Problem 60 - Prealgebra

Question

Julia and Catelyn are using the Distributive Property to simplify \(3(x+2) .\) Who is correct? Explain your reasoning. $$\begin{array}{c}\text { Julia } \\\3(x+2)=3 x+2\end{array}$$ $$\begin{array}{c}\text { Catelyn } \\\3(x+2)=3 x+6\end{array}$$

Step-by-Step Solution

Answer

Catelyn is correct; the expression simplifies to \( 3x + 6 \).

Step 1: Understand the Distributive Property

The Distributive Property of multiplication states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b+c) = ab + ac \). In this exercise, identify \( a = 3 \), \( b = x \), and \( c = 2 \).

Step 2: Apply the Distributive Property

Distribute \( 3 \) to both \( x \) and \( 2 \). This means you multiply \( 3 \) times \( x \) and then \( 3 \) times \( 2 \). So, \( 3(x+2) = 3 \cdot x + 3 \cdot 2 \).

Step 3: Simplify the Expression

Calculate both products: \( 3 \cdot x = 3x \) and \( 3 \cdot 2 = 6 \). Therefore, the expression simplifies to \( 3x + 6 \).

Step 4: Compare the Solutions

Compare both Julia's and Catelyn's solutions to the simplified expression. Julia's statement is \( 3x + 2 \), while Catelyn's is \( 3x + 6 \).

Step 5: Verify the Correct Solution

Catelyn's solution \( 3x + 6 \) matches the correctly simplified expression after applying the Distributive Property, so Catelyn is correct.

Key Concepts

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra that involves condensing mathematical statements to their simplest form. This process often requires using various algebraic properties and techniques, one of which is the Distributive Property.
Simplifying allows for easier manipulation and understanding of expressions. In the case of the expression \(3(x+2)\), the task is to break down and simplify the expression by distributing the coefficient 3. This step involves multiplying 3 by each term inside the parenthesis.
Once simplification occurs, you achieve a more streamlined expression that can be more readily used in further calculations or problem-solving applications.

• Recognizing common expressions like \(a(b+c)\) where the Distributive Property can be used helps in identifying how to approach simplification efficiently.
• Through simplification, complex expressions become manageable and their interpretations become clear.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (such as addition and multiplication). These expressions form the basis of algebra, allowing us to generalize arithmetic operations by using symbols to represent numbers.
In this exercise, \(3(x+2)\) is an algebraic expression. It includes a variable \(x\), a constant 2, and a coefficient 3 which needs to be distributed according to arithmetic rules.
Understanding algebraic expressions involves recognizing the parts of the expression, such as:

• Coefficients: Numbers that multiply the variables, here 3 is the coefficient.
• Variables: Symbols such as \(x\) that represent numbers we don't know yet.
• Constants: Numbers on their own, such as 2 in the expression.

Working with algebraic expressions requires applying efficient strategies to transform them, make calculations easier, and solve equations by using techniques like substitution, factoring, and, as seen here, distribution.

Mathematical Reasoning

Mathematical reasoning is the logical foundation that supports problem-solving in mathematics. This involves making sound judgements about how to tackle and solve a given problem, using principles and properties such as the Distributive Property in this case.
When deciding how to distribute terms in an expression such as \(3(x+2)\), mathematical reasoning guides you through recognizing the correct property to apply and executing the appropriate operations.
By examining both Julia's and Catelyn's solutions, we use mathematical reasoning to evaluate which expression is simplified correctly.
Through this type of reasoning, it is evident that:

• The Distributive Property ensures each term inside the parentheses must be multiplied by 3.
• Comparing the results to check for consistency establishes which solution is mathematically sound, as done here, confirming Catelyn's approach.

Mathematical reasoning empowers learners to trust their judgement, verify results, and make sure processes are followed accurately with logical consistency.