Problem 2 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$3 x+7 x$$

Step-by-Step Solution

Answer

The expression combines to \(10x\).

Step 1: Identify Similar Terms

In the expression \(3x + 7x\), both terms \(3x\) and \(7x\) are similar because they share the same variable part \(x\).

Step 2: Apply the Distributive Property

The distributive property allows us to factor out the common variable part. Thus, \(3x + 7x\) can be written as \((3 + 7)x\).

Step 3: Simplify the Coefficients

Add the coefficients of the similar terms: \(3 + 7 = 10\).

Step 4: Combine the Terms

Multiply the sum of the coefficients by the common variable part: \(10x\).

Key Concepts

Understanding Similar Terms

In algebra, similar terms are terms that have the exact same variable component raised to the same power. This means they essentially have the same 'type' of variable. For example, in the expression \(3x + 7x\), both terms are considered similar because they share the variable \(x\). It's important to notice that coefficients (the numbers in front of the variables) do not need to be the same for terms to be classed as similar; only the variable part needs to match.
This concept is critical in algebra because combining similar terms helps simplify expressions, making them easier to work with. Keep in mind that similar terms can only be added or subtracted—not divided unless you factor first. By mastering the identification of similar terms, you can more efficiently perform operations that simplify algebraic expressions.

Mastering Factoring

Factoring is a process used in algebra to simplify expressions or solve equations by identifying a common factor among terms. In the context of similar terms, factoring allows you to "factor out" the shared component. For example, in the expression \(3x + 7x\), both terms share a common factor, which is \(x\). By applying the distributive property, we can factor out \(x\), turning the expression into \((3 + 7)x\).
Factoring is a powerful tool because it can transform seemingly complex expressions into simpler forms, highlighting the mathematical relationships within. Additionally, factoring makes it easier to perform multiplication and division operations, as well as solve equations. Understanding how to identify and factor similar terms is essential for success in more advanced algebra.

Simplifying Expressions Effectively

Simplifying expressions involves transforming an expression into its simplest form by performing operations such as combining similar terms and factoring. In the expression \(3x + 7x\), we started by identifying the similar terms \(3x\) and \(7x\). By factoring, we wrote it as \((3 + 7)x\). The next step in simplifying involves calculating \(3 + 7\) to get \(10\). This simplification process reduces the original expression to just \(10x\).
Simplifying expressions makes them easier to read and work with, particularly when solving equations or evaluating expressions. It reduces clutter and highlights the key components of the problem. Simplified expressions are often necessary for graphing functions, understanding relationships between variables, and performing further algebraic operations. By practicing the identification of similar terms and applying the distributive property for factoring, you can efficiently simplify expressions to their cleanest and most manipulable form.