Problem 1 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$2 x+8 x$$

Step-by-Step Solution

Answer

The combined expression is \(10x\).

Step 1: Identify Similar Terms

In the expression \(2x + 8x\), identify the terms that contain the same variable raised to the same power. Here, both terms are coefficients of \(x\) and are similar.

Step 2: Apply the Distributive Property

Use the distributive property which states that \(a \cdot b + a \cdot c = a(b + c)\). In this case, factor \(x\) out of both terms: \(2x + 8x = (2 + 8)x\).

Step 3: Combine Coefficients

Add the coefficients inside the parentheses. Calculate \(2 + 8 = 10\).

Step 4: Write the Combined Expression

Multiply the summed coefficient by the variable: \(10x\).

Key Concepts

Understanding Similar Terms

When working with algebraic expressions, it's essential to recognize similar terms, as these are the foundation for simplification efforts. Similar terms share the same variables raised to the same power, which means their components must exactly match except for their coefficients. For example, the terms \(2x\) and \(8x\) are considered similar because both terms include the variable \(x\) to the first power.
Different variables or powers mean the terms aren't similar, such as \(2x\) and \(8x^2\), where the second power of \(x\) introduces a key difference.
Identifying similar terms is crucial because it allows you to effectively group them together and makes the process of simplification much easier.

Combining Coefficients

Once similar terms are identified, the next step is to combine them by manipulating their coefficients. Coefficients are simply the numerical part of the term. In the expression \(2x + 8x\), the coefficients are \(2\) and \(8\). You add these coefficients together, which is possible because their variable portions are identical. Here, adding \(2\) and \(8\) gives you \(10\).
This step reduces the expression from having multiple similar terms to a single term with the summed coefficient, making further calculations or evaluations easy to handle.

Introducing Factorization

Factorization is a useful strategy that includes rewriting expressions as the product of simpler ones. The distributive property plays a critical role here. It states that you can express \(a \cdot b + a \cdot c\) as \(a(b + c)\). For \(2x + 8x\), factorization lets you transform it into \((2 + 8)x\), effectively pulling out the common factor, \(x\).
This allows you to focus on the coefficients inside the parentheses \((2 + 8)\) in order to simplify further into \(10x\).
Factorization simplifies and tidies up expressions, making it easier to explore more complex algebraic manipulations later on. It is a powerful tool that enhances your problem-solving toolkit by breaking down expressions into manageable parts.