Problem 32 - Prealgebra

Question

Simplify expression. \(8+x-5 x\)

Step-by-Step Solution

Answer

The simplified expression is \(8 - 4x\).

Step 1: Identify Like Terms

In the expression \(8 + x - 5x\), there are two terms that contain the variable \(x\): \(x\) and \(-5x\). These are like terms and can be combined.

Step 2: Combine Like Terms

Combine the like terms \(x\) and \(-5x\) by performing the operation: \(x - 5x\). This simplifies to \(-4x\).

Step 3: Simplify the Expression

Replace the combined like terms in the original expression with the resulting term from Step 2. The expression \(8 + x - 5x\) simplifies to \(8 - 4x\).

Key Concepts

Like Terms

When we talk about like terms in algebra, we mean terms that have the same variable raised to the same power. For example, in the expression \(8 + x - 5x\), the terms \(x\) and \(-5x\) are like terms because they both have the variable \(x\) raised to the same first power. This means they can be combined together into a single term.
When identifying like terms, remember that the coefficients (the numbers in front of the variables) do not need to be the same. Only the variables and their powers must match. Here's a simple bullet point guide:

• Look for terms with identical variable parts.
• Ignore the coefficients (at first), and focus on the variables.
• Ensure the variables are raised to the same powers.

This concept is crucial for simplifying algebraic expressions, as it allows you to consolidate terms and make expressions easier to work with.

Combining Like Terms

One of the key skills in simplifying algebraic expressions is combining like terms. This process makes solving equations much simpler and less cluttered.
To combine like terms in an expression like \(8 + x - 5x\):

• First, identify the like terms. In this expression, they are \(x\) and \(-5x\).
• Add or subtract the coefficients of the like terms. Here, you combine \(x\) and \(-5x\) by subtracting 5 from 1 (since \(x\) is the same as \(1x\)), giving you \(-4x\).

Combining like terms simplifies the expression into a clearer form: in this case, it helps reduce \(8 + x - 5x\) into \(8 - 4x\). This is why combining like terms is essential—it reduces complexity and reveals the structure of the expression more clearly.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols like addition or subtraction. For instance, \(8 + x - 5x\) is an algebraic expression.
Understanding algebraic expressions is a foundation of algebra. It's important for several reasons:

• They allow us to represent real-world situations in mathematical form.
• They help us to simplify, solve equations, and understand how different quantities relate to each other.

When working with algebraic expressions, it's essential to know how to identify terms, understand variables, and apply operations like addition and subtraction. Learning to manipulate these expressions, through processes like combining like terms, is a critical step in developing algebraic thinking.