Problem 48 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$2(x+y)$$

Step-by-Step Solution

Answer

The expression \(2(x+y)\) is equivalent to \(2x + 2y\).

Step 1: Identify the Expression

The expression given is \(2(x+y)\). We need to apply the Distributive Property to this expression.

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b + c) = ab + ac\). In the expression \(2(x+y)\), apply this by multiplying \(2\) by both \(x\) and \(y\).

Step 3: Distribute the Coefficient

Multiply \(2\) by each term inside the parentheses: \(2 \times x + 2 \times y\).

Step 4: Simplify the Expression

Perform the multiplication to get: \(2x + 2y\). This is the equivalent algebraic expression.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They don't have an equality sign, which means they aren't equations. These expressions can represent real-world situations or abstract math problems. Let's break down an expression like \(2(x+y)\):

• The number \(2\) in front of the parentheses is known as a coefficient.
• The letters \(x\) and \(y\) are variables. They can stand in for different values.
• Inside the parentheses, the \((x+y)\) indicates that whatever operation is performed on \(x\) should also be performed on \(y\).

Algebraic expressions can often be simplified by applying various mathematical properties, such as the Distributive Property, to create new forms of the same expression.

Exploring Equivalent Expressions

Equivalent expressions are different expressions that have the same value for all values of the variables involved. Simplifying equations using different algebraic techniques often leads to finding these equivalent forms. For instance, the expression \(2(x+y)\) is equivalent to \(2x + 2y\). While they look different, they will produce the same result for any value of \(x\) and \(y\).
Equivalent expressions are critical when solving equations, as they allow you to work with simpler forms or reformat expressions to suit different purposes without changing their core value. This property is especially useful when moving between steps in a solution and ensuring that transformations maintain the equation's integrity.

Step-by-Step Solution: Breaking Down the Distributive Property

Solving an expression using the Distributive Property involves understanding each step clearly.
1. **Identify the Expression**: Recognize the entire problem that you need to solve, such as \(2(x+y)\).
2. **Apply the Distributive Property**: This property allows us to multiply a single term across terms within parentheses. In mathematical terms, \(a(b+c) = ab + ac\).
3. **Distribute the Coefficient**: Multiply each part of the sum within the parentheses by the outside number. Thus, \(2\times x\) plus \(2\times y\).
4. **Simplify the Expression**: Complete the multiplication to end with \(2x + 2y\), which is your sought solution.
By following a step-by-step approach, you not only perform operations correctly but gain a deeper understanding of the relationships within algebraic expressions.

Basic Prealgebra Concepts: Building Blocks for Algebra

Prealgebra serves as the foundation for established algebra principles. At this stage, you learn about numbers, variables, and fundamental operations, getting comfortable manipulating and transforming expressions.

• **Operations**: Ensure familiarity with addition, subtraction, multiplication, and division.
• **Variables**: These letters (like \(x\) or \(y\)) express generality and are essential in forming algebraic expressions.
• **Properties**: Discover properties like the Distributive Property as vital tools for simplifying expressions.

Connecting these prealgebra concepts to more complex operations in algebra provides the tools necessary for handling variables and equations effectively. Understanding each piece and how they work together is key to successfully transitioning from arithmetic to algebra.