Problem 40 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -3(b+8) $$

Step-by-Step Solution

Answer

Question: Simplify the expression -3(b+8). Answer: The simplified expression is -3b - 24.

Step 1: Distribute -3 to both terms inside the parenthesis

To distribute -3, multiply -3 with each term inside the parenthesis: $$ -3(b+8) = -3 \cdot b + (-3) \cdot 8 $$

Step 2: Perform the multiplication

Multiply -3 with the terms inside the parenthesis: $$ -3 \cdot b + (-3) \cdot 8 = -3b - 24 $$

Step 3: Simplify the expression

In this case, there are no like terms to combine, so the simplified expression is: $$ -3b - 24 $$

Key Concepts

Distributive Property

The distributive property is a fundamental principle of algebra used to simplify expressions and solve equations. It allows us to multiply each term inside a set of parentheses by a number outside the parentheses. This property is especially useful when dealing with expressions that include addition or subtraction within the parentheses.

For example, given the expression \(-3(b+8)\), we apply the distributive property by multiplying \(-3\) with each term inside the parentheses. This results in:

• The multiplication \(-3 \cdot b\)
• And the multiplication \((-3) \cdot 8\)

The expression then becomes \(-3b - 24\). Using the distributive property helps break down complex expressions into simpler parts so they are easier to manage.

Like Terms

Like terms are terms in an algebraic expression that have identical variables raised to the same power. They can be combined by adding or subtracting their coefficients.

In our example with \(-3b - 24\), we notice two terms:

• a term with the variable \(b\): \(-3b\)
• a constant term with no variables: \(-24\)

These terms are unlike because only one involves the variable \(b\). Therefore, they cannot be combined further. Recognizing like terms is important for simplifying expressions and solving equations efficiently.

Simplifying Expressions

Simplifying expressions means reducing them to their most concise form without changing their value. The main goal is to make expressions as straightforward as possible.

In algebra, this often involves using the distributive property and combining like terms.In the example problem, after distributing \(-3\) and completing the multiplications, we arrived at a simplified version of the expression:
\(-3b - 24\).This expression is already simple, with no further like terms to combine. To simplify properly, one must:

• Apply the distributive property when needed.
• Combine like terms when possible.
• Keep an eye out for common mistakes such as forgetting to distribute negatives.

Simplifying makes it easier to understand, interpret, and use algebraic expressions in problem-solving.