Problem 23 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$2(2 y+1)$$

Step-by-Step Solution

Answer

The expression rewritten without parentheses using distributive property is \(4y + 2\).

Step 1: Distribute the Multiplication

Distribute the outer term, 2, to each term inside the parentheses: \[2 \cdot 2y + 2 \cdot 1\]

Step 2: Simplify the Terms

Perform the multiplication for each term: \[4y + 2\]

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operational symbols. It is like a sentence in mathematics, that speaks through symbols rather than words. Algebraic expressions are foundational in algebra, allowing you to express a variety of problems and their solutions.
For example, in the expression \(2(2y + 1)\), we see numbers '2' and '1' and the variable 'y'. The variable acts like a letter standing in for a number we don't know yet.

• Variables: symbols like \(y\) that can represent numbers
• Numbers: constants like 2
• Operations: includes addition, subtraction, multiplication, and division symbolized here by \( + \) and parentheses

Each part of the expression has a vital role, similar to how words and punctuation create meaning in a sentence. Understanding how these elements interact is key to manipulating and solving algebraic expressions.

Simplifying Expressions

Simplifying expressions means making them as neat and simple as possible without removing their value. This process is essential when working with algebra as it makes calculations easier and expressions clearer. For the expression \(2(2y + 1)\), simplifying involves using the distributive property to expand the brackets and eliminate them.

• Identify how terms in the expression can be combined or simplified
• Apply mathematical rules like distribution or combining like terms
• Try to achieve the simplest form, where no more operations can be performed

Example of Simplifying

To simplify \(2(2y + 1)\), we use distribution to rewrite it as \(4y + 2\). Here, the expression is simplified because:

• No parentheses remain
• No further operations can combine or remove these terms

Understanding and practicing simplifying expressions helps in solving more complicated equations later on.

Multiplication in Algebra

Multiplication in algebra introduces a slightly different approach than regular numerical multiplication because it often involves variables. When multiplying in algebra, you follow similar rules, but you have to include variables in your products. For instance, multiplying \(2\) with a variable term like \(2y\) in the expression \(2\times2y\) means you're distributing that multiplication through variables.

Distributing Multiplication

Let's look at the example again: distributing \(2\) over \(2y + 1\).- First, multiply \(2\) by \(2y\) resulting in \(4y\).- Next, multiply \(2\) by \(1\), resulting in \(2\).Thus, the expression after applying multiplication is \(4y + 2\). This practice not only simplifies expressions but sets the stage for understanding more complex algebraic manipulations. It is like spreading the multiplication across all parts inside the parentheses, ensuring each component is multiplied before moving forward.