Problem 72 - Prealgebra

Question

Simplify each of the following expressions as much as possible. $$8(3 x-2)+4 x$$

Step-by-Step Solution

Answer

The simplified expression is \(28x - 16\).

Step 1: Distribute the Constant

Start by distributing the constant 8 across the terms in the parenthesis. This will eliminate the parentheses in the expression. Apply the distributive property: \[ 8(3x - 2) = 8 \times 3x - 8 \times 2 \]Calculate each multiplication:- \(8 \times 3x = 24x\)- \(8 \times 2 = 16\)After distribution, the expression becomes:\[ 24x - 16 + 4x \]

Step 2: Combine Like Terms

Now, identify and combine like terms in the expression. Like terms are terms that have the same variable raised to the same power.In this expression, the like terms are \(24x\) and \(4x\). Add these together:\[ 24x + 4x = 28x \]Write the resulting expression, remembering to bring down the constant:\[ 28x - 16 \]

Step 3: Expression Simplified

After combining like terms, the expression is simplified. There are no further terms to combine or coefficients to distribute.The final simplified expression is:\[ 28x - 16 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions. It allows us to multiply a single term outside of a parenthesis by each term inside the parenthesis. This process eliminates the parentheses and often makes the expression easier to work with.

In our example, we used the distributive property with the expression \(8(3x - 2)\). To distribute, you multiply the 8 by each term inside:

• First, multiply 8 by \(3x\), giving \(24x\).
• Next, multiply 8 by -2, giving -16.

After applying these steps, you transition from an expression with parentheses to \(24x - 16\). The distributive property not only makes algebraic expressions simpler but is a versatile tool throughout math.

Like Terms

In algebra, 'like terms' are a vital concept for simplifying expressions. Like terms are terms that have the same variables raised to the same power. It's important to recognize these terms because only like terms can be added or subtracted.

For instance, in the expression \(24x - 16 + 4x\), the terms \(24x\) and \(4x\) are like terms because they both have an \(x\) raised to the same power (which is 1 in this case).

To identify like terms, look for:

• Common variables (e.g., both \(x\) and \(x\) terms).
• The same power (e.g., both \(x^2\) terms).

Recognizing and understanding like terms is crucial as it sets the stage for their combination, simplifying complex algebraic expressions further.

Combining Like Terms

Once you've identified like terms, the next step is combining them, which streamlines the expression and helps solve equations more efficiently. The process involves adding or subtracting the coefficients of the like terms while keeping the common variable unchanged.

In the expression \(24x + 4x\), combining like terms means simply adding the coefficients (24 and 4) together, resulting in \(28x\). Here's how you do it:

• Identify the like terms \(24x\) and \(4x\).
• Add their coefficients: \(24 + 4 = 28\).
• Keep the common variable \(x\), giving \(28x\).

Combining like terms effectively reduces the complexity of an expression, leading to a cleaner, simpler final expression. In our example, the expression simplifies to \(28x - 16\), which is the final, streamlined result.