Problem 6 - Algebra 1: Concepts and Skills

Question

Identify the like terms in the expression. \(8-3(x+4)+3 x\)

Step-by-Step Solution

Answer

The like terms in the expression \(8-3(x+4)+3x\) are 8 and -12, as well as 3x and -3x.

Step 1: Simplify the expression

To simplify the expression, first distribute the -3 to the terms inside the parenthesis. This gives: \(8-3x-12+3x\).

Step 2: Rearrange the expression

Rearrange the terms to group like terms together. This gives: \(8 - 12 + 3x - 3x\).

Step 3: Identify Like Terms

Like terms are terms with the same variable raised to the same power. Here, 8 and -12 are like terms because they are constants (no variable). Also, 3x and -3x are like terms because they are both terms with the variable x raised to the power of 1.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of making them easier to understand and solve. The goal is to combine and reduce everything as much as possible without changing the expression’s value. To start, look for 'like terms' – these are terms that have the same variable raised to the same power.

In our example, we simplify the expression by first distributing any coefficients across parentheses. This means multiplying through by any numbers in front of parentheses, a step often required before you can identify and combine like terms. A simplified expression is easier to work with and lays the groundwork for solving equations.

Distribution First

Before combining like terms, you might need to work through any multiplication indicated by the distributive property. Once that's complete, the expression will be in a form where like terms are visible and can be combined to simplify.

Distributive Property

The distributive property is a useful tool in algebra that allows you to multiply a single term by each term within a parenthesis. This property is represented as: \(a(b + c) = ab + ac\). It’s essential for simplifying expressions because it breaks down complex, grouped terms into simpler components.

In our exercise, -3 is distributed across \(x+4\), transforming the expression into \(8-3x-12+3x\). Understanding the distributive property means recognizing that it applies to subtraction as well as addition, which is often a point of confusion. You can think of subtraction as adding a negative, which means it distributes in the same way.

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variables and powers. Remember, only the coefficients of like terms are combined; the variable part remains the same.

For example, \(3x\) and \(\-3x\) are like terms, and they can be combined by subtracting the coefficients to get 0 (since 3 - 3 = 0). This is how we simplify \(8-3x-12+3x\) to \(\-4\), by combining the constants (8 and -12) and the x terms (3x and -3x).

Checking Your Work

After combining like terms, it’s a good practice to double-check your work to ensure all like terms have been correctly identified and combined. If there's any doubt, reorder the terms to place like terms next to each other, which can make the combination process more transparent.