Problem 12 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$9(a-10)$$

Step-by-Step Solution

Answer

The equivalent expression is \( 9a - 90 \).

Step 1: Understand the Distributive Property

The Distributive Property states that for any numbers or variables, such as \( a, b, \) and \( c \), the expression \( a(b+c) \) is equivalent to \( ab + ac \). We will apply this property to the given expression \( 9(a-10) \).

Step 2: Apply the Distributive Property

In the expression \( 9(a-10) \), we need to multiply the number outside the parentheses (9 in this case) by each term inside the parentheses. This means multiplying 9 by \( a \) and 9 by \( -10 \).

Step 3: Perform the Multiplications

Calculate each multiplication: - First, calculate \( 9 \times a = 9a \).- Next, calculate \( 9 \times (-10) = -90 \).

Step 4: Write the Equivalent Expression

Combine the results of the multiplications from Step 3 to write the equivalent expression: \( 9a - 90 \). This is the simplified, equivalent algebraic expression using the Distributive Property.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions are fundamental in algebra, as they represent real-world problems and phenomena. For example, consider the expression \( 9(a-10) \). Here, we have:

• **A coefficient**: \(9\) - a number that multiplies the variable, indicating how many times the variable is being taken.
• **A variable**: \(a\) - a symbol which stands for a number that might change or be unknown.
• **A constant**: \(-10\) - a fixed value that doesn't change.

Understanding these parts is crucial because they help in structuring and manipulating mathematical models into meaningful forms that can be solved or simplified.

Simplifying Expressions with the Distributive Property

Simplifying expressions is a key skill in algebra. It involves reducing expressions to their simplest form. The Distributive Property is an essential tool to achieve this. This property states that you can distribute multiplication over addition or subtraction within parentheses. Using it helps break down complex expressions into simpler pieces that are easier to understand and manipulate.

When we have an expression like \( 9(a-10) \), we apply the Distributive Property as follows:

• Multiply the coefficient \(9\) by each term inside the parentheses.
• This results in two separate terms: \(9 \times a = 9a\) and \(9 \times (-10) = -90\).
• Finally, combine these terms into one expression: \(9a - 90\).

This leads us to a simplified expression that is often easier to work with in further mathematical operations.

The Process of Multiplying Variables

Multiplying variables with numbers is a fundamental operation in algebra that allows us to rearrange and simplify expressions. This process involves several key steps:

• **Identify the elements**: Locate the number and variable that are being multiplied. In \(9(a-10)\), we multiply \(9\) by \(a\).
• **Use the multiplication rule**: Multiply the number (coefficient) directly by the variable. For instance, \(9 \times a = 9a\).
• **Incorporate constants correctly**: If the variable is associated with a constant inside parentheses, account for it during multiplication. For instance, \(9 \times (-10) = -90\).

This method of multiplying variables helps us expand expressions to make them easier to interpret and solve. The approach also reinforces the structure needed to handle more complex algebraic manipulations.