Problem 56 - Prealgebra

Question

Use the distributive property to combine similar terms. \(5 x-11 x\)

Step-by-Step Solution

Answer

The expression simplifies to \(-6x\).

Step 1: Understand the Distributive Property

The distributive property allows us to multiply a number by a group of numbers added together. It is represented as: \(a(b + c) = ab + ac\). We apply the property to factor an expression.

Step 2: Identify Similar Terms

In the expression \(5x - 11x\), both terms \(5x\) and \(11x\) have the variable \(x\). This makes them similar terms, which can be combined by adjusting their coefficients.

Step 3: Apply the Distributive Property

Factor out the common variable \(x\) from both terms: \((5 - 11)x\). This uses the fact that \(5x - 11x\) is equivalent to \((5-11)x\) by the distributive property.

Step 4: Simplify the Expression

Subtract \(11\) from \(5\): \(5 - 11 = -6\). The expression now simplifies to \(-6x\).

Key Concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is an essential step in simplifying equations and making them easier to understand. Like terms are terms that contain the same variables raised to the same power. For instance, in the expression \(5x - 11x\), both terms have the variable \(x\). This makes them like terms.

To combine these like terms, you focus on their coefficients—the numerical parts of the terms. Here, the coefficients are \(5\) and \(-11\). You simply add or subtract these coefficients as indicated by the signs in front of them. Thus, \(5x - 11x\) turns into \((5-11)x\), which simplifies to \(-6x\).

By combining like terms, you effectively reduce the complexity of the expression and bring terms involving the same variable together. This process is crucial for solving equations and is a building block for more advanced algebra topics.

Factoring Expressions

Factoring is the process of breaking down an expression into a product of its factors. It's somewhat like finding the reverse of multiplication. When we factor, we look for common elements that can be taken out of each term. For example, in the expression \(5x - 11x\), the common factor is \(x\).

By factoring out \(x\), we rewrite \(5x - 11x\) as \((5 - 11)x\). This shows the intuitive aspect of factoring where we use the distributed property backward to extract the greatest common factor. Factoring is not only a way of simplifying expressions but is also useful for finding solutions to equations. It allows us to set up simpler, more manageable equations by grouping terms into factors.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of making them as straightforward as possible. It often involves a combination of operations such as combining like terms, factoring, and using arithmetic operations.

Let's consider the expression \(5x - 11x\). First, recognize and combine the like terms: \(5x\) and \(-11x\) have the same variable. By combining them, we get \(5x - 11x = -6x\).

• Identify like terms
• Combine coefficients
• Factor when necessary

This process of simplification helps with solving equations and understanding algebraic expressions. It reduces expressions to their most basic form and makes further mathematical procedures more manageable.