Problem 13 - 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. $$4 x+2 x+3+8$$

Step-by-Step Solution

Answer

The simplified expression is \(6x + 11\).

Step 1: Identify Like Terms

In the expression \(4x + 2x + 3 + 8\), the like terms are the terms that contain the same variable raised to the same power. Here, the like terms are \(4x\) and \(2x\) because they both contain \(x\). The constants 3 and 8 are also like terms but of a different kind, without variables.

Step 2: Use Commutative Property

The commutative property of addition allows us to rearrange the terms in any order. Therefore, we can reorder the expression for easier grouping: \(4x + 2x + 3 + 8 = 4x + 2x + 8 + 3\).

Step 3: Combine Like Terms with Variables

Add the coefficients of the \(x\) terms. That is \(4x + 2x = (4 + 2)x = 6x\). The expression now becomes \(6x + 8 + 3\).

Step 4: Combine Constant Terms

Add the constant terms: \(8 + 3 = 11\). Thus, the expression simplifies to \(6x + 11\).

Key Concepts

Understanding the Commutative Property

The commutative property is one of the fundamental rules of arithmetic that makes working with numbers much easier. It applies to both addition and multiplication. Simply put, the commutative property states that you can change the order of the numbers you're adding or multiplying without changing the result.

For example:

• In addition: if you have the numbers 3 and 5, it doesn't matter if you write them as \(3 + 5\) or \(5 + 3\). The sum is still 8.
• In multiplication: \(2 \times 3\) is the same as \(3 \times 2\), both resulting in 6.

Using this property in algebraic expressions is crucial, especially when you aim to group like terms together to simplify expressions. It allows us to rearrange the terms so we can clearly see which terms can be combined with one another. This step is particularly helpful before you begin combining like terms in expressions such as \(4x + 2x + 3 + 8\). Rearranging to \(4x + 2x + 8 + 3\) makes the process straightforward.

The Process of Simplifying Expressions

Simplifying an expression means reducing it to its simplest form. The goal is to make the expression as compact and manageable as possible, while retaining its original value.

The process involves combining like terms—terms that have the same variable raised to the same power, or pure constants. For instance, in the expression \(4x + 2x + 3 + 8\), the terms \(4x\) and \(2x\) are like terms because they both involve the variable \(x\). The numbers 3 and 8 are constants and are also considered like terms, although they differ from terms with variables.

To simplify, you:

• Identify like terms.
• Use the commutative property to rearrange terms if needed.
• Add coefficients of like terms with variables, e.g., \(4x + 2x = (4 + 2)x = 6x\).
• Combine constant terms, e.g., \(3 + 8 = 11\).

After applying these steps, the expression \(4x + 2x + 3 + 8\) simplifies to \(6x + 11\).

Basics of Pre-Algebra Concepts

Pre-algebra serves as the foundation of all higher-level math courses. It introduces students to basic mathematical operations involving unknowns or variables. One central objective of pre-algebra is to develop a keen understanding of algebraic expressions and the rules governing them.

This includes several key ideas:

• Variables: Symbols that stand for unknown numbers or values. They help generalize math problems and make them easier to solve.
• Expressions: Combinations of numbers, variables, and operations. For example, \(4x + 3\) is an algebraic expression.
• Combining Like Terms: A concept where terms with the same variable and power are added or subtracted. This streamlines calculations and is critical for solving equations.

Pre-algebra not only helps with simplifying expressions but also sets the stage for understanding equations and inequalities, making it a crucial stepping stone in a student's mathematical journey.