Problem 24 - Prealgebra

Question

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property. $$2 x-x-3-8$$

Step-by-Step Solution

Answer

The expression simplifies to \(x - 11\).

Step 1: Identify Like Terms

First, identify the like terms in the expression. In this expression, the terms involving \(x\) are \(2x\) and \(-x\). The constant terms are \(-3\) and \(-8\).

Step 2: Group Like Terms Together

Rearrange the expression to group like terms together using the commutative property. The expression becomes \((2x - x) + (-3 - 8)\).

Step 3: Combine Like Terms (for x)

Combine the \(x\) terms: \(2x - x = x\).

Step 4: Combine Constant Terms

Combine the constant terms: \(-3 - 8 = -11\).

Step 5: Write the Simplified Expression

Combine the results from Steps 3 and 4 to write the simplified expression: \(x - 11\).

Key Concepts

Understanding Like Terms

In algebra, recognizing and working with like terms is crucial for simplifying expressions easily. Like terms are terms in an expression that have the same variable raised to the same power. For instance, in the expression \(2x - x - 3 - 8\), the like terms are \(2x\) and \(-x\). These terms have the variable \(x\) raised to the same power, which is 1 in this case. It means they can be combined together.

• Why are like terms special? Because they can be added or subtracted from each other, allowing for simpler expressions.
• Example: In \(3y + 2y\), both terms are like terms as they both have \(y\) as the variable. Together, they simplify to \(5y\).

Understanding this concept helps in simplifying algebraic expressions by making them easier to handle.

Utilizing the Commutative Property

The commutative property is a fundamental aspect of mathematics that is extremely helpful in rearranging and simplifying expressions. It states that the order in which two numbers are added or multiplied does not change the result.

• Addition Example: \(a + b = b + a\)
• Multiplication Example: \(a \times b = b \times a\)

In our original expression \(2x - x - 3 - 8\), rearrange the terms to make computation easier: \((2x - x) + (-3 - 8)\). This is a crucial step that can simplify calculations by helping to group like terms effectively.
Understanding and applying the commutative property is vital in rearranging terms, leading to an organized and simplified solution.

Techniques to Combine Like Terms

Combining like terms is a technique that helps to make expressions simpler and more manageable. This involves grouping and simplifying terms that have the same variables and powers. Let’s break it down.

• Identifying Like Terms: Always start by identifying which terms can be combined. For instance, \(2x\) and \(-x\) share the same variable, making them combinable.
• Combining Steps: Once grouped, perform the addition or subtraction. Here, \(2x - x\) equals \(x\), and the constants \(-3 - 8\) simplify to \(-11\).
• Result: Write down your newly simplified expression, which in our example is \(x - 11\).

This process of combining like terms reduces complexity and leads to a cleaner, more understandable expression. It’s a foundational tool in algebra and will make solving equations much easier.