve equations, and verify if different-looking expressions are truly equivalent.