Problem 38 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 5(8 m-6) $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: 5(8m - 6) Answer: 40m - 30

Step 1: Apply distribution property

In order to perform the multiplication, we need to distribute the 5 to both terms within the parentheses. $$ 5(8m - 6) = 5 \cdot 8m - 5 \cdot 6 $$

Step 2: Simplify the expression

Now, we'll perform the multiplications. $$ 5 \cdot 8m - 5 \cdot 6 = 40m - 30 $$

Step 3: Combine like terms (if applicable)

In this case, there are no like terms to combine since 40m and 30 are not like terms (one is a constant term and the other is a variable term). So, the final simplified expression is: $$ 40m - 30 $$

Key Concepts

Understanding the Distributive Property

When faced with algebraic multiplication involving parentheses, the distributive property becomes your best friend. This property allows you to multiply a single term across terms within the parentheses, ensuring that each term is accounted for. The formula is simple:

• If you have an expression like \( a(b + c) \), you distribute \( a \) to both \( b \) and \( c \).
• It becomes \( ab + ac \).

Using the distributive property helps simplify expressions and prepares them for further operations, like combining like terms. Let's see it in action with our exercise: by distributing the \( 5 \) to both terms inside the parentheses (\( 8m \) and \( -6 \)), we end up with \( 5 \cdot 8m - 5 \cdot 6 \). This crucial step helps break down the expression, setting the stage for simplification.

Simplifying Expressions Through Multiplication

Once the distributive property is applied, each multiplication operation can be carried out. Simplifying involves transforming these products into simpler terms.

• In our exercise example, we multiply \( 5 \times 8m \) to get \( 40m \), and \( 5 \times -6 \) to get \( -30 \).
• This breaks the original expression into the simplified form \( 40m - 30 \).

In algebra, simplifying expressions aims to condense the complexity without losing the equivalent values. This is essential for keeping equations manageable and making them easier to solve or further manipulate.

Combining Like Terms for Simplification

Combining like terms is the final step in simplifying expressions. Like terms are terms that have the same variables raised to the same powers. These can be combined by adding or subtracting their coefficients.

• For instance, \( 2x \) and \( 3x \) are like terms and can be combined into \( 5x \).
• However, numbers with different variables or constant terms cannot be combined, like \( 40m \) and \( -30 \) in our problem.

In our specific exercise, \( 40m \) and \( -30 \) are unlike terms (one with \( m \) and the other a constant). Thus, our final expression remains \( 40m - 30 \) as no further combining is possible. Recognizing and correctly handling like terms ensures your expressions are in their simplest form, ready for solving.