Problem 49 - Prealgebra

Question

Use the distributive property to combine similar terms. \(-2 x+6 x\)

Step-by-Step Solution

Answer

The expression simplifies to \(4x\).

Step 1: Identify the Terms

The expression given is \(-2x + 6x\). In this expression, the terms \(-2x\) and \(6x\) are similar because they both have \(x\) as a variable.

Step 2: Apply the Distributive Property

The distributive property allows us to factor out common terms to simplify expressions. Here, \(x\) is the common variable, so rewrite the expression as \(x(-2 + 6)\).

Step 3: Perform the Calculation Inside the Parentheses

Calculate the expression inside the parentheses: \(-2 + 6 = 4\).

Step 4: Simplify the Expression

Now, multiply the factored out term \(x\) by the result from Step 3: \(x \times 4 = 4x\). The expression simplifies to \(4x\).

Key Concepts

Combining Like Terms

When dealing with algebraic expressions, combining like terms is a crucial skill. Like terms are terms that have the same variable raised to the same power. In the expression \(-2x + 6x\), both terms have the variable \(x\). This commonality permits us to combine them.
By adding or subtracting the coefficients of these terms, we merge them into a single term. Here, you add the coefficients \(-2\) and \(6\) to get \(4\). Therefore, \(-2x + 6x = 4x\).
This method not only simplifies the expression but also prepares it for further mathematical operations.

Factoring Expressions

Factoring expressions involves breaking down an expression into multiple components that can multiply together to form the original expression. In the context of combining like terms, this means identifying common factors and simplifying the expression.
Within \(-2x + 6x\), \(x\) is a common factor. You can rewrite the expression to emphasize this factor: \(x(-2 + 6)\). This application of the distributive property allows us to express the sum more compactly.
Factoring is useful for recognizing common patterns in expressions, making solving algebraic equations much more manageable.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This process often includes combining like terms, factoring, and performing basic arithmetic operations. The goal is to make the expression more accessible for additional calculations.
For example, in \(-2x + 6x\), after combining like terms, you end up with \(4x\), which is simpler and easier to work with.
Whenever you're tasked with simplifying, look for opportunities to apply these techniques. This not only clears up the expression but also clarifies an otherwise complex problem, allowing you to focus on solving it efficiently.