Problem 45

Question

For the following problems, perform the multiplications and combine any like terms. $$ -5(2 a+1) $$

Step-by-Step Solution

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Answer

Answer: The simplified result of the expression is -10a - 5.

1Step 1: Multiply the constant outside the parentheses by each term inside

We have to distribute the constant outside the parentheses, which is -5, to each term inside the parentheses, so we get: $$ -5(2a + 1) = -5(2a) + (-5)(1) $$

2Step 2: Perform the multiplication

Now, we need to multiply -5 by each term: $$ -5(2a) = -10a \\ (-5)(1) = -5 $$

3Step 3: Combine the results

The multiplication has been completed and there are no like terms to combine, so the final answer is: $$ -10a - 5 $$

Key Concepts

Distributive PropertyMultiplying NegativesCombining Like Terms

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to break down expressions with parentheses. When you see an expression such as $-5(2a + 1)$, the distributive property tells us to apply the multiplication of $-5$ individually to each term inside the parentheses.
In this case, the distribution step looks like this:

Multiply $-5$ by $2a$: this becomes $-5 \times 2a = -10a$.
Then multiply $-5$ by $1$: this becomes $-5 \times 1 = -5$.

Once you distribute $-5$ across both terms, the expression $-5(2a + 1)$ transforms into two distinct terms: $-10a$ and $-5$. This approach not only simplifies expressions but also paves the way for solving equations efficiently. Breaking it down step by step with the distributive property makes it easier to manage and solve.

Multiplying Negatives

Multiplying negative numbers with positive numbers is an essential skill in solving algebraic expressions. It's important to remember that when you multiply a negative number with a positive number, the result is a negative number. This rule is vital to ensure accuracy in problem-solving.
For example:

From our original problem, multiplying $-5$ by $2a$ gives $-10a$.
Similarly, multiplying $-5$ by $1$ results in $-5$.

Keeping this rule in mind, note that if you were to multiply two negative numbers, like $-5$ and another negative number, the product would be positive. Such rules are crucial as you advance in algebra and come across more complex expressions and equations.

Combining Like Terms

Combining like terms is the process of merging terms in an expression that share the same variables raised to the same power. It simplifies algebraic expressions, making them easier to work with. In the expression $-10a - 5$, there are no terms to combine because each term is different.
If we had another term with $a$, like $-3a$, we would add it to $-10a$ for simplification:

$-10a - 3a = -13a$

However, in our specific solution $-10a - 5$, terms are already distinct:

$-10a$ has the variable $a$.
$-5$ is a constant without a variable.

Always ensure to only combine terms that are truly like, sharing both the variable and its corresponding exponent, to keep expressions accurate and simplified.

Problem 45

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