Problem 37 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(3(2 x+5)\)

Step-by-Step Solution

Answer

The simplified expression is \( 6x + 15 \).

Step 1: Apply Distributive Property

The distributive property states that for any three numbers a, b, and c, the equation \( a(b + c) = ab + ac \) holds true. Here, you have \( 3(2x + 5) \). This means you need to distribute the 3 to both terms inside the parentheses: \( 3 \times 2x + 3 \times 5 \).

Step 2: Solve Multiplication Inside Parentheses

Now carry out the multiplication: \( 3 \times 2x = 6x \) and \( 3 \times 5 = 15 \). So the expression inside the parentheses becomes \( 6x + 15 \).

Step 3: Simplify the Expression

There are no like terms here to combine further, so the expression \( 6x + 15 \) is already in its simplest form. Therefore, \( 6x + 15 \) is your simplified expression.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in algebra that helps you work with and understand equations more effectively. When you simplify an expression, you reduce it to its simplest form, which often makes solving the equation easier. Let's break it down using the example of the expression \(3(2x + 5)\).

To simplify this expression, you initially use the distributive property to remove the parentheses. Once the multiplication is complete, you look for any like terms to combine. Like terms are terms that have the same variables raised to the same powers. In our given expression \(6x + 15\), there are no like terms to combine, so it is already simplified. This helps in quickly understanding and solving the expression in algebraic problems.

When you simplify algebraic expressions, you make them cleaner and easier to interpret, which is a valuable skill when dealing with more complex equations later on.

Multiplication in Algebra

Multiplication in algebra is a core operation, similar to multiplication in arithmetic, but it involves variables and expressions. In algebra, multiplication is often used in combination with the distributive property to simplify expressions.

Let's use our sample problem to illustrate this. We have \(3(2x + 5)\). The process involves distributing the number 3 to each term in the parenthesis:

• Multiply 3 by \(2x\), which gives you \(6x\).
• Multiply 3 by 5, resulting in 15.

So, the multiplication process allows you to transform \(3(2x + 5)\) into \(6x + 15\). Remember, the key here is to multiply each term in the parentheses by the number outside the parentheses accurately. This step is crucial because it sets the foundation for simplifying the entire expression.

Prealgebra Concepts

Prealgebra serves as the groundwork for all higher-level math courses, and it introduces fundamental concepts that are used throughout mathematics. Understanding these concepts is crucial for mastering algebra and beyond.

In the context of our exercise, the distributive property is a prealgebra concept. The property indicates that you can multiply a single term outside the parentheses by each term inside it. This allows you to simplify expressions step-by-step by eliminating the parentheses.

Another vital prealgebra concept is identifying and working with variables. In our example \(3(2x + 5)\), the variable is \(x\). Recognizing variables and knowing how to manipulate them while preserving equality is central to prealgebra and all algebraic equations. By mastering these foundational concepts, you'll be better prepared to tackle more advanced topics in mathematics.