Problem 41 - College Algebra

Question

For the following exercises, simplify the expression. $$ 8 b-4 b(3)+1 $$

Step-by-Step Solution

Answer

The simplified expression is \\(-4b + 1\\).

Step 1: Distribute the Multiplication

The expression provided is simplified by first applying distribution. The term \(4b(3)\) means that 4 is multiplied by each term inside the parenthesis, which in this case is only 3. So, \(4b(3) = 12b\). Thus, rewrite the expression as: \(8b - 12b + 1\).

Step 2: Combine Like Terms

The next step is to combine like terms. The like terms in the expression are \(8b\) and \(-12b\), as both include the variable \(b\). Combining these gives: \((8b - 12b) = -4b\). The expression now reads: \-4b + 1\.

Step 3: State the Simplified Expression

After combining the like terms, we have reached the simplest form of the expression, which is \(-4b + 1\). No further simplification is possible.

Key Concepts

Distribution in Algebra

In algebra, distribution involves multiplying each term inside a set of parentheses by a coefficient placed outside the parentheses. This principle is known as the distributive property. It's like spreading or "distributing" the multiplier to every element within the group.
The distributive property follows the rule:

• For any numbers or expressions, say \( a(b + c) = ab + ac \).

Let's simplify it in simple words: you basically multiply the term outside by each term inside the parentheses.
In our exercise, we applied distribution to the term \( 4b(3) \). Here, \( 4b \) was multiplied by \( 3 \), yielding \( 12b \). Thereby, transforming the original expression \( 8b - 4b(3) + 1 \) into \( 8b - 12b + 1 \). This application of distribution is crucial in simplifying expressions, as it allows for further simplification like combining terms.

Combining Like Terms

After distribution, you often find terms that share the same variable part. These are called "like terms" since they can be combined. To "combine like terms," you essentially add or subtract their coefficients while keeping the variable part unchanged.
When solving expressions, identifying and combining like terms leads to a clean and easier-to-understand form. For example:

• Like terms such as \( 5x \) and \( -3x \) combine to form \( (5 - 3)x = 2x \).

In the provided exercise, the like terms were \( 8b \) and \(-12b \). By performing \( 8b - 12b \), we obtained \(-4b\).
This step eliminates unnecessary complexity, consolidating the expression into a more concise format, progressing us towards a simpler solution.

Algebraic Simplification Steps

Simplifying algebraic expressions involves a systematic approach: breaking down the expression step by step to its simplest form.

The process usually involves three main steps:

• First, distribute any terms within parentheses.
• Second, identify like terms and combine them appropriately.
• Lastly, check if any further simplification is possible and write down the result.

In our exercise, these steps applied as follows: Once distribution was complete, yielding \( 8b - 12b + 1 \), we combined the like terms \( 8b \) and \( -12b \) to obtain \( -4b \). The final expression, \( -4b + 1 \), reflected its simplest form, with no further reduction needed.
Adopting this structured methodology not only ensures accuracy but also makes algebraic problems manageable, promoting an understanding of how each step interrelates within the larger problem.